National Workshop on LaTeX and MATLAB for Beginners

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1 BITS Pilani Pilani Campus National Workshop on LaTeX and MATLAB for Beginners December, 2014 BITS Pilani, Partially Supported by DST, Rajasthan

2 BITS Pilani Pilani Campus Lecture - 13 : STATISTICAL COMMANDS Dr. Shivi Agarwal

3 Measures of Central Tendency (Location) Measures of Dispersion Probability Distribution Linear Models Covariance Correlation Linear Regression Hypothesis Testing 3

4 Measures of Central Tendency (Location) To locate the data values on the number line. Measures of location. geomean(x) harmmean(x) mean(x) median(x) max(x) min(x) : Geometric mean : Harmonic mean : Arithmetic average : 50 th percentile (in MATLAB) : Maximum value : Minimum value 4

5 Measures of Central Tendency (Location) Example: x = [ones(1,6) 100] x = locate = [geomean(x), harmmean(x), mean(x), median(x)] locate =

6 Measures of Dispersion To find out how spread out the data values are on the number line. Measures of spread. iqr(x) mad(x) range(x) std(x) var(x) Interquartile Range Mean Absolute Deviation Range Standard deviation Variance 6

7 Measures of Dispersion Range: difference between the maximum and minimum values; simplest measure of spread; not robust to outliers. Interquartile Range (IQR): difference between the 75 th and 25 th percentile of the data. Since only the middle 50% of the data affects this measure, it is robust to outliers. 7

8 Measures of Dispersion Example: x = [ones(1,6) 100] x = stats = [iqr(x), mad(x), range(x), std(x)] stats =

9 prctile(x,p) : p th Percentile of a data set X, p [0,100] quantile(x,p) : p th Quantile of a data set X, p [0,1] skewness(x) : skewness of sample X kurtosis(x) : kurtosis of sample X corr(x) : pairwise linear correlation coefficient 9

10 Probability Distribution Random variable 10

11 Probability Distribution Probability distributions arise from experiments where the outcome is subject to chance. The nature of the experiment dictates which probability distributions may be appropriate for modeling the resulting random outcomes. 11

12 Probability Distribution The uncertain behavior of the random variable is predicted by: (i) Probability density function f(x) (ii) Cumulative distribution function F(x) 12

13 Probability Distribution The Statistics Toolbox supports 20 probability distributions. For each distribution there are several associated functions. They are: Probability density function (pdf) Cumulative distribution function (cdf) Random number generator 13

14 Probability Distribution Mean and variance as a function of the parameters Parameter estimates and Confidence intervals for distributions (binomial, Poisson, uniform, gamma, exponential, normal, beta, and Weibull),. 14

15 Probability Distribution Discrete Binomial Distribution Poisson Distribution Discrete Uniform Distribution Continuous Continuous Uniform Distribution Gamma Distribution Exponential Distribution Normal Distribution 15

16 Discrete Probability Distribution The density function of a discrete random variable X is defined by f (x) = P(X=x) for all real x. The cumulative distribution function F of a discrete random variable X, is defined by F(x) P(X x) k x f(k) for any real number x, here f denote the density of X. 16

17 Binomial Distribution A discrete random variable X has binomial distribution with parameters n and p, n is a positive integer and 0 < p < 1, if its density function is f ( x) n x n x p (1 p) ; x 0,1,2,..., n x 0 otherwise. E[ X ] np and Var[ X ] npq. 17

18 18

19 Binomial Distribution A = binopdf(x,n,p) B= binocdf(x,n,p) n: must be positive integers, p: values must lie on the interval [0,1]. 19

20 Binomial Distribution Example: A Quality Assurance inspector tests 200 circuit boards a day. If 2% of the boards have defects, what is the probability that the inspector will find no defective boards on any given day? >>binopdf(0,200,0.02) 20

21 Binomial Distribution [M,V]=binostat(n,p) returns the mean and variance for the binomial distribution with parameters specified by n and p. 21

22 Binomial Distribution Ex. If a baseball team plays 162 games in a season and has a chance of winning any game, then the mean and the variance is : >> [M,V]=binostat(162,0.5) M = 81 V =

23 Binomial Distribution R = binornd(n,p) Description : generates random number from the binomial distribution with parameters specified by n and p. 23

24 Binomial Distribution R = binornd(n,p,m) Description : generates a matrix of size m*m containing random numbers from the binomial distribution with parameters n and p 24

25 Binomial Distribution R = binornd(n,p,m,k) Description : generates an m-by-k matrix containing random numbers from the binomial distribution with parameters n and p. 25

26 Binomial Distribution phat = binofit(x,n) Description: returns a maximum likelihood estimate of the probability of success in a given binomial trial based on the number of successes, x, observed in n independent trials. 26

27 Binomial Distribution [phat,pci] = binofit(x,n) Description: returns the probability estimate, phat, and the 95% confidence intervals, pci. 27

28 Binomial Distribution [phat,pci] = binofit(x,n,alpha) Description: returns the 100(1-alpha)% confidence intervals. For example, alpha = 0.01 yields 99% confidence intervals. 28

29 Binomial Distribution Example: First generate a binomial sample of 100 elements, where the probability of success in a given trial is 0.6. Then, estimate this probability from the outcomes in the sample. >> r = binornd(100,0.6); >> [phat,pci] = binofit(r,100) phat = pci =

30 Poisson Distribution This distribution is named on the French mathematician Simeon Denis Poisson. Let λ > 0 be a constant and, for any real number x, x e ; for x 0,1,2,... f ( x) x! 0 otherwise 30

31 Y = poisspdf(x,l); L > 0 and x 0 and integer. P = poisscdf(x,l) R = poissrnd(l) R = poissrnd(l,m) R = poissrnd(l,m,n) lambdahat = poissfit(x) [lambdahat,lambdaci] = poissfit(x) [lambdahat,lambdaci] = poissfit(x,alpha) [M,V] = poisstat(l) Poisson Distribution 31

32 Poisson Distribution Examples: A computer hard disk manufacturer has observed that flaws occur randomly in the manufacturing process at the average rate of two flaws in a 4 GB hard disk and has found this rate to be acceptable. What is the probability that a disk will be manufactured with no defects? λ= 2 and x = 0. >> p = poisspdf(0,2) p =

33 Discrete Uniform Distribution Y = unidpdf(x,n) P = unidcdf(x,n) N > 0 and integer R = unidrnd(n) R = unidrnd(n,m) R = unidrnd(n,m,k) For discrete uniform pdf is = 1/N; The mean is = (N+1)/2 The variance is = (N 2-1)/12. [M,V] = unidstat(n) 33

34 Discrete Uniform Distribution Examples: For fixed n, the uniform discrete pdf is a constant. >> y = unidpdf(1:6,10) y =

35 Discrete Probability Distribution Geometric Distribution Hypergeometric Distribution Negative Binomial Distribution 35

36 Continuous Density (Probability density function ) Definition: Let X be a continuous random variable. A function f(x) is called continuous density (probability density function i.e. pdf ) iff f(x) integral converges. f ( x ) 0 dx 1 36

37 Cumulative Distribution Function Let X be the continuous r.v. with density f(x). The cumulative distribution function (cdf) for X, denoted by F(X), is defined by F(X) = P ( X x ), all x = x f ( t ) dt 37

38 Continuous uniform distribution A random variable X is said to be uniformly distributed over an interval (a, c) if its density is given by f ( x E Var ( ) X 1 c a, ( ) X ) a a c 2 ( c 12 a x ) 2 c 38

39 Continuous uniform distribution Y = unifpdf(x,a,b); B > A P = unifcdf(x,a,b) R = unifrnd(a,b) R = unifrnd(a,b,m) R = unifrnd(a,b,m,k) [M,V] = unifstat(a,b) 39

40 Continuous uniform distribution [ahat,bhat] = unifit(x) [ahat,bhat,aci,bci] = unifit(x) [ahat,bhat,aci,bci] = unifit(x,alpha) 40

41 Gamma Distribution A random variable X with density function 1 f ( x ) x e ( ) is said to have a Gamma Distribution with parameters and,for x > 0, > 0, > 0. E[X] = Mean = X = Var(X) = 2 = 2 1 x / 0, f o r x 0, 41

42 Gamma distribution Y = gampdf(x,a,b); A,B > 0 and X [0, ) P = gamcdf(x,a,b) R = gamrnd(a,b) R = gamrnd(a,b,m) R = gamrnd(a,b,m,n) [M,V] = gamstat(a,b) phat = gamfit(x) [phat,pci] = gamfit(x) [phat,pci] = gamfit(x,alpha) 42

43 Exponential distribution In Gamma Distribution, put = 1, we get f ( x ) 1 0 e, x, x elsewhere 0, 0 Mean = =, var (X) = 2, 43

44 Exponential distribution Y = exppdf(x,mu) P = expcdf(x,mu) R = exprnd(mu) R = exprnd(mu,m) R = exprnd(mu,m,n) [M,V] = expstat(mu) muhat = expfit(x) [muhat,muci] = expfit(x) [muhat,muci] = expfit(x,alpha) 44

45 Normal distribution A random variable X with density f(x) is said to have normal distribution with parameters and > 0, where f(x) is given by: f ( x) 1 e 2 x 2 2 2, x, (, ); 0. 45

46 Mean and Standard deviation for Normal distribution Let X be a normal random variable with parameters µ and. Then µ is the mean of X and is its standard deviation. 46

47 The density Curves 47

48 Cumulative Distribution Function 48

49 Normal distribution Y = normpdf(x,mu,sigma) P = normcdf(x,mu,sigma) [M,V] = normstat(mu,sigma) R = normrnd(mu,sigma) R = normrnd(mu,sigma,m) R = normrnd(mu,sigma,m,n) 49

50 Normal distribution [mhat,sigmahat,muci,sigmaci] = normfit(x) [mhat,sigmahat,muci,sigmaci] = normfit(x,alpha) Example: >> x = [-3:0.1:3]; >> f = normpdf(x,0,1) >> p = normcdf(x,0,1) 50

51 Continuous Probability Distribution Beta Distribution Lognormal Distribution Rayleigh Distribution Weibull Distribution 51

52 Continuous Statistics Probability Distribution Chi-square Distribution Non-central Chi-square Distribution F Distribution Non-central F Distribution T Distribution Non-central t Distribution 52

53 Linear Models p = anova1(x) Description: performs a balanced one-way ANOVA for comparing the means of two or more columns of data in the m-by-n matrix X, where each column represents an independent sample containing m mutually independent observations. The function returns the p-value for the null hypothesis that all samples in X are drawn from the same population (or from different populations with the same mean). 53

54 Linear Models >> X = meshgrid(1:5); >> X = X + normrnd(0,1,5); >> p = anova1(x) p = e-006 If the p-value is near zero, this suggests that at least one sample mean is significantly different than the other sample means. It is common to declare a result significant if the p- value is less than 0.05 or

55 Linear Models Mean Squares (MS) for each source, = SS/df p-value, which is derived from the cdf of F. As F increases, the p- value decreases. degrees of freedom (df) source of the variability associated with each source Sum of Squares (SS) due to each source the F statistic, which is the ratio of the MS s. 55

56 Linear Models The second figure displays box plots of each column of X. Large differences in the center lines of the box plots correspond to large values of F and correspondingly small p- values. 56

57 Linear Models p = anova2(x,reps) Description: performs a balanced two-way ANOVA for comparing the means of two or more columns and two or more rows of the observations in X. The data in different columns represent changes in factor A. The data in different rows represent changes in factor B. If there is more than one observation for each combination of factors, input reps indicates the number of replicates in each cell, which much be constant. 57

58 Linear Models p = anovan(x,group) Description: performs a balanced or unbalanced multi-way ANOVA for comparing the means of the observations in vector X with respect to N different factors. The factors and factor levels of the observations in X are assigned by the cell array group. 58

59 Covariance C = cov(x) Description: computes the covariance matrix C = cov(x,y) Description: cov(x,y), where x and y are column vectors of equal length, gives the same result as cov([x y]) 59

60 Correlation R = corrcoef(x) Description: returns a matrix of correlation coefficients calculated from an input matrix whose rows are observations and whose columns are variables. 60

61 Linear Regression [b,bint,r,rint,stats] = regress(y,x) Description: b: an estimate of ; bint: a 95% confidence interval for in the p-by-2 vector. r: residuals rint: a 95% confidence interval for each residual in the n-by-2 vector. stats: contains the R 2 statistic along with the F and p values for the regression. 61

62 Linear Regression [b,bint,r,rint,stats] = regress(y,x,alpha) Description: b: an estimate of ; bint: a 100(1-alpha)% confidence interval for in the p-by-2 vector. r: residuals rint: a 100(1-alpha)% confidence interval for each residual in the n-by-2 vector. stats: contains the R 2 statistic along with the F and p values for the regression. 62

63 Hypothesis testing Hypothesis testing for the mean of one sample with known variance Hypothesis testing for a single sample mean when the standard deviation is unknown. 63

64 Hypothesis testing Hypothesis testing for the mean of one sample with known variance Description: h = ztest(x,m,sigma) performs determine whether a sample x from a normal distribution with standard deviation sigma could a Z test at significance level 0.05 to have mean m. 64

65 Hypothesis testing Hypothesis testing for the mean of one sample with known variance h = ztest(x,m,sigma) h = 1, you can reject the null hypothesis at the significance level h = 0, you cannot reject the null hypothesis at the significance level

66 Hypothesis testing [h,sig,ci,zval] = ztest(x,m,sigma,alpha,tail) Description: sig: probability that the observed value of Z could be as large or larger by chance under the null hypothesis that the mean of x is equal to m. ci:(1-alpha) confidence interval for the true mean x-m z = σ n zval: the value of the Z statistic 66

67 Hypothesis testing [h,sig,ci,zval] = ztest(x,m,sigma,alpha,tail) Description: alpha: the significance level alpha tail: tail = 0 specifies the alternative x m (default) tail = 1 specifies the alternative x > m tail = -1 specifies the alternative x < m 67

68 Hypothesis testing >> x = normrnd(0,1,100,1); >> m = mean(x) m = >> [h,sig,ci] = ztest(x,0,1) h = 0; % we cannot reject the null hypothesis sig = ci =

69 Hypothesis testing Hypothesis testing for a single sample mean when the standard deviation is unknown h = ttest(x,m) performs a t-test at significance level 0.05 to determine whether a sample from a normal distribution (in x) could have mean m when the standard deviation is unknown. 69

70 Hypothesis testing Hypothesis testing for a single sample mean when the standard deviation is unknown h = ttest(x,m,alpha) performs a t-test at significance level alpha to determine whether a sample from a normal distribution (in x) could have mean m when the standard deviation is unknown 70

71 Hypothesis testing Hypothesis testing for a single sample mean when the standard deviation is unknown [h,sig,ci] = ttest(x,m,alpha,tail) Description: sig: p-value associated with the T-statistic ci: (1-alpha) confidence interval for the true mean t = x-m s n 71

72 Hypothesis testing Hypothesis testing for a single sample mean when the standard deviation is unknown [h,sig,ci] = ttest(x,m,alpha,tail) Description: alpha: the significance level alpha tail: tail = 0 specifies the alternative x m (default) tail = 1 specifies the alternative x > m tail = -1 specifies the alternative x < m 72

73 Hypothesis testing jbtest: kstest: kstest2: lillietest: ranksum: signrank: signtest: ttest2: Normal distribution for one sample Any specified distribution for one sample Equal distributions for two samples Normal distribution for one sample Median of two unpaired samples Median of two paired samples Median of two paired samples Mean of two normal samples 73

74 Cluster Analysis Multivariate Statistics PCA Multivariate Analysis of Variance Statistical Plots Box Plots Distribution Plots Scatter Plots Design of Experiments (DOE) 74

Economics 573 Problem Set 4 Fall 2002 Due: 20 September

Economics 573 Problem Set 4 Fall 2002 Due: 20 September 1. Ten students selected at random have the following "final averages" in physics and economics. Students 1 2 3 4 5 6 7 8 9 10 Physics 66 70 50 80