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BITS PilaniPilani Campus

National Workshop on LaTeX and MATLAB for Beginners

24 - 28 December, 2014BITS Pilani,

Partially Supported by DST, Rajasthan

BITS PilaniPilani Campus

Lecture - 13: STATISTICAL

COMMANDSDr. Shivi Agarwal

BITS Pilani, Pilani Campus

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• Measures of Central Tendency(Location)

• Measures of Dispersion• Probability Distribution• Linear Models• Covariance• Correlation• Linear Regression• Hypothesis TestingDecember 28, 2014 Dr. Shivi Agarwal, NWLMB-2014


Measures of Central Tendency (Location)

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• To locate the data values on the number line.• Measures of location.

geomean(x) : Geometric meanharmmean(x) : Harmonic meanmean(x) : Arithmetic average median(x) : 50th percentile (in MATLAB)max(x) : Maximum valuemin(x) : Minimum value

December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

BITS Pilani, Pilani CampusBITS Pilani, Pilani Campus

Example:x = [ones(1,6) 100]x = 1 1 1 1 1 1 100locate = [geomean(x), harmmean(x), mean(x), median(x)]

locate =1.9307 1.1647 15.1429 1.0000

5December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

Measures of Central Tendency (Location)


Measures of Dispersion

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• To find out how spread out the data values are onthe number line.

• Measures of spread.

iqr(x) Interquartile Rangemad(x) Mean Absolute Deviationrange(x) Rangestd(x) Standard deviation var(x) Variance



Range: difference between the maximum and minimum values; simplest measure of spread; not robust to outliers.

Interquartile Range (IQR): difference between the 75th and 25th percentile of the data. Since only the middle 50% of the data affects this measure, it is robust to outliers.

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Example: x = [ones(1,6) 100]x =1 1 1 1 1 1 100stats = [iqr(x), mad(x), range(x), std(x)]stats =0 24.2449 99.0000 37.4185




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prctile(X,p) : pth Percentile of a data set X,p [0,100]

quantile(X,p) : pth Quantile of a data set X,p [0,1]

skewness(X) : skewness of sample Xkurtosis(X) : kurtosis of sample Xcorr(X) : pairwise linear correlation

coefficient



Random variableProbability Distribution


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Probability distributions arise fromexperiments where the outcome is subject tochance.The nature of the experiment dictates whichprobability distributions may be appropriatefor modeling the resulting random outcomes.

Probability Distribution


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The uncertain behavior of the randomvariable is predicted by:

(i) Probability density function f(x)(ii) Cumulative distribution function

F(x)



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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



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• 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),.



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Discrete• Binomial

Distribution• Poisson

Distribution• Discrete Uniform

Distribution

Continuous• Continuous Uniform

Distribution• Gamma Distribution• Exponential

Distribution• Normal Distribution



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The density function of a discrete randomvariable X is defined by

f (x) = P(X=x) for all real x.

Discrete Probability Distribution

xk

f(k)x)P(XF(x)

The cumulative distribution function F of adiscrete random variable X, is defined by

for any real number x, here f denote the density ofX.December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

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A discrete random variable X has binomialdistribution with parameters n and p, n is a positiveinteger and 0 < p < 1, if its density function is

(1 ) ; 0,1,2,...,( )

0 otherwise.

n x n xp p x nf x x

Binomial Distribution

[ ] and [ ] .E X np Var X npq December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

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BITS Pilani, Pilani CampusBITS Pilani, Pilani CampusDecember 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

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n: must be positive integers,p: values must lie on the interval [0,1].

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A = binopdf(x,n,p)B= binocdf(x,n,p)




Example: A Quality Assurance inspector tests200 circuit boards a day. If 2% of the boardshave defects, what is the probability that theinspector will find no defective boards on anygiven day?

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>>binopdf(0,200,0.02)



[M,V]=binostat(n,p)

returns the mean and variance for the binomial distribution with parameters specified by n and p.

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Ex. If a baseball team plays 162 games in a season and has a 50-50 chance of winning any game, then the mean and the variance is :

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>> [M,V]=binostat(162,0.5)M = 81V = 40.5



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

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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

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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.

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phat = binofit(x,n)Description:

returns a maximum likelihoodestimate of the probability ofsuccess in a given binomial trialbased on the number ofsuccesses, x, observed in nindependent trials.

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[phat,pci] = binofit(x,n)Description:

returns the probability estimate, phat, and the 95% confidence intervals, pci.

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

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Example: First generate a binomial sample of 100elements, where the probability of success in agiven trial is 0.6. Then, estimate this probabilityfrom the outcomes in the sample.

>> r = binornd(100,0.6);>> [phat,pci] = binofit(r,100)

phat = 0.5800pci = 0.4771 0.6780

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This distribution is named on the Frenchmathematician Simeon Denis Poisson.Let λ > 0 be a constant and, for any real number x,

Poisson Distribution

; for 0,1,2,...( ) !0 otherwise

xe xf x x


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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)December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

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Examples: A computer hard disk manufacturerhas observed that flaws occur randomly in themanufacturing process at the average rate oftwo flaws in a 4 GB hard disk and has foundthis rate to be acceptable. What is theprobability that a disk will be manufacturedwith no defects?

λ= 2 and x = 0.>> p = poisspdf(0,2)

p = 0.1353December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

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Discrete Uniform DistributionFor discrete uniform • pdf is = 1/N; • The mean is = (N+1)/2• The variance is =

(N2-1)/12.

Y = unidpdf(X,N)P = unidcdf(X,N)

N > 0 and integer

R = unidrnd(N)R = unidrnd(N,m)R = unidrnd(N,m,k)[M,V] = unidstat(N)


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Examples: For fixed n, the uniform discrete pdf is a constant.

>> y = unidpdf(1:6,10)y =0.1000 0.1000 0.1000 0.1000

0.1000 0.1000

Discrete Uniform Distribution


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Geometric Distribution Hypergeometric DistributionNegative Binomial Distribution

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Discrete Probability Distribution



Definition: Let X be a continuous randomvariable. A function f(x) is called continuousdensity (probability density function i.e. pdf ) iff

integral converges.

Continuous Density (Probability density function )

1)(.2

0f(x)1.

dxxf


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Let X be the continuous r.v. with density f(x). Thecumulative distribution function (cdf) for X, denotedby F(X) , is defined by

F(X) = P ( X x ) , all x

=

Cumulative Distribution Function

x

dttf )(


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Continuous uniform distributionA random variable X is said to be uniformlydistributed over an interval (a, c) if its densityis given by

cxaac

xf

,1)(

12)()(

2)(

2acXVar

caXE


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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)

Continuous uniform distribution


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[ahat,bhat] = unifit(X)[ahat,bhat,ACI,BCI] = unifit(X)[ahat,bhat,ACI,BCI] = unifit(X,alpha)

Continuous uniform distribution


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A random variable X with density function

is said to have a Gamma Distribution with parameters and ,for x > 0, > 0, > 0.

E[X] = Mean = X = Var(X) = 2 = 2

Gamma Distribution

1 /1( )( )

0 , f o r x 0 ,

xf x x e


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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)

Gamma distribution


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In Gamma Distribution, put = 1, we get

Mean = = , var (X) = 2,

elsewhere

xexf

x

,0

0,0,1)(

Exponential distribution


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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)

Exponential distribution


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A random variable X with density f(x) issaid to have normal distribution withparameters and > 0, where f(x) isgiven by:

.0);,(,,21)(

2

2

2

xexfx

Normal distribution


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Mean and Standard deviation for Normal distribution

Let X be a normal random variable withparameters µ and . Then µ is the meanof X and is its standard deviation.


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The density Curves

December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014 47


Cumulative Distribution Function

December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014 48


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)

Normal distribution


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[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)

Normal distribution


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Beta Distribution Lognormal Distribution Rayleigh Distribution Weibull Distribution

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Continuous Probability Distribution



Chi-square DistributionNon-central Chi-square DistributionF DistributionNon-central F DistributionT DistributionNon-central t Distribution

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Continuous Statistics Probability Distribution



p = anova1(X)Description:performs a balanced one-way ANOVA forcomparing the means of two or more columns ofdata in the m-by-n matrix X, where each columnrepresents an independent sample containing mmutually independent observations. The functionreturns the p-value for the null hypothesis that allsamples in X are drawn from the same population(or from different populations with the same mean).53

Linear Models



>> X = meshgrid(1:5);>> X = X + normrnd(0,1,5);>> p = anova1(X)

p = 1.5283e-006If 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 0.01. 54

Linear Models



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source of the variabilitySum of Squares (SS) due to each source

degrees of freedom (df) associated with each source

Mean Squares (MS) for each source, = SS/df

the F statistic, which is the ratio of the MS’s.

p-value, which is derived from the cdf of F. As F increases, the p-value decreases.

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.

Linear Models

56December 28, 2014 Dr. Shivi Agarwal, NWLMB-2014


p = anova2(X,reps)Description:performs a balanced two-way ANOVA for comparing themeans 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.

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Linear Models



p = anovan(X,group)Description:

performs a balanced or unbalanced multi-wayANOVA for comparing the means of theobservations in vector X with respect to Ndifferent factors. The factors and factor levels ofthe observations in X are assigned by the cellarray group.

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Linear Models



C = cov(X)Description:computes the covariance matrix

C = cov(x,y)Description:cov(x,y), where x and y are column vectorsof equal length, gives the same result ascov([x y]) 59

Covariance



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

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Correlation



[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 R2 statistic along with the F and

p values for the regression.61

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 R2 statistic along with the F and

p values for the regression.62

Linear Regression



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Hypothesis testing

•Hypothesis testing for the mean of onesample with known variance

•Hypothesis testing for a single samplemean when the standard deviation isunknown.



Hypothesis testing for the mean of one sample with known variance

h = ztest(x,m,sigma)Description: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.

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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 0.05.

h = 0, you cannot reject the null hypothesis at the significance level 0.05.

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Hypothesis testing



[h,sig,ci,zval] = ztest(x,m,sigma,alpha,tail)Description:

sig: probability that the observed value of Zcould be as large or larger by chance under thenull hypothesis that the mean of x is equal tom.

ci:(1-alpha) confidence interval for the true mean

zval: the value of the Z statistic 66

Hypothesis testing

x-mz = σn



[h,sig,ci,zval] = ztest(x,m,sigma,alpha,tail)Description:alpha: the significance level alphatail:

•tail = 0 specifies the alternative x ≠ m (default)

•tail = 1 specifies the alternative x > m•tail = -1 specifies the alternative x < m 67

Hypothesis testing



>> x = normrnd(0,1,100,1);>> m = mean(x)m = 0.0727>> [h,sig,ci] = ztest(x,0,1)h = 0; % we cannot reject the null hypothesissig = 0.4669ci = -0.1232 0.2687

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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 todetermine whether a sample from a normaldistribution (in x) could have mean m whenthe standard deviation is unknown.

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Hypothesis testing




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

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Hypothesis testing




[h,sig,ci] = ttest(x,m,alpha,tail)Description:

sig: p-value associated with the T-statisticci: (1-alpha) confidence interval for the true

mean

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Hypothesis testing

x-m=n

t s




[h,sig,ci] = ttest(x,m,alpha,tail)Description:alpha: the significance level alphatail: •tail = 0 specifies the alternative x ≠ m (default)

•tail = 1 specifies the alternative x > m•tail = -1 specifies the alternative x < m

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Hypothesis testing



jbtest: Normal distribution for one samplekstest: Any specified distribution for one samplekstest2: Equal distributions for two sampleslillietest: Normal distribution for one sampleranksum: Median of two unpaired samplessignrank: Median of two paired samplessigntest: Median of two paired samplesttest2: Mean of two normal samples

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Hypothesis testing



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• Cluster Analysis• Multivariate Statistics

–PCA–Multivariate Analysis of Variance

• Statistical Plots–“Box Plots”–“Distribution Plots”–“Scatter Plots”

• Design of Experiments (DOE)


BITS Pilani, Pilani CampusDecember 28, 2014 Dr. Shivi Agarwal, NWLMB-2014

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