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11.4: Numerical Summaries of Data - Mathematics

11.4: Numerical Summaries of Data - Mathematics



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It is often desirable to use a few numbers to summarize a distribution. One important aspect of a distribution is where its center is located. Measures of central tendency are discussed first. A second aspect of a distribution is how spread out it is. In other words, how much the data in the distribution vary from one another. The second section describes measures of variability.


Importance of Numerical Data Analysis

Numerical data is of paramount importance in the world of mathematics. At times, it can be difficult to identify numerical data. In this article, we are going to take a look at the importance of numerical data analysis. This will help you improve your understanding of this type of approach to data. Read on to find out more.

As far as mathematics is concerned, data refers to the collected information. In most cases, this information is used to discuss a hypothesis or make a scientific guess in an experiment. For example, this information may be related to the number of movies, the number of topics, hair color, and so on. Typically, it is possible to categorize data into different groups based on many factors. Let's go into details.

What Is Numerical Data?

In simple words, numerical data refers to information that can be measured. In most cases, it is given in the form of numbers. However, different types of data can be found in the form of numbers. For example, it may refer to the number of people who watch a movie in a theatre over a week or month.

There are many ways to identify this type of information. For example, you can find out if the data can be added to the database you already have. The beauty of data is that you can perform different types of mathematical operations on it. Another sign of numerical data is that it is possible to represent answers in decimal or fraction form. Similarly, if the information can be categorized, it will be called categorical info.

For example, if you have the measurement of 6 ladders, you can get an average height or you can just give the height information in descending or ascending order. The reason is that this type of data is numerical.

Importance of Numerical Data in Research?

Researchers give a lot of importance to this type of information. The reason is that it is compatible with the majority of statistical techniques. Apart from this, it can help make the research process a lot easier. During product development, researchers make use of TURF analysis in order to find out if a product or service satisfies the target market.

What is the Numerical Data Analysis?

This type of environment analysis involves the use of mathematical calculations. The idea is to get approximate solutions and reduce the chances of errors. Apart from this, numerical data is used in a lot of fields including physical science and engineering.

How is Analysis Useful?

In many businesses, analysis of data is of paramount importance to get a better understanding of organizational problems. Apart from days, it helps explore data in several ways. In its basic form, data refers to only facts and figures. Once it is analyzed, it provides useful information for different applications.

Is it an Ongoing Process?

This type of analysis is an ongoing interactive process. The collection and analysis of data is done almost simultaneously.

In short, hopefully, this article will help you understand the importance of numerical data analysis.

If you want know more about numbers information and information about numbers, you can check out Numbers Data.


Where might you face data interpretation questions?

You might expect such questions in numerical reasoning tests when applying for graduate or management level roles, especially if you are applying to jobs in the corporate, financial or consulting sectors.

That said, date interpretation is a widely used and required skill across pretty much all industries, so it is best to be prepared.

You may also expect to face a data interpretation test if the role you apply for includes any data analysis or strategic decision making, or data handling. This could include marketing, clerical and administrative roles, as well as managerial positions where you will have decision-making responsibilities.


History of statistics

In the 9th century, the Islamic mathematician, Al-Kindi, was the first to use statistics to decipher encrypted messages and developed the first code-breaking algorithm in the House of Wisdom in Baghdad, based on frequency analysis. He wrote a book entitled Manuscript on Deciphering Cryptographic Messages, containing detailed discussions on statistics. Ώ] It covers methods of cryptanalysis, encipherments, cryptanalysis of certain encipherments, and statistical analysis of letters and letter combinations in Arabic. ΐ]

In the early 11th century, Al-Biruni's scientific method emphasized repeated experimentation. Biruni was concerned with how to conceptualize and prevent both systematic errors and observational biases, such as "errors caused by the use of small instruments and errors made by human observers." He argued that if instruments produce errors because of their imperfections or idiosyncratic qualities, then multiple observations must be taken, analyzed qualitatively, and on this basis, arrive at a "common-sense single value for the constant sought", whether an arithmetic mean or a "reliable estimate." Α]

Modern history

The Word statistics have been derived from Latin word “Status” or the Italian word “Statista”, meaning of these words is “Political State” or a Government. Shakespeare used a word Statist is his drama Hamlet (1602). In the past, the statistics was used by rulers. The application of statistics was very limited but rulers and kings needed information about lands, agriculture, commerce, population of their states to assess their military potential, their wealth, taxation and other aspects of government.

Gottfried Achenwall used the word statistik at a German University in 1749 which means that political science of different countries. In 1771 W. Hooper (Englishman) used the word statistics in his translation of Elements of Universal Erudition written by Baron B.F Bieford, in his book statistics has been defined as the science that teaches us what is the political arrangement of all the modern states of the known world. There is a big gap between the old statistics and the modern statistics, but old statistics also used as a part of the present statistics.

During the 18th century the English writer have used the word statistics in their works, so statistics has developed gradually during last few centuries. A lot of work has been done in the end of the nineteenth century.

At the beginning of the 20th century, William S Gosset was developed the methods for decision making based on small set of data. During the 20th century several statistician are active in developing new methods, theories and application of statistics. Now these days the availability of electronics computers is certainly a major factor in the modern development of statistics.


Dr. Yumin Cheng is Professor of the Shanghai Institute of Applied Mathematics and Mechanics of Shanghai University. He received his bachelor&rsquos degree of mathematics from Shanxi University of China, and PhD degree of computational mechanics from Xi&rsquoan Jiaotong University of China. His research interests include the meshless method, boundary element method, and scientific and engineering computing. He has published more than 180 journal papers with 4600 citations. His h-index in scopus.com is 44. He has been honored with Fellow of The International Association of Applied Mechanics (IAAM), Fellow of International Association of Advanced materials and VEBLEO Fellow awards. He is an Executive Committee member of IAAM and Chairman of Committee of IAAM Standards and Codes. He has been Lead Guest Editor of the Special Issue of Mathematical Problems in Engineering (SI: Mathematical Aspects of Meshless Methods SI: New Trends in Numerical Simulation and Data Analysis), and he is Associate Editor of the International Journal of Computers, Editor and Member of the Editorial Board of the International Journal of Applied Mechanics, and Member of the Editorial Board of the International Journal of Computational Materials Science and Engineering, Journal of Computational Engineering, International Journal of Applied & Experimental Mathematics, and International Journal of Mathematical Physics and Video Proceedings of Advanced Materials.

Based on numerical methods such as the finite element method, boundary element method, and meshless method, numerical simulations for various problems in science, engineering, and society fields have developed rapidly in the recent decades. Various numerical methods are presented for solving problems in different fields, and the corresponding computational efficiency, accuracy, and convergence are studied as well. With the development of big data, numerical simulation combined data analysis will play a more important role in studying problems in the science, engineering, and society fields.

In this Special Issue, we particularly take an interest in manuscripts that report the relevance of numerical computation and data analysis for mathematical and engineering problems. The Special Issue will become an international forum for researchers to summarize the most recent developments of numerical simulations and data analysis within the last five years, especially for new problems. Moreover, manuscripts on the mathematical theories of numerical computation and data analysis for complicated science, engineering or social problems are welcome. We also concern the development of the corresponding aspects based on big data, including the corresponding theory, numerical method, and applications.

Software is an important part of numerical computation and data analysis in mathematics and engineering. This Special Issue also concerns the developments of the software of numerical methods, including the finite element method, boundary element method, and meshless method, and the ones for data analysis.

Prof. Dr. Yumin Cheng
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Solution

Using the median to describe typical speed, we would say that typical speed is about the same (median of 65 mph for northbound and 63.5 mph for southbound) for northbound cars and southbound cars. One noticeable difference between the two speed distributions is that the southbound speeds are more variable than the northbound speeds. This means that the northbound speeds tended to be more consistent than the southbound speeds, which tended to differ more from one car to another. Other than the outlier in the northbound speeds, both speed distributions appear to be approximately symmetric.


Connections to data science

Although the quantity and complexity of data available to researchers continues to increase in many application domains, there are many important scenarios in science and engineering where there is a lack of data, leading to uncertainty. Mathematical and statistical tools that make the best use of limited data to make predictions, and that can inform us how best to gather more data (if possible) in order to gain improved estimates of quantities of interest, are essential.

UQ intersects with data science in many ways. A natural example is in the numerical solution of Bayesian inverse problems, where there is a need to develop statistical sampling methods to efficiently estimate posterior distributions of uncertain model inputs. As more and more data becomes available, developing hybrid approaches to modelling that combine classical mechanistic models with new data-driven and machine learning techniques is also an important challenge.


Graphs and Numerical Summaries

Videos and solutions to help grade 6 students learn how to match the graphical representations and numerical summaries of a distribution, Matches involve dot plots, histograms, and summary statistics.

New York State Common Core Math Grade 6, Module 6, Lesson 18

Lesson 18 Student Outcomes

&bullStudents match the graphical representations and numerical summaries of a distribution. Matches involve dot plots, histograms, and summary statistics.

Generally, we can compute or approximate many values in a numerical summary for a data set by looking at a histogram or a dot plot for the data set. Thus, we can generally match a histogram or a dot plot to summary measures provided.
When making a histogram and a dot plot for the same data set, the two graphs will have similarities. However, some information may be more easily communicated by one graph as opposed to the other.

1. The following histogram shows the amount of coal produced (by state) for the 20 largest coal-producing states in 2011. Many of these states produced less than 50 million tons of coal, but one state produced over 400 million tons (Wyoming). For the histogram, which one of the three sets of summary measures could match the graph? For each choice that you eliminate, give at least one reason for eliminating the choice.
a. Minimum = 1, Q1 = 12, Median = 36, Q3 = 57, Maximum = 410 Mean = 33, MAD = 2.76
b. Minimum = 2, Q1 = 13.5, Median = 27.5, Q3 = 44, Maximum = 439 Mean = 54.6, MAD = 52.36
c. Minimum = 10, Q1 = 37.5, Median = 62, Q3 = 105, Maximum = 439 Mean = 54.6, MAD = 52.36

2. The heights (rounded to the nearest inch) of the 41 members of the 2012–2013 University of Texas Men’s Swimming and Diving Team are shown in the dot plot below.
a. Use the dot plot to determine the 5-number summary (minimum, lower quartile, median, upper quartile, and maximum) for the data set.
b. Based on this dot plot, make a histogram of the heights using the following intervals: 66 to < 68 inches, 68 to < 70 inches, and so on.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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Practice: Numerical Summaries of Data Score: 0/10 0/10 answered Question 9 < > Find the 5 number summary for the data shown X 7

Practice: Numerical Summaries of Data Score: 0/10 0/10 answered Question 9 < > Find the 5 number summary for the data shown X 7 7.7 9.2 15.2 15.3 16.4 18.1 20.9 21.3 26.2 5 number summary: Use the Locator/Percentile method described in your book, not your calculator. Question Help: D Video 1 D Video 2 Message instructor Calculator

2. x = 1 3. x = 2 7. x=7 8. x = -1 9. x = -3 4. x = 3 10. x = -5 5. x = 4 6. x = 6 f(6169 11. x = 10 12. x = -10 Directions Evaluate the function f(x) = () * for the values of x given. 13. x = 0 19. x = -2 14. x = 1 20. x = -3 15. x = 2 21. x = -4 16. x = 3 22. x = -5 17. x = 4 23. x = 10 18. x = -1 24. x = -10 Directions Draw the graph of the points in problems 1–6. Connect the points with a smooth curve. 25. y -- 60 --50 --40 -30 --20 ---10 X Yo 1 3 4 5 Algebra 2 Algebra 2 CAGS Publishing. Permission is granted to reproduce for classroom use only.

Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 3 of 20 Savo Submit Let E represent the set of all of the prime numbers that are less than 20. Let Trepresent the set of the factors of six. If x represents the amount of elements that are in Eur , find the value of -3x+11 0-22 0-31 0-19 A 0-13 O 28

[(4, -3), (-1,4), (2,0), (0,-3), (-1,-5), (-2,5)> Using the relation above, choose the explanation that describes why the relation is or is not a function. th A) The relation is a function because each output has only one input B) The relation is a function because each input has only one output. C) The relation is not a function because each output has more than one input. D) The relation is not a function because each input has more than one output.


11.4: Numerical Summaries of Data - Mathematics

Chapter 1: MATHEMATICAL MODELING 1
1.1 Modeling in Computer Animation 2
1.1.1 A Model Robe 2
1.2 Modeling in Physics: Radiation Transport 4
1.3 Modeling in Sports 6
1.4 Ecological Models 8
1.5 Modeling a Web Surfer and Google 11
1.5.1 The Vector Space Model 11
1.5.2 Google's PageRank 13
1.6 Chapter 1 Exercises 14

Chapter 2: BASIC OPERATIONS WITH MATLAB 19
2.1 Launching MATLAB 19
2.2 Vectors 20
2.3 Getting Help 22
2.4 Matrices 23
2.5 Creating and Running .m Files 24
2.6 Comments 25
2.7 Plotting 25
2.8 Creating Your Own Functions 27
2.9 Printing 28
2.10 More Loops and Conditionals 29
2.11 Clearing Variables 31
2.12 Logging Your Session 31
2.13 More Advanced Commands 31
2.14 Chapter 2 Exercises 32

Chapter 3: MONTE CARLO METHODS 41
3.1 A Mathematical Game of Cards 41
3.1.1 The Odds in Texas Holdem 42
3.2 Basic Statistics 46
3.2.1 Discrete Random Variables 48
3.2.2 Continuous Random Variables 51
3.2.3 The Central Limit Theorem 53
3.3 Monte Carlo Integration 56
3.3.1 Buffon's Needle 56
3.3.2 Estimating p 58
3.3.3 Another Example of Monte Carlo Integration 60
3.4 Monte Carlo Simulation of Web Surfing 64
3.5 Chapter 3 Exercises 67

Chapter 4: SOLUTION OF A SINGLE NONLINEAR EQUATION IN ONE UNKNOWN 71
4.1 Bisection 75
4.2 Taylor's Theorem 80
4.3 Newton's Method 83
4.4 Quasi-Newton Methods 89
4.4.1 Avoiding Derivatives 89
4.4.2 Constant Slope Method 89
4.4.3 Secant Method 90
4.5 Analysis of Fixed Point Methods 93
4.6 Fractals, Julia Sets, and Mandelbrot Sets 98
4.7 Chapter 4 Exercises 102

Chapter 5: FLOATING-POINT ARITHMETIC 107
5.1 Costly Disasters Caused by Rounding Errors 108
5.2 Binary Representation and Base 2 Arithmetic 110
5.3 Floating-Point Representation 112
5.4 IEEE Floating-Point Arithmetic 114
5.5 Rounding 116
5.6 Correctly Rounded Floating-Point Operations 118
5.7 Exceptions 119
5.8 Chapter 5 Exercises 120

Chapter 6: CONDITIONING OF PROBLEMS STABILITY OF ALGORITHMS 124
6.1 Conditioning of Problems 125
6.2 Stability of Algorithms 126
6.3 Chapter 6 Exercises 129

Chapter 7: DIRECT METHODS FOR SOLVING LINEAR SYSTEMS AND LEAST SQUARES PROBLEMS 131
7.1 Review of Matrix Multiplication 132
7.2 Gaussian Elimination 133
7.2.1 Operation Counts 137
7.2.2 LU Factorization 139
7.2.3 Pivoting 141
7.2.4 Banded Matrices and Matrices for Which Pivoting Is Not Required 144
7.2.5 Implementation Considerations for High Performance 148
7.3 Other Methods for Solving Ax = b 151
7.4 Conditioning of Linear Systems 154
7.4.1 Norms 154
7.4.2 Sensitivity of Solutions of Linear Systems 158
7.5 Stability of Gaussian Elimination with Partial Pivoting 164
7.6 Least Squares Problems 166
7.6.1 The Normal Equations 167
7.6.2 QR Decomposition 168
7.6.3 Fitting Polynomials to Data 171
7.7 Chapter 7 Exercises 175

Chapter 8: POLYNOMIAL AND PIECEWISE POLYNOMIAL INTERPOLATION 181
8.1 The Vandermonde System 181
8.2 The Lagrange Form of the Interpolation Polynomial 181
8.3 The Newton Form of the Interpolation Polynomial 185
8.3.1 Divided Differences 187
8.4 The Error in Polynomial Interpolation 190
8.5 Interpolation at Chebyshev Points and chebfun 192
8.6 Piecewise Polynomial Interpolation 197
8.6.1 Piecewise Cubic Hermite Interpolation 200
8.6.2 Cubic Spline Interpolation 201
8.7 Some Applications 204
8.8 Chapter 8 Exercises 206

Chapter 9: NUMERICAL DIFFERENTIATION AND RICHARDSON EXTRAPOLATION 212
9.1 Numerical Differentiation 213
9.2 Richardson Extrapolation 221
9.3 Chapter 9 Exercises 225

Chapter 10: NUMERICAL INTEGRATION 227
10.1 Newton-Cotes Formulas 227
10.2 Formulas Based on Piecewise Polynomial Interpolation 232
10.3 Gauss Quadrature 234
10.3.1 Orthogonal Polynomials 236
10.4 Clenshaw-Curtis Quadrature 240
10.5 Romberg Integration 242
10.6 Periodic Functions and the Euler-Maclaurin Formula 243
10.7 Singularities 247
10.8 Chapter 10 Exercises 248

Chapter 11: NUMERICAL SOLUTION OF THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS 251
11.1 Existence and Uniqueness of Solutions 253
11.2 One-Step Methods 257
11.2.1 Euler's Method 257
11.2.2 Higher-Order Methods Based on Taylor Series 262
11.2.3 Midpoint Method 262
11.2.4 Methods Based on Quadrature Formulas 264
11.2.5 Classical Fourth-Order Runge-Kutta and Runge-Kutta-Fehlberg Methods 265
11.2.6 An Example Using MATLAB's ODE Solver 267
11.2.7 Analysis of One-Step Methods 270
11.2.8 Practical Implementation Considerations 272
11.2.9 Systems of Equations 274
11.3 Multistep Methods 275
11.3.1 Adams-Bashforth and Adams-Moulton Methods 275
11.3.2 General Linear m-Step Methods 277
11.3.3 Linear Difference Equations 280
11.3.4 The Dahlquist Equivalence Theorem 283
11.4 Stiff Equations 284
11.4.1 Absolute Stability 285
11.4.2 Backward Differentiation Formulas (BDF Methods) 289
11.4.3 Implicit Runge-Kutta (IRK) Methods 290
11.5 Solving Systems of Nonlinear Equations in Implicit Methods 291
11.5.1 Fixed Point Iteration 292
11.5.2 Newton's Method 293
11.6 Chapter 11 Exercises 295

Chapter 12: MORE NUMERICAL LINEAR ALGEBRA: EIGENVALUES AND ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS 300
12.1 Eigenvalue Problems 300
12.1.1 The Power Method for Computing the Largest Eigenpair 310
12.1.2 Inverse Iteration 313
12.1.3 Rayleigh Quotient Iteration 315
12.1.4 The QR Algorithm 316
12.1.5 Google's PageRank 320
12.2 Iterative Methods for Solving Linear Systems 327
12.2.1 Basic Iterative Methods for Solving Linear Systems 327
12.2.2 Simple Iteration 328
12.2.3 Analysis of Convergence 332
12.2.4 The Conjugate Gradient Algorithm 336
12.2.5 Methods for Nonsymmetric Linear Systems 334
12.3 Chapter 12 Exercises 345

Chapter 13: NUMERICAL SOLUTION OF TWO-POINT BOUNDARY VALUE PROBLEMS 350
13.1 An Application: Steady-State Temperature Distribution 350
13.2 Finite Difference Methods 352
13.2.1 Accuracy 354
13.2.2 More General Equations and Boundary Conditions 360
13.3 Finite Element Methods 365
13.3.1 Accuracy 372
13.4 Spectral Methods 374
13.5 Chapter 13 Exercises 376

Chapter 14: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 379
14.1 Elliptic Equations 381
14.1.1 Finite Difference Methods 381
14.1.2 Finite Element Methods 386
14.2 Parabolic Equations 388
14.2.1 Semidiscretization and the Method of Lines 389
14.2.2 Discretization in Time 389
14.3 Separation of Variables 396
14.3.1 Separation of Variables for Difference Equations 400
14.4 Hyperbolic Equations 402
14.4.1 Characteristics 402
14.4.2 Systems of Hyperbolic Equations 403
14.4.3 Boundary Conditions 404
14.4.4 Finite Difference Methods 404
14.5 Fast Methods for Poisson's Equation 409
14.5.1 The Fast Fourier Transform 411
14.6 Multigrid Methods 414
14.7 Chapter 14 Exercises 418

APPENDIX A REVIEW OF LINEAR ALGEBRA 421
A.1 Vectors and Vector Spaces 421
A.2 Linear Independence and Dependence 422
A.3 Span of a Set of Vectors Bases and Coordinates Dimension of a Vector Space 423
A.4 The Dot Product Orthogonal and Orthonormal Sets the Gram-Schmidt Algorithm 423
A.5 Matrices and Linear Equations 425
A.6 Existence and Uniqueness of Solutions the Inverse Conditions for Invertibility 427
A.7 Linear Transformations the Matrix of a Linear Transformation 431
A.8 Similarity Transformations Eigenvalues and Eigenvectors 432


Watch the video: Statistics - Module 3 Video 1 - Numerical Summaries (August 2022).