# 2.3: Data, Sampling, and Variation in Data and Sampling - Mathematics

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Data may come from a population or from a sample. Small letters like (x) or (y) generally are used to represent data values. Most data can be put into the following categories:

• Qualitative
• Quantitative

Qualitative data are the result of categorizing or describing attributes of a population. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type.

Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous.

All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three.

All data that are the result of measuring are quantitative continuous data assuming that we can measure accurately. Measuring angles in radians might result in such numbers as (frac{pi}{6}), (frac{pi}{3}), (frac{pi}{2}), (pi), (frac{3pi}{4}), and so on. If you and your friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data.

Sample of Quantitative Discrete Data

The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. The numbers of books (three, four, two, and one) are the quantitative discrete data.

Exercise (PageIndex{1})

The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this?

quantitative discrete data

Sample of Quantitative Continuous Data

The data are the weights of backpacks with books in them. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data because weights are measured.

Exercise (PageIndex{2})

The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. feet, 160 sq. feet, 190 sq. feet, 180 sq. feet, and 210 sq. feet. What type of data is this?

quantitative continuous data

Exercise (PageIndex{3})

You go to the supermarket and purchase three cans of soup (19 ounces) tomato bisque, 14.1 ounces lentil, and 19 ounces Italian wedding), two packages of nuts (walnuts and peanuts), four different kinds of vegetable (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces Cherry Garcia ice cream and two pounds (32 ounces chocolate chip cookies).

Name data sets that are quantitative discrete, quantitative continuous, and qualitative.

Solution

One Possible Solution:

• The three cans of soup, two packages of nuts, four kinds of vegetables and two desserts are quantitative discrete data because you count them.
• The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure weights as precisely as possible.
• Types of soups, nuts, vegetables and desserts are qualitative data because they are categorical.

Try to identify additional data sets in this example.

Sample of qualitative data

The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data.

Exercise (PageIndex{4})

The data are the colors of houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this?

qualitative data

Collaborative Exercise (PageIndex{1})

Work collaboratively to determine the correct data type (quantitative or qualitative). Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words "the number of."

1. the number of pairs of shoes you own
2. the type of car you drive
3. where you go on vacation
4. the distance it is from your home to the nearest grocery store
5. the number of classes you take per school year.
6. the tuition for your classes
7. the type of calculator you use
8. movie ratings
9. political party preferences
10. weights of sumo wrestlers
11. amount of money (in dollars) won playing poker
12. number of correct answers on a quiz
13. peoples’ attitudes toward the government
14. IQ scores (This may cause some discussion.)

Items a, e, f, k, and l are quantitative discrete; items d, j, and n are quantitative continuous; items b, c, g, h, i, and m are qualitative.

Exercise (PageIndex{5})

Determine the correct data type (quantitative or qualitative) for the number of cars in a parking lot. Indicate whether quantitative data are continuous or discrete.

quantitative discrete

Exercise (PageIndex{6})

A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart Figure (PageIndex{1}). What type of data does this graph show?

Figure (PageIndex{1})

This pie chart shows the students in each year, which is qualitative data.

Exercise (PageIndex{7})

The registrar at State University keeps records of the number of credit hours students complete each semester. The data he collects are summarized in the histogram. The class boundaries are 10 to less than 13, 13 to less than 16, 16 to less than 19, 19 to less than 22, and 22 to less than 25.

Figure (PageIndex{2})

What type of data does this graph show?

A histogram is used to display quantitative data: the numbers of credit hours completed. Because students can complete only a whole number of hours (no fractions of hours allowed), this data is quantitative discrete.

## Qualitative Data Discussion

Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.

De Anza CollegeFoothill College
Table (PageIndex{1}): Fall Term 2007 (Census day)
NumberPercentNumberPercent
Full-time9,20040.9%Full-time4,05928.6%
Part-time13,29659.1%Part-time10,12471.4%
Total22,496100%Total14,183100%

Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative data are pie charts and bar graphs.

• In a pie chart, categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category.
• In a bar graph, the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.
• A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).

Look at Figures (PageIndex{3}) and (PageIndex{4}) and determine which graph (pie or bar) you think displays the comparisons better.

Figure (PageIndex{3}): Pie Charts

It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the “best” graph depending on the data and the context. Our choice also depends on what we are using the data for.

Figure (PageIndex{4}): Bar chart

## Percentages That Add to More (or Less) Than 100%

Sometimes percentages add up to be more than 100% (or less than 100%). In the graph, the percentages add to more than 100% because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100%.

Characteristic/CategoryPercent
Table (PageIndex{2}): De Anza College Spring 2010
Full-Time Students40.9%
Students who intend to transfer to a 4-year educational institution48.6%
Students under age 2561.0%
TOTAL150.5%

Figure (PageIndex{2}): Bar chart of data in Table (PageIndex{2}).

## Omitting Categories/Missing Data

The table displays Ethnicity of Students but is missing the "Other/Unknown" category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.

Table (PageIndex{2}): Ethnicity of Students at De Anza College Fall Term 2007 (Census Day)
FrequencyPercent
Asian8,79436.1%
Black1,4125.8%
Filipino1,2985.3%
Hispanic4,18017.1%
Native American1460.6%
Pacific Islander2361.0%
White5,97824.5%
TOTAL22,044 out of 24,38290.4% out of 100%

Figure (PageIndex{3}): Enrollment of De Anza College (Spring 2010)

The following graph is the same as the previous graph but the “Other/Unknown” percent (9.6%) has been included. The “Other/Unknown” category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0%). This is important to know when we think about what the data are telling us.

This particular bar graph in Figure (PageIndex{4}) can be difficult to understand visually. The graph in Figure (PageIndex{5}) is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.

Figure (PageIndex{4}): Bar Graph with Other/Unknown Category

Figure (PageIndex{5}): Pareto Chart With Bars Sorted by Size

## Pie Charts: No Missing Data

The following pie charts have the “Other/Unknown” category included (since the percentages must add to 100%). The chart in Figure (PageIndex{6}) is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure (PageIndex{6}).

Figure (PageIndex{6}).

## Sampling

Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Any group of n individuals is equally likely to be chosen by any other group of n individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in Table (PageIndex{2}):

Table (PageIndex{3}): Class Roster
IDNameIDNameIDName
00Anselmo11King21Roquero
01Bautista12Legeny22Roth
02Bayani13Lundquist23Rowell
03Cheng14Macierz24Salangsang
05Cuningham16Okimoto26Stratcher
06Fontecha17Patel27Tallai
07Hong18Price28Tran
08Hoobler19Quizon29Wai
09Jiao20Reyes30Wood
10Khan

Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows:

0.94360; 0.99832; 0.14669; 0.51470; 0.40581; 0.73381; 0.04399

Lisa reads two-digit groups until she has chosen three class members (that is, she reads 0.94360 as the groups 94, 43, 36, 60). Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers.

The random numbers 0.94360 and 0.99832 do not contain appropriate two digit numbers. However the third random number, 0.14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Macierz, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Marcierz, Cuningham, and Cuarismo.

To generate random numbers:

• Press MATH.
• Arrow over to PRB.
• Press 5:randInt(. Enter 0, 30).
• Press ENTER for the first random number.
• Press ENTER two more times for the other 2 random numbers. If there is a repeat press ENTER again.

Note: randInt(0, 30, 3) will generate 3 random numbers.

Figure (PageIndex{7})

Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.

To choose a stratified sample, divide the population into groups called strata and then take a proportionate number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members who make up the stratified sample.

To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your college population, the four departments make up the cluster sample. Divide your college faculty by department. The departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.

To choose a systematic sample, randomly select a starting point and take every nth piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings. You must choose 400 names for the sample. Number the population 1–20,000 and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of 400 names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is a simple method.

A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others.

Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased (they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.

True random sampling is done with replacement. That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done without replacement. Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low.

In a college population of 10,000 people, suppose you want to pick a sample of 1,000 randomly for a survey. For any particular sample of 1,000, if you are sampling with replacement,

• the chance of picking the first person is 1,000 out of 10,000 (0.1000);
• the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999);
• the chance of picking the same person again is 1 out of 10,000 (very low).

If you are sampling without replacement,

• the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000);
• the chance of picking a different second person is 999 out of 9,999 (0.0999);
• you do not replace the first person before picking the next person.

Compare the fractions 999/10,000 and 999/9,999. For accuracy, carry the decimal answers to four decimal places. To four decimal places, these numbers are equivalent (0.0999).

Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is 25 people, the sample is ten, and you are sampling with replacement for any particular sample, then the chance of picking the first person is ten out of 25, and the chance of picking a different second person is nine out of 25 (you replace the first person).

If you sample without replacement, then the chance of picking the first person is ten out of 25, and then the chance of picking the second person (who is different) is nine out of 24 (you do not replace the first person).

Compare the fractions 9/25 and 9/24. To four decimal places, 9/25 = 0.3600 and 9/24 = 0.3750. To four decimal places, these numbers are not equivalent.

When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error.

In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error.

In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others (remember, each member of the population should have an equally likely chance of being chosen). When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied.

Exercise (PageIndex{8})

A study is done to determine the average tuition that San Jose State undergraduate students pay per semester. Each student in the following samples is asked how much tuition he or she paid for the Fall semester. What is the type of sampling in each case?

1. A sample of 100 undergraduate San Jose State students is taken by organizing the students’ names by classification (freshman, sophomore, junior, or senior), and then selecting 25 students from each.
2. A random number generator is used to select a student from the alphabetical listing of all undergraduate students in the Fall semester. Starting with that student, every 50th student is chosen until 75 students are included in the sample.
3. A completely random method is used to select 75 students. Each undergraduate student in the fall semester has the same probability of being chosen at any stage of the sampling process.
4. The freshman, sophomore, junior, and senior years are numbered one, two, three, and four, respectively. A random number generator is used to pick two of those years. All students in those two years are in the sample.
5. An administrative assistant is asked to stand in front of the library one Wednesday and to ask the first 100 undergraduate students he encounters what they paid for tuition the Fall semester. Those 100 students are the sample.

a. stratified; b. systematic; c. simple random; d. cluster; e. convenience

Example (PageIndex{9}): Calculator

You are going to use the random number generator to generate different types of samples from the data. This table displays six sets of quiz scores (each quiz counts 10 points) for an elementary statistics class.

#1#2#3#4#5#6
5710983
1059876
9108679
91010989
789574
9991087
7710988
8891088
978778
8810987

Instructions: Use the Random Number Generator to pick samples.

1. Create a stratified sample by column. Pick three quiz scores randomly from each column.
• Number each row one through ten.
• On your calculator, press Math and arrow over to PRB.
• For column 1, Press 5:randInt( and enter 1,10). Press ENTER. Record the number. Press ENTER 2 more times (even the repeats). Record these numbers. Record the three quiz scores in column one that correspond to these three numbers.
• Repeat for columns two through six.
• These 18 quiz scores are a stratified sample.
2. Create a cluster sample by picking two of the columns. Use the column numbers: one through six.
• Press MATH and arrow over to PRB.
• Press 5:randInt( and enter 1,6). Press ENTER and record that number.
• The two numbers are for two of the columns.
• The quiz scores (20 of them) in these 2 columns are the cluster sample.
3. Create a simple random sample of 15 quiz scores.
• Use the numbering one through 60.
• Press MATH. Arrow over to PRB. Press 5:randInt( and enter 1, 60).
• Press ENTER 15 times and record the numbers.
• Record the quiz scores that correspond to these numbers.
• These 15 quiz scores are the systematic sample.
4. Create a systematic sample of 12 quiz scores.
• Use the numbering one through 60.
• Press MATH. Press 5:randInt( and enter 1, 60).
• Press ENTER. Record the number and the first quiz score. From that number, count ten quiz scores and record that quiz score. Keep counting ten quiz scores and recording the quiz score until you have a sample of 12 quiz scores. You may wrap around (go back to the beginning).

Example (PageIndex{10})

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

1. A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group of boys aged 11 to 12, and three players from a group of boys aged 13 to 14 to form a recreational soccer team.
2. A pollster interviews all human resource personnel in five different high tech companies.
3. A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers.
4. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.
5. A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.
6. A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on the average.

a. cluster; c. stratified; d. systematic; e. simple random; f.convenience

Exercise (PageIndex{11})

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

A high school principal polls 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors regarding policy changes for after school activities.

stratified

If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural.

Example (PageIndex{12}): Sampling

Suppose ABC College has 10,000 part-time students (the population). We are interested in the average amount of money a part-time student spends on books in the fall term. Asking all 10,000 students is an almost impossible task. Suppose we take two different samples.

First, we use convenience sampling and survey ten students from a first term organic chemistry class. Many of these students are taking first term calculus in addition to the organic chemistry class. The amount of money they spend on books is as follows:

$128;$87; $173;$116; $130;$204; $147;$189; $93;$153

The second sample is taken using a list of senior citizens who take P.E. classes and taking every fifth senior citizen on the list, for a total of ten senior citizens. They spend:

$50;$40; $36;$15; $50;$100; $40;$53; $22;$22

a. Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population?

a. No. The first sample probably consists of science-oriented students. Besides the chemistry course, some of them are also taking first-term calculus. Books for these classes tend to be expensive. Most of these students are, more than likely, paying more than the average part-time student for their books. The second sample is a group of senior citizens who are, more than likely, taking courses for health and interest. The amount of money they spend on books is probably much less than the average parttime student. Both samples are biased. Also, in both cases, not all students have a chance to be in either sample.

b. Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population?

b. For these samples, each member of the population did not have an equally likely chance of being chosen.

Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. (We assume that these are the only disciplines in which part-time students at ABC College are enrolled and that an equal number of part-time students are enrolled in each of the disciplines.) Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he or she has a corresponding number. The students spend the following amounts:

$180;$50; $150;$85; $260;$75; $180;$200; $200;$150

c. Is the sample biased?

Students often ask if it is "good enough" to take a sample, instead of surveying the entire population. If the survey is done well, the answer is yes.

Exercise (PageIndex{12})

A local radio station has a fan base of 20,000 listeners. The station wants to know if its audience would prefer more music or more talk shows. Asking all 20,000 listeners is an almost impossible task.

The station uses convenience sampling and surveys the first 200 people they meet at one of the station’s music concert events. 24 people said they’d prefer more talk shows, and 176 people said they’d prefer more music.

Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population?

The sample probably consists more of people who prefer music because it is a concert event. Also, the sample represents only those who showed up to the event earlier than the majority. The sample probably doesn’t represent the entire fan base and is probably biased towards people who would prefer music.

Collaborative Exercise (PageIndex{8})

As a class, determine whether or not the following samples are representative. If they are not, discuss the reasons.

1. To find the average GPA of all students in a university, use all honor students at the university as the sample.
2. To find out the most popular cereal among young people under the age of ten, stand outside a large supermarket for three hours and speak to every twentieth child under age ten who enters the supermarket.
3. To find the average annual income of all adults in the United States, sample U.S. congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every U.S. congressman in the cluster.
4. To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you.
5. To determine the average cost of a two-day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling.

## Variation in Data

Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:

15.8; 16.1; 15.2; 14.8; 15.8; 15.9; 16.0; 15.5

Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range.

Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy.

## Variation in Samples

It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their college sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however.

Think about what contributes to making Doreen’s and Jung’s samples different.

If Doreen and Jung took larger samples (i.e. the number of data values is increased), their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This variability in samples cannot be stressed enough.

## Size of a Sample

The size of a sample (often called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200 to 1,500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals.

Be aware that many large samples are biased. For example, call-in surveys are invariably biased, because people choose to respond or not.

Collaborative Exercise (PageIndex{8})

Divide into groups of two, three, or four. Your instructor will give each group one six-sided die. Try this experiment twice. Roll one fair die (six-sided) 20 times. Record the number of ones, twos, threes, fours, fives, and sixes you get in the following table (“frequency” is the number of times a particular face of the die occurs):

First Experiment (20 rolls)Second Experiment (20 rolls)
Face on DieFrequencyFace on DieFrequency
1
2
3
4
5
6

Did the two experiments have the same results? Probably not. If you did the experiment a third time, do you expect the results to be identical to the first or second experiment? Why or why not?

Which experiment had the correct results? They both did. The job of the statistician is to see through the variability and draw appropriate conclusions.

## Critical Evaluation

We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include

• Problems with samples: A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not valid.
• Self-selected samples: Responses only by people who choose to respond, such as call-in surveys, are often unreliable.
• Sample size issues: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples: crash testing cars or medical testing for rare conditions
• Undue influence: collecting data or asking questions in a way that influences the response
• Non-response or refusal of subject to participate: The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can affect the results.
• Causality: A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable.
• Self-funded or self-interest studies: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done.
• Misleading use of data: improperly displayed graphs, incomplete data, or lack of context
• Confounding: When the effects of multiple factors on a response cannot be separated. Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor.

## References

1. Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/default.asp (accessed May 1, 2013).
2. Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/methodology.asp (accessed May 1, 2013).
3. Gallup-Healthways Well-Being Index. http://www.gallup.com/poll/146822/ga...questions.aspx (accessed May 1, 2013).
4. Data from www.bookofodds.com/Relationsh...-the-President
5. Dominic Lusinchi, “’President’ Landon and the 1936 Literary Digest Poll: Were Automobile and Telephone Owners to Blame?” Social Science History 36, no. 1: 23-54 (2012), ssh.dukejournals.org/content/36/1/23.abstract (accessed May 1, 2013).
6. “The Literary Digest Poll,” Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/data/LiteraryDigest.html (accessed May 1, 2013).
7. “Gallup Presidential Election Trial-Heat Trends, 1936–2008,” Gallup Politics http://www.gallup.com/poll/110548/ga...9362004.aspx#4 (accessed May 1, 2013).
8. The Data and Story Library, lib.stat.cmu.edu/DASL/Datafiles/USCrime.html (accessed May 1, 2013).
9. LBCC Distance Learning (DL) program data in 2010-2011, http://de.lbcc.edu/reports/2010-11/f...hts.html#focus (accessed May 1, 2013).
10. Data from San Jose Mercury News

## Review

Data are individual items of information that come from a population or sample. Data may be classified as qualitative, quantitative continuous, or quantitative discrete.

Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data.

Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted.

1. lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013).
2. Scott Keeter et al., “Gauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,” Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (accessed May 1, 2013).
3. Frequently Asked Questions, Pew Research Center for the People & the Press, www.people-press.org/methodol...wer-your-polls (accessed May 1, 2013).

## Glossary

Cluster Sampling
a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters. Every individual in the chosen clusters is included in the sample.
Continuous Random Variable
a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV.
Convenience Sampling
a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data.
Discrete Random Variable
a random variable (RV) whose outcomes are counted
Nonsampling Error
an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis.
Qualitative Data
See Data.
Quantitative Data
See Data.
Random Sampling
a method of selecting a sample that gives every member of the population an equal chance of being selected.
Sampling Bias
not all members of the population are equally likely to be selected
Sampling Error
the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error.
Sampling with Replacement
Once a member of the population is selected for inclusion in a sample, that member is returned to the population for the selection of the next individual.
Sampling without Replacement
A member of the population may be chosen for inclusion in a sample only once. If chosen, the member is not returned to the population before the next selection.
Simple Random Sampling
a straightforward method for selecting a random sample; give each member of the population a number. Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample.
Stratified Sampling
a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum.
Systematic Sampling
a method for selecting a random sample; list the members of the population. Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with the one that was randomly selected. If necessary, return to the beginning of the population list to complete your sample.

## Measures of variability

Published on September 7, 2020 by Pritha Bhandari. Revised on October 26, 2020.

Variability describes how far apart data points lie from each other and from the center of a distribution. Along with measures of central tendency, measures of variability give you descriptive statistics that summarize your data.

Variability is also referred to as spread, scatter or dispersion. It is most commonly measured with the following:

• Range: the difference between the highest and lowest values
• Interquartile range: the range of the middle half of a distribution
• Standard deviation: average distance from the mean
• Variance: average of squared distances from the mean

## Sampling Methods and Bias

### Selecting a Population

Suppose we are hired by a politician to determine the amount of support he has among the electorate should he decide to run for another term. What population should we study? Every person in the district? Not every person is eligible to vote, and regardless of how strongly someone likes or dislikes the candidate, they don’t have much to do with him being re-elected if they are not able to vote.

What about eligible voters in the district? That might be better, but if someone is eligible to vote but does not register by the deadline, they won’t have any say in the election either. What about registered voters? Many people are registered but choose not to vote. What about “likely voters?”

This is the criteria used in much political polling, but it is sometimes difficult to define a “likely voter.” Is it someone who voted in the last election? In the last general election? In the last presidential election? Should we consider someone who just turned 18 a “likely voter?” They weren’t eligible to vote in the past, so how do we judge the likelihood that they will vote in the next election?

In November 1998, former professional wrestler Jesse “The Body” Ventura was elected governor of Minnesota. Up until right before the election, most polls showed he had little chance of winning. There were several contributing factors to the polls not reflecting the actual intent of the electorate:

• Ventura was running on a third-party ticket and most polling methods are better suited to a two-candidate race.
• Many respondents to polls may have been embarrassed to tell pollsters that they were planning to vote for a professional wrestler.
• The mere fact that the polls showed Ventura had little chance of winning might have prompted some people to vote for him in protest to send a message to the major-party candidates.

But one of the major contributing factors was that Ventura recruited a substantial amount of support from young people, particularly college students, who had never voted before and who registered specifically to vote in the gubernatorial election. The polls did not deem these young people likely voters (since in most cases young people have a lower rate of voter registration and a turnout rate for elections) and so the polling samples were subject to sampling bias: they omitted a portion of the electorate that was weighted in favor of the winning candidate.

### Sampling bias

A sampling method is biased if every member of the population doesn’t have equal likelihood of being in the sample.

So even identifying the population can be a difficult job, but once we have identified the population, how do we choose an appropriate sample? Remember, although we would prefer to survey all members of the population, this is usually impractical unless the population is very small, so we choose a sample. There are many ways to sample a population, but there is one goal we need to keep in mind: we would like the sample to be representative of the population.

Returning to our hypothetical job as a political pollster, we would not anticipate very accurate results if we drew all of our samples from among the customers at a Starbucks, nor would we expect that a sample drawn entirely from the membership list of the local Elks club would provide a useful picture of district-wide support for our candidate.

One way to ensure that the sample has a reasonable chance of mirroring the population is to employ randomness. The most basic random method is simple random sampling.

### Simple random sample

A random sample is one in which each member of the population has an equal probability of being chosen. A simple random sample is one in which every member of the population and any group of members has an equal probability of being chosen.

### Example

If we could somehow identify all likely voters in the state, put each of their names on a piece of paper, toss the slips into a (very large) hat and draw 1000 slips out of the hat, we would have a simple random sample.

In practice, computers are better suited for this sort of endeavor than millions of slips of paper and extremely large headgear.

It is always possible, however, that even a random sample might end up not being totally representative of the population. If we repeatedly take samples of 1000 people from among the population of likely voters in the state of Washington, some of these samples might tend to have a slightly higher percentage of Democrats (or Republicans) than does the general population some samples might include more older people and some samples might include more younger people etc. In most cases, this sampling variability is not significant.

### Sampling variability

The natural variation of samples is called sampling variability.

This is unavoidable and expected in random sampling, and in most cases is not an issue.

To help account for variability, pollsters might instead use a stratified sample.

### Stratified sampling

In stratified sampling, a population is divided into a number of subgroups (or strata). Random samples are then taken from each subgroup with sample sizes proportional to the size of the subgroup in the population.

### Example

Suppose in a particular state that previous data indicated that the electorate was comprised of 39% Democrats, 37% Republicans and 24% independents. In a sample of 1000 people, they would then expect to get about 390 Democrats, 370 Republicans and 240 independents. To accomplish this, they could randomly select 390 people from among those voters known to be Democrats, 370 from those known to be Republicans, and 240 from those with no party affiliation.

Stratified sampling can also be used to select a sample with people in desired age groups, a specified mix ratio of males and females, etc. A variation on this technique is called quota sampling.

### Quota sampling

Quota sampling is a variation on stratified sampling, wherein samples are collected in each subgroup until the desired quota is met.

### Example

Suppose the pollsters call people at random, but once they have met their quota of 390 Democrats, they only gather people who do not identify themselves as a Democrat.

You may have had the experience of being called by a telephone pollster who started by asking you your age, income, etc. and then thanked you for your time and hung up before asking any “real” questions. Most likely, they already had contacted enough people in your demographic group and were looking for people who were older or younger, richer or poorer, etc. Quota sampling is usually a bit easier than stratified sampling, but also does not ensure the same level of randomness.

Another sampling method is cluster sampling, in which the population is divided into groups, and one or more groups are randomly selected to be in the sample.

### Cluster sampling

In cluster sampling, the population is divided into subgroups (clusters), and a set of subgroups are selected to be in the sample.

### Example

If the college wanted to survey students, since students are already divided into classes, they could randomly select 10 classes and give the survey to all the students in those classes. This would be cluster sampling.

Other sampling methods include systematic sampling.

### Systematic sampling

In systematic sampling, every n th member of the population is selected to be in the sample.

### Example

To select a sample using systematic sampling, a pollster calls every 100th name in the phone book.

Systematic sampling is not as random as a simple random sample (if your name is Albert Aardvark and your sister Alexis Aardvark is right after you in the phone book, there is no way you could both end up in the sample) but it can yield acceptable samples.

### The Worst Way to Sample

Perhaps the worst types of sampling methods are convenience samples and voluntary response samples.

### Convenience sampling and voluntary response sampling

Convenience sampling is the practice of samples chosen by selecting whoever is convenient.

Voluntary response sampling is allowing the sample to volunteer.

### Example

A pollster stands on a street corner and interviews the first 100 people who agree to speak to him. Which sampling method is represented by this scenario?

A website has a survey asking readers to give their opinion on a tax proposal. Which sampling method is represented?

This is a self-selected sample, or voluntary response sample, in which respondents volunteer to participate.

Usually voluntary response samples are skewed towards people who have a particularly strong opinion about the subject of the survey or who just have way too much time on their hands and enjoy taking surveys.

Watch the following video for an overview of all the sampling methods discussed so far.

### Try It Now

In each case, indicate what sampling method was used

a. Every 4th person in the class was selected

b. A sample was selected to contain 25 men and 35 women

c. Viewers of a new show are asked to vote on the show’s website

d. A website randomly selects 50 of their customers to send a satisfaction survey to

e. To survey voters in a town, a polling company randomly selects 10 city blocks, and interviews everyone who lives on those blocks.

### Problematic Sampling and Surveying

There are number of ways that a study can be ruined before you even start collecting data. The first we have already explored – sampling or selection bias, which is when the sample is not representative of the population. One example of this is voluntary response bias, which is bias introduced by only collecting data from those who volunteer to participate. This is not the only potential source of bias.

### Sources of bias

• Sampling bias – when the sample is not representative of the population
• Voluntary response bias – the sampling bias that often occurs when the sample is volunteers
• Self-interest study – bias that can occur when the researchers have an interest in the outcome
• Response bias – when the responder gives inaccurate responses for any reason
• Perceived lack of anonymity – when the responder fears giving an honest answer might negatively affect them
• Loaded questions – when the question wording influences the responses
• Non-response bias – when people refusing to participate in the study can influence the validity of the outcome

### Examples

Consider a recent study which found that chewing gum may raise math grades in teenagers [1] . This study was conducted by the Wrigley Science Institute, a branch of the Wrigley chewing gum company. Identify the type of sampling bias found in this example.

A survey asks people “when was the last time you visited your doctor?” What type of sampling bias might this lead to?

This might suffer from response bias, since many people might not remember exactly when they last saw a doctor and give inaccurate responses.

Sources of response bias may be innocent, such as bad memory, or as intentional as pressuring by the pollster. Consider, for example, how many voting initiative petitions people sign without even reading them.

A survey asks participants a question about their interactions with members of other races. Which sampling bias might occur for this survey strategy?

An employer puts out a survey asking their employees if they have a drug abuse problem and need treatment help. Which sampling bias may occur in this scenario?

A survey asks “do you support funding research of alternative energy sources to reduce our reliance on high-polluting fossil fuels?” Which sampling bias may result from this survey?

Loaded questions can occur intentionally by pollsters with an agenda, or accidentally through poor question wording. Also a concern is question order, where the order of questions changes the results. A psychology researcher provides an example [2] :

“My favorite finding is this: we did a study where we asked students, ‘How satisfied are you with your life? How often do you have a date?’ The two answers were not statistically related – you would conclude that there is no relationship between dating frequency and life satisfaction. But when we reversed the order and asked, ‘How often do you have a date? How satisfied are you with your life?’ the statistical relationship was a strong one. You would now conclude that there is nothing as important in a student’s life as dating frequency.”

A telephone poll asks the question “Do you often have time to relax and read a book?”, and 50% of the people called refused to answer the survey. Which sampling bias is represented by this survey?

These problematic scenarios for statistics gathering are discussed further in the following video.

### Try It Now

In each situation, identify a potential source of bias

a. A survey asks how many sexual partners a person has had in the last year

c. A substitute teacher wants to know how students in the class did on their last test. The teacher asks the 10 students sitting in the front row to state their latest test score.

d. High school students are asked if they have consumed alcohol in the last two weeks.

e. The Beef Council releases a study stating that consuming red meat poses little cardiovascular risk.

f. A poll asks “Do you support a new transportation tax, or would you prefer to see our public transportation system fall apart?”

## NORC's 1990 Sampling Design

### Summary

The 1990 National Sampling Frame for area probability studies, as described in this article, was a multistage cluster design with systematic sampling of geographies and housing units. The sampling stages are summarized in Fig. 4 . The geographical sampling units in the first two stages were selected with probability proportional to the number of housing units. Lists of addresses were collected for the smallest geographical sampling units. The address listings comprised the frame for nationally representative samples of housing units requiring face-to-face interviewing. When HUs were selected, the sample was allocated to the geographies such that all HUs had an equal probability of selection. The National Sampling Frame was the basis for a number of high profile studies throughout the past decade.

Figure 4 . Summary of the multistage sample design in NORC's 1990 National Sampling Frame.

While the basic principles of area probability designs have changed little over the years, technological advancements are making implementation quicker and cheaper. Organizations such as NORC can take advantage of these developments to reduce costs and to increase the statistical efficiency of the basic designs.

## Data, Sampling, and Variation in Data and Sampling

Data may come from a population or from a sample. Lowercase letters like or generally are used to represent data values. Most data can be put into the following categories:

Qualitative data are the result of categorizing or describing attributes of a population. Qualitative data are also often called categorical data . Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type.

Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous .

All data that are the result of counting are called quantitative discrete data . These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three.

Data that are not only made up of counting numbers, but that may include fractions, decimals, or irrational numbers, are called quantitative continuous data . Continuous data are often the results of measurements like lengths, weights, or times. A list of the lengths in minutes for all the phone calls that you make in a week, with numbers like 2.4, 7.5, or 11.0, would be quantitative continuous data.

The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. The numbers of books (three, four, two, and one) are the quantitative discrete data.

The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this?

The data are the weights of backpacks with books in them. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data.

The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. feet, 160 sq. feet, 190 sq. feet, 180 sq. feet, and 210 sq. feet. What type of data is this?

You go to the supermarket and purchase three cans of soup (19 ounces tomato bisque, 14.1 ounces lentil, and 19 ounces Italian wedding), two packages of nuts (walnuts and peanuts), four different kinds of vegetable (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces pistachio ice cream and 32 ounces chocolate chip cookies).

Name data sets that are quantitative discrete, quantitative continuous, and qualitative.

• The three cans of soup, two packages of nuts, four kinds of vegetables and two desserts are quantitative discrete data because you count them.
• The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure weights as precisely as possible.
• Types of soups, nuts, vegetables and desserts are qualitative data because they are categorical.

Try to identify additional data sets in this example.

The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data.

The data are the colors of houses. You sample five houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this?

You may collect data as numbers and report it categorically. For example, the quiz scores for each student are recorded throughout the term. At the end of the term, the quiz scores are reported as A, B, C, D, or F.

Work collaboratively to determine the correct data type (quantitative or qualitative). Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words “the number of.”

1. the number of pairs of shoes you own
2. the type of car you drive
3. the distance it is from your home to the nearest grocery store
4. the number of classes you take per school year.
5. the type of calculator you use
6. weights of sumo wrestlers
7. number of correct answers on a quiz
8. IQ scores (This may cause some discussion.)

Items a, d, and g are quantitative discrete items c, f, and h are quantitative continuous items b and e are qualitative, or categorical.

Determine the correct data type (quantitative or qualitative) for the number of cars in a parking lot. Indicate whether quantitative data are continuous or discrete.

A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart (Figure). What type of data does this graph show?

This pie chart shows the students in each year, which is qualitative (or categorical) data.

The registrar at State University keeps records of the number of credit hours students complete each semester. The data he collects are summarized in the histogram. The class boundaries are 10 to less than 13, 13 to less than 16, 16 to less than 19, 19 to less than 22, and 22 to less than 25.

What type of data does this graph show?

### Qualitative Data Discussion

Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.

Fall Term 2007 (Census day)
De Anza College Foothill College
Number Percent Number Percent
Full-time 9,200 40.9% Full-time 4,059 28.6%
Part-time 13,296 59.1% Part-time 10,124 71.4%
Total 22,496 100% Total 14,183 100%

Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative data are pie charts and bar graphs.

In a pie chart , categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category.

In a bar graph , the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.

A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).

Look at (Figure) and (Figure) and determine which graph (pie or bar) you think displays the comparisons better.

It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the “best” graph depending on the data and the context. Our choice also depends on what we are using the data for.

#### Percentages That Add to More (or Less) Than 100%

Sometimes percentages add up to be more than 100% (or less than 100%). In the graph, the percentages add to more than 100% because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100%.

De Anza College Spring 2010
Characteristic/Category Percent
Full-Time Students 40.9%
Students who intend to transfer to a 4-year educational institution 48.6%
Students under age 25 61.0%
TOTAL 150.5%

#### Omitting Categories/Missing Data

The table displays Ethnicity of Students but is missing the “Other/Unknown” category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.

Ethnicity of Students at De Anza College Fall Term 2007 (Census Day)
Frequency Percent
Asian 8,794 36.1%
Black 1,412 5.8%
Filipino 1,298 5.3%
Hispanic 4,180 17.1%
Native American 146 0.6%
Pacific Islander 236 1.0%
White 5,978 24.5%
TOTAL 22,044 out of 24,382 90.4% out of 100%

The following graph is the same as the previous graph but the “Other/Unknown” percent (9.6%) has been included. The “Other/Unknown” category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0%). This is important to know when we think about what the data are telling us.

This particular bar graph in (Figure) can be difficult to understand visually. The graph in (Figure) is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.

#### Pie Charts: No Missing Data

The following pie charts have the “Other/Unknown” category included (since the percentages must add to 100%). The chart in (Figure) is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in (Figure).

### Sampling

Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Any group of n individuals is equally likely to be chosen as any other group of n individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in (Figure):

Class Roster
ID Name ID Name ID Name
00 Anselmo 11 King 21 Roquero
01 Bautista 12 Legeny 22 Roth
02 Bayani 13 Lundquist 23 Rowell
03 Cheng 14 Macierz 24 Salangsang
04 Cuarismo 15 Motogawa 25 Slade
05 Cuningham 16 Okimoto 26 Stratcher
06 Fontecha 17 Patel 27 Tallai
07 Hong 18 Price 28 Tran
08 Hoobler 19 Quizon 29 Wai
09 Jiao 20 Reyes 30 Wood
10 Khan

Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows:

0.94360 0.99832 0.14669 0.51470 0.40581 0.73381 0.04399

Lisa reads two-digit groups until she has chosen three class members (that is, she reads 0.94360 as the groups 94, 43, 36, 60). Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers.

The random numbers 0.94360 and 0.99832 do not contain appropriate two digit numbers. However the third random number, 0.14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Macierz, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Marcierz, Cuningham, and Cuarismo.

To generate random numbers:

• Press MATH.
• Arrow over to PRB.
• Press 5:randInt(. Enter 0, 30).
• Press ENTER for the first random number.
• Press ENTER two more times for the other 2 random numbers. If there is a repeat press ENTER again.

Note: randInt(0, 30, 3) will generate 3 random numbers.

Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.

To choose a stratified sample, divide the population into groups called strata and then take a proportionate number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members who make up the stratified sample.

To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your college population, the four departments make up the cluster sample. Divide your college faculty by department. The departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.

To choose a systematic sample, randomly select a starting point and take every n th piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings. You must choose 400 names for the sample. Number the population 1–20,000 and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of 400 names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is a simple method.

A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others.

Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased (they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.

True random sampling is done with replacement. That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done without replacement. Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low.

In a college population of 10,000 people, suppose you want to pick a sample of 1,000 randomly for a survey. For any particular sample of 1,000, if you are sampling with replacement,

• the chance of picking the first person is 1,000 out of 10,000 (0.1000)
• the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999)
• the chance of picking the same person again is 1 out of 10,000 (very low).

If you are sampling without replacement,

• the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000)
• the chance of picking a different second person is 999 out of 9,999 (0.0999)
• you do not replace the first person before picking the next person.

Compare the fractions 999/10,000 and 999/9,999. For accuracy, carry the decimal answers to four decimal places. To four decimal places, these numbers are equivalent (0.0999).

Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is 25 people, the sample is ten, and you are sampling with replacement for any particular sample, then the chance of picking the first person is ten out of 25, and the chance of picking a different second person is nine out of 25 (you replace the first person).

If you sample without replacement, then the chance of picking the first person is ten out of 25, and then the chance of picking the second person (who is different) is nine out of 24 (you do not replace the first person).

Compare the fractions 9/25 and 9/24. To four decimal places, 9/25 = 0.3600 and 9/24 = 0.3750. To four decimal places, these numbers are not equivalent.

When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error.

In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error.

In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others (remember, each member of the population should have an equally likely chance of being chosen). When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied.

#### Critical Evaluation

We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include

• Problems with samples: A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not valid.
• Self-selected samples: Responses only by people who choose to respond, such as call-in surveys, are often unreliable.
• Sample size issues: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples: crash testing cars or medical testing for rare conditions
• Undue influence: collecting data or asking questions in a way that influences the response
• Non-response or refusal of subject to participate: The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can affect the results.
• Causality: A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable.
• Self-funded or self-interest studies: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done.
• Misleading use of data: improperly displayed graphs, incomplete data, or lack of context
• Confounding: When the effects of multiple factors on a response cannot be separated. Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor.

As a class, determine whether or not the following samples are representative. If they are not, discuss the reasons.

1. To find the average GPA of all students in a university, use all honor students at the university as the sample.
2. To find out the most popular cereal among young people under the age of ten, stand outside a large supermarket for three hours and speak to every twentieth child under age ten who enters the supermarket.
3. To find the average annual income of all adults in the United States, sample U.S. congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every U.S. congressman in the cluster.
4. To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you.
5. To determine the average cost of a two-day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling.

A study is done to determine the average tuition that San Jose State undergraduate students pay per semester. Each student in the following samples is asked how much tuition he or she paid for the Fall semester. What is the type of sampling in each case?

1. A sample of 100 undergraduate San Jose State students is taken by organizing the students’ names by classification (freshman, sophomore, junior, or senior), and then selecting 25 students from each.
2. A random number generator is used to select a student from the alphabetical listing of all undergraduate students in the Fall semester. Starting with that student, every 50th student is chosen until 75 students are included in the sample.
3. A completely random method is used to select 75 students. Each undergraduate student in the fall semester has the same probability of being chosen at any stage of the sampling process.
4. The freshman, sophomore, junior, and senior years are numbered one, two, three, and four, respectively. A random number generator is used to pick two of those years. All students in those two years are in the sample.
5. An administrative assistant is asked to stand in front of the library one Wednesday and to ask the first 100 undergraduate students he encounters what they paid for tuition the Fall semester. Those 100 students are the sample.

a. stratified b. systematic c. simple random d. cluster e. convenience

You are going to use the random number generator to generate different types of samples from the data.

This table displays six sets of quiz scores (each quiz counts 10 points) for an elementary statistics class.

#1 #2 #3 #4 #5 #6
5 7 10 9 8 3
10 5 9 8 7 6
9 10 8 6 7 9
9 10 10 9 8 9
7 8 9 5 7 4
9 9 9 10 8 7
7 7 10 9 8 8
8 8 9 10 8 8
9 7 8 7 7 8
8 8 10 9 8 7

Instructions: Use the Random Number Generator to pick samples.

1. Create a stratified sample by column. Pick three quiz scores randomly from each column.
• Number each row one through ten.
• On your calculator, press Math and arrow over to PRB.
• For column 1, Press 5:randInt( and enter 1,10). Press ENTER. Record the number. Press ENTER 2 more times (even the repeats). Record these numbers. Record the three quiz scores in column one that correspond to these three numbers.
• Repeat for columns two through six.
• These 18 quiz scores are a stratified sample.
2. Create a cluster sample by picking two of the columns. Use the column numbers: one through six.
• Press MATH and arrow over to PRB.
• Press 5:randInt( and enter 1,6). Press ENTER. Record the number. Press ENTER and record that number.
• The two numbers are for two of the columns.
• The quiz scores (20 of them) in these 2 columns are the cluster sample.
3. Create a simple random sample of 15 quiz scores.
• Use the numbering one through 60.
• Press MATH. Arrow over to PRB. Press 5:randInt( and enter 1, 60).
• Press ENTER 15 times and record the numbers.
• Record the quiz scores that correspond to these numbers.
• These 15 quiz scores are the systematic sample.
4. Create a systematic sample of 12 quiz scores.
• Use the numbering one through 60.
• Press MATH. Arrow over to PRB. Press 5:randInt( and enter 1, 60).
• Press ENTER. Record the number and the first quiz score. From that number, count ten quiz scores and record that quiz score. Keep counting ten quiz scores and recording the quiz score until you have a sample of 12 quiz scores. You may wrap around (go back to the beginning).

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

1. A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group of boys aged 11 to 12, and three players from a group of boys aged 13 to 14 to form a recreational soccer team.
2. A pollster interviews all human resource personnel in five different high tech companies.
3. A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers.
4. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.
5. A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers.
6. A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on the average.

a. stratified b. cluster c. stratified d. systematic e. simple random f.convenience

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

A high school principal polls 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors regarding policy changes for after school activities.

If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural.

Suppose ABC College has 10,000 part-time students (the population). We are interested in the average amount of money a part-time student spends on books in the fall term. Asking all 10,000 students is an almost impossible task.

Suppose we take two different samples.

First, we use convenience sampling and survey ten students from a first term organic chemistry class. Many of these students are taking first term calculus in addition to the organic chemistry class. The amount of money they spend on books is as follows:

?128 ?87 ?173 ?116 ?130 ?204 ?147 ?189 ?93 ?153

The second sample is taken using a list of senior citizens who take P.E. classes and taking every fifth senior citizen on the list, for a total of ten senior citizens. They spend:

?50 ?40 ?36 ?15 ?50 ?100 ?40 ?53 ?22 ?22

It is unlikely that any student is in both samples.

a. Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population?

a. No. The first sample probably consists of science-oriented students. Besides the chemistry course, some of them are also taking first-term calculus. Books for these classes tend to be expensive. Most of these students are, more than likely, paying more than the average part-time student for their books. The second sample is a group of senior citizens who are, more than likely, taking courses for health and interest. The amount of money they spend on books is probably much less than the average parttime student. Both samples are biased. Also, in both cases, not all students have a chance to be in either sample.

b. Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population?

b. No. For these samples, each member of the population did not have an equally likely chance of being chosen.

Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. (We assume that these are the only disciplines in which part-time students at ABC College are enrolled and that an equal number of part-time students are enrolled in each of the disciplines.) Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he or she has a corresponding number. The students spend the following amounts:

?180 ?50 ?150 ?85 ?260 ?75 ?180 ?200 ?200 ?150

c. The sample is unbiased, but a larger sample would be recommended to increase the likelihood that the sample will be close to representative of the population. However, for a biased sampling technique, even a large sample runs the risk of not being representative of the population.

Students often ask if it is “good enough” to take a sample, instead of surveying the entire population. If the survey is done well, the answer is yes.

A local radio station has a fan base of 20,000 listeners. The station wants to know if its audience would prefer more music or more talk shows. Asking all 20,000 listeners is an almost impossible task.

The station uses convenience sampling and surveys the first 200 people they meet at one of the station’s music concert events. 24 people said they’d prefer more talk shows, and 176 people said they’d prefer more music.

Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population?

### Variation in Data

Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:

15.8 16.1 15.2 14.8 15.8 15.9 16.0 15.5

Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range.

Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy.

### Variation in Samples

It was mentioned previously that two or more samples from the same population , taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their college sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen’s sample will be different from Jung’s sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however.

Think about what contributes to making Doreen’s and Jung’s samples different.

If Doreen and Jung took larger samples (i.e. the number of data values is increased), their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This variability in samples cannot be stressed enough.

#### Size of a Sample

The size of a sample (often called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200 to 1,500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals.

Be aware that many large samples are biased. For example, call-in surveys are invariably biased, because people choose to respond or not.

Divide into groups of two, three, or four. Your instructor will give each group one six-sided die. Try this experiment twice. Roll one fair die (six-sided) 20 times. Record the number of ones, twos, threes, fours, fives, and sixes you get in (Figure) and (Figure) (“frequency” is the number of times a particular face of the die occurs):

First Experiment (20 rolls)
Face on Die Frequency
1
2
3
4
5
6
Second Experiment (20 rolls)
Face on Die Frequency
1
2
3
4
5
6

Did the two experiments have the same results? Probably not. If you did the experiment a third time, do you expect the results to be identical to the first or second experiment? Why or why not?

Which experiment had the correct results? They both did. The job of the statistician is to see through the variability and draw appropriate conclusions.

### References

Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/default.asp (accessed May 1, 2013).

Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/methodology.asp (accessed May 1, 2013).

Gallup-Healthways Well-Being Index. http://www.gallup.com/poll/146822/gallup-healthways-index-questions.aspx (accessed May 1, 2013).

Data from http://www.bookofodds.com/Relationships-Society/Articles/A0374-How-George-Gallup-Picked-the-President

Dominic Lusinchi, “’President’ Landon and the 1936 Literary Digest Poll: Were Automobile and Telephone Owners to Blame?” Social Science History 36, no. 1: 23-54 (2012), http://ssh.dukejournals.org/content/36/1/23.abstract (accessed May 1, 2013).

“The Literary Digest Poll,” Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/data/LiteraryDigest.html (accessed May 1, 2013).

“Gallup Presidential Election Trial-Heat Trends, 1936–2008,” Gallup Politics http://www.gallup.com/poll/110548/gallup-presidential-election-trialheat-trends-19362004.aspx#4 (accessed May 1, 2013).

The Data and Story Library, http://lib.stat.cmu.edu/DASL/Datafiles/USCrime.html (accessed May 1, 2013).

LBCC Distance Learning (DL) program data in 2010-2011, http://de.lbcc.edu/reports/2010-11/future/highlights.html#focus (accessed May 1, 2013).

Data from San Jose Mercury News

### Chapter Review

Data are individual items of information that come from a population or sample. Data may be classified as qualitative(categorical), quantitative continuous, or quantitative discrete.

Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data.

Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted.

### Practice

“Number of times per week” is what type of data?

a. qualitative(categorical) b. quantitative discrete c. quantitative continuous

Use the following information to answer the next four exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Antonio, Texas. The first house in the neighborhood around the park was selected randomly, and then the resident of every eighth house in the neighborhood around the park was interviewed.

a. simple random b. systematic c. stratified d. cluster

“Duration (amount of time)” is what type of data?

a. qualitative(categorical) b. quantitative discrete c. quantitative continuous

The colors of the houses around the park are what kind of data?

a. qualitative(categorical) b. quantitative discrete c. quantitative continuous

The population is ______________________

(Figure) contains the total number of deaths worldwide as a result of earthquakes from 2000 to 2012.

Year Total Number of Deaths
2000 231
2001 21,357
2002 11,685
2003 33,819
2004 228,802
2005 88,003
2006 6,605
2007 712
2008 88,011
2009 1,790
2010 320,120
2011 21,953
2012 768
Total 823,856

Use (Figure) to answer the following questions.

1. What is the proportion of deaths between 2007 and 2012?
2. What percent of deaths occurred before 2001?
3. What is the percent of deaths that occurred in 2003 or after 2010?
4. What is the fraction of deaths that happened before 2012?
5. What kind of data is the number of deaths?
6. Earthquakes are quantified according to the amount of energy they produce (examples are 2.1, 5.0, 6.7). What type of data is that?
7. What contributed to the large number of deaths in 2010? In 2004? Explain.
1. 0.5242
2. 0.03%
3. 6.86%
4. quantitative discrete
5. quantitative continuous
6. In both years, underwater earthquakes produced massive tsunamis.

For the following four exercises, determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).

A group of test subjects is divided into twelve groups then four of the groups are chosen at random.

A market researcher polls every tenth person who walks into a store.

The first 50 people who walk into a sporting event are polled on their television preferences.

A computer generates 100 random numbers, and 100 people whose names correspond with the numbers on the list are chosen.

Use the following information to answer the next seven exercises: Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected.

Researcher A: 3 4 11 15 16 17 22 44 37 16 14 24 25 15 26 27 33 29 35 44 13 21 22 10 12 8 40 32 26 27 31 34 29 17 8 24 18 47 33 34

Researcher B: 3 14 11 5 16 17 28 41 31 18 14 14 26 25 21 22 31 2 35 44 23 21 21 16 12 18 41 22 16 25 33 34 29 13 18 24 23 42 33 29

Complete the tables using the data provided:

Researcher A
Survival Length (in months) Frequency Relative Frequency Cumulative Relative Frequency
0.5–6.5
6.5–12.5
12.5–18.5
18.5–24.5
24.5–30.5
30.5–36.5
36.5–42.5
42.5–48.5
Researcher B
Survival Length (in months) Frequency Relative Frequency Cumulative Relative Frequency
0.5–6.5
6.5–12.5
12.5–18.5
18.5–24.5
24.5–30.5
30.5–36.5
36.5-45.5

Determine what the key term data refers to in the above example for Researcher A.

values for X, such as 3, 4, 11, and so on

List two reasons why the data may differ.

<!– <solution Answers will vary. Sample answer: One reason may be the average age of the individuals in the two samples. Or, perhaps the drug affects men and women differently. If the ratio of men and women aren’t the same in both sample groups, then the data would differ. –>

Can you tell if one researcher is correct and the other one is incorrect? Why?

No, we do not have enough information to make such a claim.

Would you expect the data to be identical? Why or why not?

<!– <solution Since the treatment is not the same the data might be different unless neither treatment has an effect. –>

Suggest at least two methods the researchers might use to gather random data.

Take a simple random sample from each group. One way is by assigning a number to each patient and using a random number generator to randomly select patients.

Suppose that the first researcher conducted his survey by randomly choosing one state in the nation and then randomly picking 40 patients from that state. What sampling method would that researcher have used?

<!– <solution He has used a simple random sample method. –>

Suppose that the second researcher conducted his survey by choosing 40 patients he knew. What sampling method would that researcher have used? What concerns would you have about this data set, based upon the data collection method?

This would be convenience sampling and is not random.

Use the following data to answer the next five exercises: Two researchers are gathering data on hours of video games played by school-aged children and young adults. They each randomly sample different groups of 150 students from the same school. They collect the following data.

Researcher A
Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency
0–2 26 0.17 0.17
2–4 30 0.20 0.37
4–6 49 0.33 0.70
6–8 25 0.17 0.87
8–10 12 0.08 0.95
10–12 8 0.05 1
Researcher B
Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency
0–2 48 0.32 0.32
2–4 51 0.34 0.66
4–6 24 0.16 0.82
6–8 12 0.08 0.90
8–10 11 0.07 0.97
10–12 4 0.03 1

Give a reason why the data may differ.

<!– <solution The researchers are studying different groups, so there will be some variation in the data. –>

Would the sample size be large enough if the population is the students in the school?

Yes, the sample size of 150 would be large enough to reflect a population of one school.

Would the sample size be large enough if the population is school-aged children and young adults in the United States?

<!– <solution There are many school-aged children and young adults in the United States, and the study was done at only one school, so the sample size is not large enough to reflect the population. –>

Researcher A concludes that most students play video games between four and six hours each week. Researcher B concludes that most students play video games between two and four hours each week. Who is correct?

Even though the specific data support each researcher’s conclusions, the different results suggest that more data need to be collected before the researchers can reach a conclusion.

As part of a way to reward students for participating in the survey, the researchers gave each student a gift card to a video game store. Would this affect the data if students knew about the award before the study?

<!– <solution Yes, people who play games more might be more likely to participate, since they would want the gift card more than a student who does not play video games. This would leave out many students who do not play games at all and skew the data. –>

Use the following data to answer the next five exercises: A pair of studies was performed to measure the effectiveness of a new software program designed to help stroke patients regain their problem-solving skills. Patients were asked to use the software program twice a day, once in the morning and once in the evening. The studies observed 200 stroke patients recovering over a period of several weeks. The first study collected the data in (Figure). The second study collected the data in (Figure).

Group Showed improvement No improvement Deterioration
Used program 142 43 15
Did not use program 72 110 18
Group Showed improvement No improvement Deterioration
Used program 105 74 19
Did not use program 89 99 12

Given what you know, which study is correct?

There is not enough information given to judge if either one is correct or incorrect.

The first study was performed by the company that designed the software program. The second study was performed by the American Medical Association. Which study is more reliable?

<!– <solution The second study is more reliable, because the company would be interested in showing results that favored a higher rate of improvement from patients using their software. The data may be skewed however, the American Medical Association is not concerned with the success of the software and so should be objective. –>

Both groups that performed the study concluded that the software works. Is this accurate?

The software program seems to work because the second study shows that more patients improve while using the software than not. Even though the difference is not as large as that in the first study, the results from the second study are likely more reliable and still show improvement.

The company takes the two studies as proof that their software causes mental improvement in stroke patients. Is this a fair statement?

<!– <solution No, the data suggest the two are correlated, but more studies need to be done to prove that using the software causes improvement in stroke patients. –>

Patients who used the software were also a part of an exercise program whereas patients who did not use the software were not. Does this change the validity of the conclusions from (Figure)?

Yes, because we cannot tell if the improvement was due to the software or the exercise the data is confounded, and a reliable conclusion cannot be drawn. New studies should be performed.

Is a sample size of 1,000 a reliable measure for a population of 5,000?

<!– <solution Yes, 1,000 represents 20% of the population and should be representative, if the population of the sample is chosen at random. –>

Is a sample of 500 volunteers a reliable measure for a population of 2,500?

No, even though the sample is large enough, the fact that the sample consists of volunteers makes it a self-selected sample, which is not reliable.

A question on a survey reads: “Do you prefer the delicious taste of Brand X or the taste of Brand Y?” Is this a fair question?

<!– <solution No, the question is creating undue influence by adding the word “delicious” to describe Brand X. The wording may influence responses. –>

Is a sample size of two representative of a population of five?

No, even though the sample is a large portion of the population, two responses are not enough to justify any conclusions. Because the population is so small, it would be better to include everyone in the population to get the most accurate data.

Is it possible for two experiments to be well run with similar sample sizes to get different data?

<!– <solution Yes, there will most likely be a degree of variation between any two studies, even if they are set up and run the same way. Each study may be affected differently by unknown factors such as location, mood of the subjects, or time of year. –>

### HOMEWORK

For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative), and give an example of the data.

number of tickets sold to a concert

quantitative discrete, 150

<!– <solution quantitative continuous, 19.2% –>

time in line to buy groceries

<!– <solution quantitative continuous, 7.2 minutes –>

number of students enrolled at Evergreen Valley College

quantitative discrete, 11,234 students

most-watched television show

<!– <solution qualitative, Dancing with the Stars –>

distance to the closest movie theatre

<!– <solution quantitative continuous, 8.32 miles –>

age of executives in Fortune 500 companies

quantitative continuous, 47.3 years

number of competing computer spreadsheet software packages

<!– <solution quantitative discrete, three –>

Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed.

“Number of times per week” is what type of data?

“Duration (amount of time)” is what type of data?

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study.

1. Using complete sentences, list three things wrong with the way the survey was conducted.
2. Using complete sentences, list three ways that you would improve the survey if it were to be repeated.
1. The survey was conducted using six similar flights.
The survey would not be a true representation of the entire population of air travelers.
Conducting the survey on a holiday weekend will not produce representative results.
2. Conduct the survey during different times of the year.
Conduct the survey using flights to and from various locations.
Conduct the survey on different days of the week.

Suppose you want to determine the mean number of students per statistics class in your state. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

Answers will vary. Sample Answer: You could use a systematic sampling method. Stop the tenth person as they leave one of the buildings on campus at 9:50 in the morning. Then stop the tenth person as they leave a different building on campus at 1:50 in the afternoon.

List some practical difficulties involved in getting accurate results from a telephone survey.

<!– <solution Answers will vary. Sample Answer: Not all people have a listed phone number. Many people hang up or do not respond to phone surveys. –>

List some practical difficulties involved in getting accurate results from a mailed survey.

Answers will vary. Sample Answer: Many people will not respond to mail surveys. If they do respond to the surveys, you can’t be sure who is responding. In addition, mailing lists can be incomplete.

With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey.

<!– <solution Ask everyone to include their age then take a random sample from the data. Include in the report how the survey was conducted and why the results may not be accurate. –>

The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. The type of sampling she used is

1. cluster sampling
2. stratified sampling
3. simple random sampling
4. convenience sampling

A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was:

Name the sampling method used in each of the following situations:

1. A woman in the airport is handing out questionnaires to travelers asking them to evaluate the airport’s service. She does not ask travelers who are hurrying through the airport with their hands full of luggage, but instead asks all travelers who are sitting near gates and not taking naps while they wait.
2. A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class.
3. The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, 100 randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest.
4. The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books.
5. A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party’s polling staff calls 1,200 randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates.

convenience cluster stratified systematic simple random

A “random survey” was conducted of 3,274 people of the “microprocessor generation” (people born since 1971, the year the microprocessor was invented). It was reported that 48% of those individuals surveyed stated that if they had ?2,000 to spend, they would use it for computer equipment. Also, 66% of those surveyed considered themselves relatively savvy computer users.

1. Do you consider the sample size large enough for a study of this type? Why or why not?
2. Based on your “gut feeling,” do you believe the percents accurately reflect the U.S. population for those individuals born since 1971? If not, do you think the percents of the population are actually higher or lower than the sample statistics? Why?
Additional information: The survey, reported by Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute’s road show called “America’s Smithsonian.”
3. With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not?
4. With the additional information, comment on how accurately you think the sample statistics reflect the population parameters.

<!– <solution Yes, in polling, samples that are from 1,200 to 1,500 observations are considered large enough and good enough if the survey is random and is well done. We do not have enough information to decide if this is a random sample from the U.S. population. No, this is a convenience sample taken from individuals who visited an exhibition in the Angeles Convention Center. This sample is not representative of the U.S. population. It is possible that the two sample statistics, 48% and 66% are larger than the true parameters in the population at large. In any event, no conclusion about the population proportions can be inferred from this convenience sample. –>

The Well-Being Index is a survey that follows trends of U.S. residents on a regular basis. There are six areas of health and wellness covered in the survey: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment, and Basic Access. Some of the questions used to measure the Index are listed below.

Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous.

1. Do you have any health problems that prevent you from doing any of the things people your age can normally do?
2. During the past 30 days, for about how many days did poor health keep you from doing your usual activities?
3. In the last seven days, on how many days did you exercise for 30 minutes or more?
4. Do you have health insurance coverage?
1. qualitative
2. quantitative discrete
3. quantitative discrete
4. qualitative

In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards.

1. Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time.
2. What effect does the low response rate have on the reliability of the sample?
3. Are these problems examples of sampling error or nonsampling error?
4. During the same year, George Gallup conducted his own poll of 30,000 prospective voters. These researchers used a method they called “quota sampling” to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module?

<!– <solution The country was in the middle of the Great Depression and many people could not afford these “luxury” items and therefore not able to be included in the survey. Samples that are too small can lead to sampling bias. sampling error stratified –>

Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI’s Uniform Crime Report. One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates.

Which of the potential problems with samples discussed in (Figure) could explain this connection?

Causality: The fact that two variables are related does not guarantee that one variable is influencing the other. We cannot assume that crime rate impacts education level or that education level impacts crime rate.

Confounding: There are many factors that define a community other than education level and crime rate. Communities with high crime rates and high education levels may have other lurking variables that distinguish them from communities with lower crime rates and lower education levels. Because we cannot isolate these variables of interest, we cannot draw valid conclusions about the connection between education and crime. Possible lurking variables include police expenditures, unemployment levels, region, average age, and size.

YouPolls is a website that allows anyone to create and respond to polls. One question posted April 15 asks:

“Do you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?” (lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: http://www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013).)

As of April 25, 11 people responded to this question. Each participant answered “NO!”

Which of the potential problems with samples discussed in this module could explain this connection?

<!– <solution Self-Selected Samples: Only people who are interested in the topic are choosing to respond. Sample Size Issues: A sample with only 11 participants will not accurately represent the opinions of a nation. Undue Influence: The question is wording in a specific way to generate a specific response. Self-Funded or Self-Interest Studies: This question was generated to support one person’s claim and it was designed to get the answer that the person desires. –>

A scholarly article about response rates begins with the following quote:

“Declining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about the validity of estimates drawn from such research.”(Scott Keeter et al., “Gauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,” Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (accessed May 1, 2013).)

The Pew Research Center for People and the Press admits:

“The percentage of people we interview – out of all we try to interview – has been declining over the past decade or more.” (Frequently Asked Questions, Pew Research Center for the People & the Press, http://www.people-press.org/methodology/frequently-asked-questions/#dont-you-have-trouble-getting-people-to-answer-your-polls (accessed May 1, 2013).)

1. What are some reasons for the decline in response rate over the past decade?
2. Explain why researchers are concerned with the impact of the declining response rate on public opinion polls.
1. Possible reasons: increased use of caller id, decreased use of landlines, increased use of private numbers, voice mail, privacy managers, hectic nature of personal schedules, decreased willingness to be interviewed
2. When a large number of people refuse to participate, then the sample may not have the same characteristics of the population. Perhaps the majority of people willing to participate are doing so because they feel strongly about the subject of the survey.

### Bringing It Together

Seven hundred and seventy-one distance learning students at Long Beach City College responded to surveys in the 2010-11 academic year. Highlights of the summary report are listed in (Figure).

1. What percent of the students surveyed do not have a computer at home?
2. About how many students in the survey live at least 16 miles from campus?
3. If the same survey were done at Great Basin College in Elko, Nevada, do you think the percentages would be the same? Why?

<!– <solution 4% 13% Not necessarily. Long beach City is the seventh largest in California the college has an enrollment of approximately 27,000 students. On the other hand, Great Basin College has its campuses in rural northeastern Nevada, and its enrollment of about 3,500 students. –>

Several online textbook retailers advertise that they have lower prices than on-campus bookstores. However, an important factor is whether the Internet retailers actually have the textbooks that students need in stock. Students need to be able to get textbooks promptly at the beginning of the college term. If the book is not available, then a student would not be able to get the textbook at all, or might get a delayed delivery if the book is back ordered.

A college newspaper reporter is investigating textbook availability at online retailers. He decides to investigate one textbook for each of the following seven subjects: calculus, biology, chemistry, physics, statistics, geology, and general engineering. He consults textbook industry sales data and selects the most popular nationally used textbook in each of these subjects. He visits websites for a random sample of major online textbook sellers and looks up each of these seven textbooks to see if they are available in stock for quick delivery through these retailers. Based on his investigation, he writes an article in which he draws conclusions about the overall availability of all college textbooks through online textbook retailers.

Write an analysis of his study that addresses the following issues: Is his sample representative of the population of all college textbooks? Explain why or why not. Describe some possible sources of bias in this study, and how it might affect the results of the study. Give some suggestions about what could be done to improve the study.

Answers will vary. Sample answer: The sample is not representative of the population of all college textbooks. Two reasons why it is not representative are that he only sampled seven subjects and he only investigated one textbook in each subject. There are several possible sources of bias in the study. The seven subjects that he investigated are all in mathematics and the sciences there are many subjects in the humanities, social sciences, and other subject areas, (for example: literature, art, history, psychology, sociology, business) that he did not investigate at all. It may be that different subject areas exhibit different patterns of textbook availability, but his sample would not detect such results.

He also looked only at the most popular textbook in each of the subjects he investigated. The availability of the most popular textbooks may differ from the availability of other textbooks in one of two ways:

• the most popular textbooks may be more readily available online, because more new copies are printed, and more students nationwide are selling back their used copies OR
• the most popular textbooks may be harder to find available online, because more student demand exhausts the supply more quickly.

In reality, many college students do not use the most popular textbook in their subject, and this study gives no useful information about the situation for those less popular textbooks.

## The Range

The first measure of variability that we discuss is the simplest.

### Definition

The range The variability of a data set as measured by the number R = x max − x min . of a data set is the number R defined by the formula

where x max is the largest measurement in the data set and x min is the smallest.

### Example 10

Find the range of each data set in Table 2.1 "Two Data Sets".

For Data Set I the maximum is 43 and the minimum is 38, so the range is R = 43 − 38 = 5 .

For Data Set II the maximum is 47 and the minimum is 33, so the range is R = 47 − 33 = 14 .

The range is a measure of variability because it indicates the size of the interval over which the data points are distributed. A smaller range indicates less variability (less dispersion) among the data, whereas a larger range indicates the opposite.

## Why does variance matter?

Variance matters for two main reasons:

• Parametric statistical tests are sensitive to variance.
• Comparing the variance of samples helps you assess group differences.

### Homogeneity of variance in statistical tests

Variance is important to consider before performing parametric tests. These tests require equal or similar variances, also called homogeneity of variance or homoscedasticity, when comparing different samples.

Uneven variances between samples result in biased and skewed test results. If you have uneven variances across samples, non-parametric tests are more appropriate.

### Using variance to assess group differences

Statistical tests like variance tests or the analysis of variance (ANOVA) use sample variance to assess group differences. They use the variances of the samples to assess whether the populations they come from differ from each other.

Research example As an education researcher, you want to test the hypothesis that different frequencies of quizzes lead to different final scores of college students. You collect the final scores from three groups with 20 students each that had quizzes frequently, infrequently, or rarely over a semester.

• Sample A: Once a week
• Sample B: Once every 3 weeks
• Sample C: Once every 6 weeks

To assess group differences, you perform an ANOVA.

The main idea behind an ANOVA is to compare the variances between groups and variances within groups to see whether the results are best explained by the group differences or by individual differences.

If there’s higher between-group variance relative to within-group variance, then the groups are likely to be different as a result of your treatment. If not, then the results may come from individual differences of sample members instead.

Research example Your ANOVA assesses whether the differences in mean final scores between groups come from the differences in the frequency of quizzes or the individual differences of the students in each group.

To do so, you get a ratio of the between-group variance of final scores and the within-group variance of final scores – this is the F-statistic. With a large F-statistic, you find the corresponding p-value, and conclude that the groups are significantly different from each other.

## Purposeful Sampling in Implementation Research

### Characteristics of Implementation Research

In implementation research, quantitative and qualitative methods often play important roles, either simultaneously or sequentially, for the purpose of answering the same question through convergence of results from different sources, answering related questions in a complementary fashion, using one set of methods to expand or explain the results obtained from use of the other set of methods, using one set of methods to develop questionnaires or conceptual models that inform the use of the other set, and using one set of methods to identify the sample for analysis using the other set of methods (Palinkas et al., 2011). A review of mixed method designs in implementation research conducted by Palinkas and colleagues (2011) revealed seven different sequential and simultaneous structural arrangements, five different functions of mixed methods, and three different ways of linking quantitative and qualitative data together. However, this review did not consider the sampling strategies involved in the types of quantitative and qualitative methods common to implementation research, nor did it consider the consequences of the sampling strategy selected for one method or set of methods on the choice of sampling strategy for the other method or set of methods. For instance, one of the most significant challenges to sampling in sequential mixed method designs lies in the limitations the initial method may place on sampling for the subsequent method. As Morse and Neihaus (2009) observe, when the initial method is qualitative, the sample selected may be too small and lack randomization necessary to fulfill the assumptions for a subsequent quantitative analysis. On the other hand, when the initial method is quantitative, the sample selected may be too large for each individual to be included in qualitative inquiry and lack purposeful selection to reduce the sample size to one more appropriate for qualitative research. The fact that potential participants were recruited and selected at random does not necessarily make them information rich.

Table 2 below provides a description of the use of different purposeful sampling strategies in mixed methods implementation studies. Criterion-i sampling was most frequently used in mixed methods implementation studies that employed a simultaneous design where the qualitative method was secondary to the quantitative method or studies that employed a simultaneous structure where the qualitative and quantitative methods were assigned equal priority. These mixed method designs were used to complement the depth of understanding afforded by the qualitative methods with the breadth of understanding afforded by the quantitative methods (n = 13), to explain or elaborate upon the findings of one set of methods (usually quantitative) with the findings from the other set of methods (n = 10), or to seek convergence through triangulation of results or quantifying qualitative data (n = 8). The process of mixing methods in the large majority (n = 18) of these studies involved embedding the qualitative study within the larger quantitative study. In one study (Goia & Dziadosz, 2008), criterion sampling was used in a simultaneous design where quantitative and qualitative data were merged together in a complementary fashion, and in two studies (Aarons et al., 2012 Zazelli et al., 2008), quantitative and qualitative data were connected together, one in sequential design for the purpose of developing a conceptual model (Zazelli et al., 2008), and one in a simultaneous design for the purpose of complementing one another (Aarons et al., 2012). Three of the six studies that used maximum variation sampling used a simultaneous structure with quantitative methods taking priority over qualitative methods and a process of embedding the qualitative methods in a larger quantitative study (Henke et al., 2008 Palinkas et al., 2010 Slade et al., 2008). Two of the six studies used maximum variation sampling in a sequential design (Aarons et al., 2009 Zazelli et al., 2008) and one in a simultaneous design (Henke et al., 2010) for the purpose of development, and three used it in a simultaneous design for complementarity (Bachman et al., 2009 Henke et al., 2008 Palinkas, Ell, Hansen, Cabassa, & Wells, 2011). The two studies relying upon intensity sampling used a simultaneous structure for the purpose of either convergence or expansion, and both studies involved a qualitative study embedded in a larger quantitative study (Aarons & Palinkas, 2007 Kramer & Burns, 2008). The single typical case study involved a simultaneous design where the qualitative study was embedded in a larger quantitative study for the purpose of complementarity (Hoagwood et al., 2007). The snowball/maximum variation study involved a sequential design where the qualitative study was merged into the quantitative data for the purpose of convergence and conceptual model development (Green & Aarons, 2011). Although not used in any of the 28 implementation studies examined here, another common sequential sampling strategy is using criteria sampling of the larger quantitative sample to produce a second-stage qualitative sample in a manner similar to maximum variation sampling, except that the former narrows the range of variation while the latter expands the range.

### Table 2

Purposeful sampling strategies and mixed method designs in implementation research

Sampling strategyStructureDesignFunction
Single stage sampling (n = 22)
Criterion
(n = 18)
Simultaneous (n = 17)
Sequential (n = 6)
Merged (n = 9)
Connected (n = 9)
Embedded (n = 14)
Convergence (n = 6)
Complementarity (n = 12)
Expansion (n = 10)
Development (n = 3)
Sampling (n = 4)
Maximum variation
(n = 4)
Simultaneous (n = 3)
Sequential (n = 1)
Merged (n = 1)
Connected (n = 1)
Embedded (n = 2)
Convergence (n = 1)
Complementarity (n = 2)
Expansion (n = 1)
Development (n = 2)
Intensity
(n = 1)
Simultaneous
Sequential
Merged
Connected
Embedded
Convergence
Complementarity
Expansion
Development
Typical case Study
(n = 1)
SimultaneousEmbeddedComplementarity
Multistage sampling (n = 4)
Criterion/maximum
variation
(n = 2)
Simultaneous
Sequential
Embedded
Connected
Complementarity
Development
Criterion/intensity
(n = 1)
SimultaneousEmbeddedConvergence
Complementarity
Expansion
Criterion/snowball
(n = 1)
SequentialConnectedConvergence
Development

Criterion-i sampling as a purposeful sampling strategy shares many characteristics with random probability sampling, despite having different aims and different procedures for identifying and selecting potential participants. In both instances, study participants are drawn from agencies, organizations or systems involved in the implementation process. Individuals are selected based on the assumption that they possess knowledge and experience with the phenomenon of interest (i.e., the implementation of an EBP) and thus will be able to provide information that is both detailed (depth) and generalizable (breadth). Participants for a qualitative study, usually service providers, consumers, agency directors, or state policy-makers, are drawn from the larger sample of participants in the quantitative study. They are selected from the larger sample because they meet the same criteria, in this case, playing a specific role in the organization and/or implementation process. To some extent, they are assumed to be “representative” of that role, although implementation studies rarely explain the rationale for selecting only some and not all of the available role representatives (i.e., recruiting 15 providers from an agency for semi-structured interviews out of an available sample of 25 providers). From the perspective of qualitative methodology, participants who meet or exceed a specific criterion or criteria possess intimate (or, at the very least, greater) knowledge of the phenomenon of interest by virtue of their experience, making them information-rich cases.

However, criterion sampling may not be the most appropriate strategy for implementation research because by attempting to capture both breadth and depth of understanding, it may actually be inadequate to the task of accomplishing either. Although qualitative methods are often contrasted with quantitative methods on the basis of depth versus breadth, they actually require elements of both in order to provide a comprehensive understanding of the phenomenon of interest. Ideally, the goal of achieving theoretical saturation by providing as much detail as possible involves selection of individuals or cases that can ensure all aspects of that phenomenon are included in the examination and that any one aspect is thoroughly examined. This goal, therefore, requires an approach that sequentially or simultaneously expands and narrows the field of view, respectively. By selecting only individuals who meet a specific criterion defined on the basis of their role in the implementation process or who have a specific experience (e.g., engaged only in an implementation defined as successful or only in one defined as unsuccessful), one may fail to capture the experiences or activities of other groups playing other roles in the process. For instance, a focus only on practitioners may fail to capture the insights, experiences, and activities of consumers, family members, agency directors, administrative staff, or state policy leaders in the implementation process, thus limiting the breadth of understanding of that process. On the other hand, selecting participants on the basis of whether they were a practitioner, consumer, director, staff, or any of the above, may fail to identify those with the greatest experience or most knowledgeable or most able to communicate what they know and/or have experienced, thus limiting the depth of understanding of the implementation process.

To address the potential limitations of criterion sampling, other purposeful sampling strategies should be considered and possibly adopted in implementation research ( Figure 1 ). For instance, strategies placing greater emphasis on breadth and variation such as maximum variation, extreme case, confirming and disconfirming case sampling are better suited for an examination of differences, while strategies placing greater emphasis on depth and similarity such as homogeneous, snowball, and typical case sampling are better suited for an examination of commonalities or similarities, even though both types of sampling strategies include a focus on both differences and similarities. Alternatives to criterion sampling may be more appropriate to the specific functions of mixed methods, however. For instance, using qualitative methods for the purpose of complementarity may require that a sampling strategy emphasize similarity if it is to achieve depth of understanding or explore and develop hypotheses that complement a quantitative probability sampling strategy achieving breadth of understanding and testing hypotheses (Kemper et al., 2003). Similarly, mixed methods that address related questions for the purpose of expanding or explaining results or developing new measures or conceptual models may require a purposeful sampling strategy aiming for similarity that complements probability sampling aiming for variation or dispersion. A narrowly focused purposeful sampling strategy for qualitative analysis that 𠇌omplements” a broader focused probability sample for quantitative analysis may help to achieve a balance between increasing inference quality/trustworthiness (internal validity) and generalizability/transferability (external validity). A single method that focuses only on a broad view may decrease internal validity at the expense of external validity (Kemper et al., 2003). On the other hand, the aim of convergence (answering the same question with either method) may suggest use of a purposeful sampling strategy that aims for breadth that parallels the quantitative probability sampling strategy.

Purposeful and Random Sampling Strategies for Mixed Method Implementation Studies

Priority and sequencing of Qualitative (QUAL) and Quantitative (QUAN) can be reversed.

Refers to emphasis of sampling strategy.

Refers to sequential structure refers to simultaneous structure.

Furthermore, the specific nature of implementation research suggests that a multistage purposeful sampling strategy be used. Three different multistage sampling strategies are illustrated in Figure 1 below. Several qualitative methodologists recommend sampling for variation (breadth) before sampling for commonalities (depth) (Glaser, 1978 Bernard, 2002) (Multistage I). Also known as a 𠇏unnel approach”, this strategy is often recommended when conducting semi-structured interviews (Spradley, 1979) or focus groups (Morgan, 1997). This approach begins with a broad view of the topic and then proceeds to narrow down the conversation to very specific components of the topic. However, as noted earlier, the lack of a clear understanding of the nature of the range may require an iterative approach where each stage of data analysis helps to determine subsequent means of data collection and analysis (Denzen, 1978 Patton, 2001) (Multistage II). Similarly, multistage purposeful sampling designs like opportunistic or emergent sampling, allow the option of adding to a sample to take advantage of unforeseen opportunities after data collection has been initiated (Patton, 2001, p. 240) (Multistage III). Multistage I models generally involve two stages, while a Multistage II model requires a minimum of 3 stages, alternating from sampling for variation to sampling for similarity. A Multistage III model begins with sampling for variation and ends with sampling for similarity, but may involve one or more intervening stages of sampling for variation or similarity as the need or opportunity arises.

Multistage purposeful sampling is also consistent with the use of hybrid designs to simultaneously examine intervention effectiveness and implementation. An extension of the concept of “practical clinical trials” (Tunis, Stryer & Clancey, 2003), effectiveness-implementation hybrid designs provide benefits such as more rapid translational gains in clinical intervention uptake, more effective implementation strategies, and more useful information for researchers and decision makers (Curran et al., 2012). Such designs may give equal priority to the testing of clinical treatments and implementation strategies (Hybrid Type 2) or give priority to the testing of treatment effectiveness (Hybrid Type 1) or implementation strategy (Hybrid Type 3). Curran and colleagues (2012) suggest that evaluation of the intervention’s effectiveness will require or involve use of quantitative measures while evaluation of the implementation process will require or involve use of mixed methods. When conducting a Hybrid Type 1 design (conducting a process evaluation of implementation in the context of a clinical effectiveness trial), the qualitative data could be used to inform the findings of the effectiveness trial. Thus, an effectiveness trial that finds substantial variation might purposefully select participants using a broader strategy like sampling for disconfirming cases to account for the variation. For instance, group randomized trials require knowledge of the contexts and circumstances similar and different across sites to account for inevitable site differences in interventions and assist local implementations of an intervention (Bloom & Michalopoulos, 2013 Raudenbush & Liu, 2000). Alternatively, a narrow strategy may be used to account for the lack of variation. In either instance, the choice of a purposeful sampling strategy is determined by the outcomes of the quantitative analysis that is based on a probability sampling strategy. In Hybrid Type 2 and Type 3 designs where the implementation process is given equal or greater priority than the effectiveness trial, the purposeful sampling strategy must be first and foremost consistent with the aims of the implementation study, which may be to understand variation, central tendencies, or both. In all three instances, the sampling strategy employed for the implementation study may vary based on the priority assigned to that study relative to the effectiveness trial. For instance, purposeful sampling for a Hybrid Type 1 design may give higher priority to variation and comparison to understand the parameters of implementation processes or context as a contribution to an understanding of effectiveness outcomes (i.e., using qualitative data to expand upon or explain the results of the effectiveness trial), In effect, these process measures could be seen as modifiers of innovation/EBP outcome. In contrast, purposeful sampling for a Hybrid Type 3 design may give higher priority to similarity and depth to understand the core features of successful outcomes only.

Finally, multistage sampling strategies may be more consistent with innovations in experimental designs representing alternatives to the classic randomized controlled trial in community-based settings that have greater feasibility, acceptability, and external validity. While RCT designs provide the highest level of evidence, “in many clinical and community settings, and especially in studies with underserved populations and low resource settings, randomization may not be feasible or acceptable” (Glasgow, et al., 2005, p. 554). Randomized trials are also “relatively poor in assessing the benefit from complex public health or medical interventions that account for individual preferences for or against certain interventions, differential adherence or attrition, or varying dosage or tailoring of an intervention to individual needs” (Brown et al., 2009, p. 2). Several alternatives to the randomized design have been proposed, such as “interrupted time series,” “multiple baseline across settings” or “regression-discontinuity” designs. Optimal designs represent one such alternative to the classic RCT and are addressed in detail by Duan and colleagues (this issue). Like purposeful sampling, optimal designs are intended to capture information-rich cases, usually identified as individuals most likely to benefit from the experimental intervention. The goal here is not to identify the typical or average patient, but patients who represent one end of the variation in an extreme case, intensity sampling, or criterion sampling strategy. Hence, a sampling strategy that begins by sampling for variation at the first stage and then sampling for homogeneity within a specific parameter of that variation (i.e., one end or the other of the distribution) at the second stage would seem the best approach for identifying an “optimal” sample for the clinical trial.

Another alternative to the classic RCT are the adaptive designs proposed by Brown and colleagues (Brown et al, 2006 Brown et al., 2008 Brown et al., 2009). Adaptive designs are a sequence of trials that draw on the results of existing studies to determine the next stage of evaluation research. They use cumulative knowledge of current treatment successes or failures to change qualities of the ongoing trial. An adaptive intervention modifies what an individual subject (or community for a group-based trial) receives in response to his or her preferences or initial responses to an intervention. Consistent with multistage sampling in qualitative research, the design is somewhat iterative in nature in the sense that information gained from analysis of data collected at the first stage influences the nature of the data collected, and the way they are collected, at subsequent stages (Denzen, 1978). Furthermore, many of these adaptive designs may benefit from a multistage purposeful sampling strategy at early phases of the clinical trial to identify the range of variation and core characteristics of study participants. This information can then be used for the purposes of identifying optimal dose of treatment, limiting sample size, randomizing participants into different enrollment procedures, determining who should be eligible for random assignment (as in the optimal design) to maximize treatment adherence and minimize dropout, or identifying incentives and motives that may be used to encourage participation in the trial itself.

Alternatives to the classic RCT design may also be desirable in studies that adopt a community-based participatory research framework (Minkler & Wallerstein, 2003), considered to be an important tool on conducting implementation research (Palinkas & Soydan, 2012). Such frameworks suggest that identification and recruitment of potential study participants will place greater emphasis on the priorities and “local knowledge” of community partners than on the need to sample for variation or uniformity. In this instance, the first stage of sampling may approximate the strategy of sampling politically important cases (Patton, 2002) at the first stage, followed by other sampling strategies intended to maximize variations in stakeholder opinions or experience.

## Definition of 'Random Sampling'

Definition: Random sampling is a part of the sampling technique in which each sample has an equal probability of being chosen. A sample chosen randomly is meant to be an unbiased representation of the total population. If for some reasons, the sample does not represent the population, the variation is called a sampling error.

Description: Random sampling is one of the simplest forms of collecting data from the total population. Under random sampling, each member of the subset carries an equal opportunity of being chosen as a part of the sampling process. For example, the total workforce in organisations is 300 and to conduct a survey, a sample group of 30 employees is selected to do the survey. In this case, the population is the total number of employees in the company and the sample group of 30 employees is the sample. Each member of the workforce has an equal opportunity of being chosen because all the employees which were chosen to be part of the survey were selected randomly. But, there is always a possibility that the group or the sample does not represent the population as a whole, in that case, any random variation is termed as a sampling error.

An unbiased random sample is important for drawing conclusions. For example when we took out the sample of 30 employees from the total population of 300 employees, there is always a possibility that a researcher might end up picking over 25 men even if the population consists of 200 men and 100 women. Hence, some variations when drawing results can come up, which is known as a sampling error. One of the disadvantages of random sampling is the fact that it requires a complete list of population. For example, if a company wants to carry out a survey and intends to deploy random sampling, in that case, there should be total number of employees and there is a possibility that all the employees are spread across different regions which make the process of survey little difficult.

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