MEASURES OF VARIABILITY

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MEASURES OF VARIABILITY

• Variance– Population variance – Sample variance

• Standard Deviation– Population standard deviation – Sample standard deviation

• Coefficient of Variation (CV)– Sample CV– Population CV

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MEASURES OF VARIABILITYPOPULATION VARIANCE

• The population variance is the mean squared deviation from the population mean:

• Where 2 stands for the population variance is the population mean• N is the total number of values in the population• is the value of the i-th observation.• represents a summation

N

xN

ii

12

)(

ix

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MEASURES OF VARIABILITYSAMPLE VARIANCE

• The sample variance is defined as follows:

• Where s2 stands for the sample variance• is the sample mean• n is the total number of values in the sample• is the value of the i-th observation.• represents a summation

112

n

xxs

N

ii )(

ix

x

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• A sample of monthly advertising expenses (in 000$) is taken. The data for five months are as follows: 2.5, 1.3, 1.4, 1.0 and 2.0. Compute the sample variance.

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• Notice that the sample variance is defined as the sum of the squared deviations divided by n-1.

• Sample variance is computed to estimate the population variance.

• An unbiased estimate of the population variance may be obtained by defining the sample variance as the sum of the squared deviations divided by n-1 rather than by n.

• Defining sample variance as the mean squared deviation from the sample mean tends to underestimate the population variance.

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• A shortcut formula for the sample variance:

• Where s2 is the sample variance• n is the total number of values in the sample• is the value of the i-th observation.• represents a summation

n

x

xn

s

n

iin

ii

2

1

1

22

1

1

ix

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• A sample of monthly sales expenses (in 000 units) is taken. The data for five months are as follows: 264, 116, 165, 101 and 209. Compute the sample variance using the short-cut formula.

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• The shortcut formula for the sample variance:

• If you have the sum of the measurements already computed, the above formula is a shortcut because you need only to compute the sum of the squares,

n

x

xn

s

n

iin

ii

2

1

1

22

1

1

n

iix

1

n

iix

1

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MEASURES OF VARIABILITY POPULATION/SAMPLE STANDARD DEVIATION

• The standard deviation is the positive square root of the variance:

Population standard deviation:

Sample standard deviation: • Compute the standard deviations of advertising and

sales.

2ss

2

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MEASURES OF VARIABILITY POPULATION/SAMPLE STANDARD DEVIATION

• Compute the sample standard deviation of advertising data: 2.5, 1.3, 1.4, 1.0 and 2.0

• Compute the sample standard deviation of sales data: 264, 116, 165, 101 and 209

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MEASURES OF VARIABILITY POPULATION/SAMPLE CV

• The coefficient of variation is the standard deviation divided by the means

Population coefficient of variation:

Sample coefficient of variation:x

scv

CV

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MEASURES OF VARIABILITY POPULATION/SAMPLE CV

• Compute the sample coefficient of variation of advertising data: 2.5, 1.3, 1.4, 1.0 and 2.0

• Compute the sample coefficient of variation of sales data: 264, 116, 165, 101 and 209

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MEASURES OF ASSOCIATION

• Scatter diagram plot provides a graphical description of positive/negative, linear/non-linear relationship

• Some numerical description of the positive/negative, linear/non-linear relationship are obtained by:– Covariance

• Population covariance• Sample covariance

– Coefficient of correlation• Population coefficient of correlation• Sample coefficient of correlation

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• A sample of monthly advertising and sales data are collected and shown below:

• How is the relationship between sales and advertising? Is the relationship linear/non-linear, positive/negative, etc.

MEASURES OF ASSOCIATION: EXAMPLE

Sales AdvertisingMonth (000 units) (000 $)

1 264 2.52 116 1.33 165 1.44 101 1.05 209 2.0

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POPULATION COVARIANCE

• The population covariance is mean of products of deviations from the population mean:

• Where COV(X,Y) is the population covariance x, y are the population means of X and Y respectively

• N is the total number of values in the population• are the values of the i-th observations of X and Y

respectively.• represents a summation

N

yxYXCOV

N

iyixi

1

),(

ii yx ,

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SAMPLE COVARIANCE

• The sample covariance is mean of products of deviations from the sample mean:

• Where cov(X,Y) is the sample covariance• are the sample means of X and Y respectively• n is the total number of values in the population• are the values of the i-th observations of X and Y

respectively.• represents a summation

1

1

1

n

yyxx)Y,Xcov(

n

iii

ii yx ,

y,x

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SAMPLE COVARIANCE

Advertising SalesMonth (in 000$) (in 000 units)

1 2.5 2642 1.3 1163 1.4 1654 1 1015 2 209

Mean 1.64 171 Total=SD 0.602495 67.18258703 cov =

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POPULATION/SAMPLE COVARIANCE

• If two variables increase/decrease together, covariance is a large positive number and the relationship is called positive.

• If the relationship is such that when one variable increases, the other decreases and vice versa, then covariance is a large negative number and the relationship is called negative.

• If two variables are unrelated, the covariance may be a small number.

• How large is large? How small is small?

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POPULATION/SAMPLE COVARIANCE

• How large is large? How small is small? A drawback of covariance is that it is usually difficult to provide any guideline how large covariance shows a strong relationship and how small covariance shows no relationship.

• Coefficient of correlation can overcome this drawback to a certain extent.

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POPULATION COEFFICIENT OF CORRELATION

• The population coefficient of correlation is the population covariance divided by the population standard deviations of X and Y:

• Where is the population coefficient of correlation• COV(X,Y) is the population covariance x, y are the population means of X and Y

respectively

yx

)Y,X(COV

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SAMPLE COEFFICIENT OF CORRELATION

• The sample coefficient of correlation is the sample covariance divided by the sample standard deviations of X and Y:

• Where r is the sample coefficient of correlation• cov(X,Y) is the sample covariance

• sx, sy are the sample means of X and Y respectively

yx

)Y,X(COV

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Advertising SalesMonth (in 000$) (in 000 units)

1 2.5 2642 1.3 1163 1.4 1654 1 1015 2 209

Mean 1.64 171 Total=SD 0.602495 67.18258703 cov =

r =

SAMPLE COEFFICIENT OF CORRELATION

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POPULATION/SAMPLE COEFFICIENT OF CORRELATION

• The coefficient of correlation is always between -1 and +1.– Values near -1 or +1 show strong relationship– Values near 0 show no relationship’– Values near 1 show strong positive linear

relationship– Values near -1 show strong negative linear

relationship

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EXAMPLE

• Salary and expenses for cultural activities, and sports related activities are collected from 100 households. Data of only 5 households shown below:

How are the relationships (linear/non-linear, positive/negative)between (i) salary and culture, (ii) salary and sports, and (iii) sports and culture?

Salary and expensesdata for 100 households

Salary Culture Sports$54,600 $1,020 $990$57,500 $1,100 $460$53,300 $900 $780$43,500 $570 $860$57,200 $900 $1,390

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SALARY-CULTURE

$0

$400

$800

$1,200

$1,600

$35,000 $55,000 $75,000 $95,000

Salary

Ex

pe

ns

es

fo

r C

ult

ura

l A

cti

viti

es

cov = 1094787, r = 0.5065 (positive, linear)

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SPORTS-CULTURE

0

400

800

1200

1600

$500 $1,000 $1,500 $2,000

Expenses for sports related activities

Ex

pe

ns

es

fo

r c

ult

ura

l a

cti

viti

es

cov = -33608, r = -0.5201 (negative, linear)

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SALARY-SPORTS

$400

$900

$1,400

$1,900

$35,000 $55,000 $75,000 $95,000

Salary

Ex

pe

ns

es

fo

r s

po

rts

re

late

d a

cti

viti

es

cov = -219026, r = -0.08122 (no linear relationship)

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MEASURES OF VARIABILITY