2 Mean Median Mode Variance

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Tendencia central y dispersiónTendencia central y dispersión

de una distribuciónde una distribución



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Review Topics

•Measures of Central Tendency Mean, Median, Mode

•Quartile

•Measures of Variation

The Range, Variance and

Standard Deviation, Coefficient of variation•Shape

Symmetric, Skewed



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Important Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

One sample

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

VariationRange



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Measures of Central Tendency

Central Tendency

Mean Median Mode

n

x n

i i

= 1

Data: You can access practice sample data on

HMO premiums here.

http://www.econ.uiuc.edu/ECON173/hmodata.xls

http://www.econ.uiuc.edu/ECON173/hmodata.xls



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With one data pointclearly the centrallocation is at the point

itself.

if the third data pointears on the left hand-side

e midrange, it should “pull”central location to the left.

Measures of Central

Location (Tendency)Usually, we focus our attention on two

aspects of measures of central location:

– Measure of the central data point (the average).

– Measure of dispersion of the data about theaverage. With two data points,

the central locationshould fall in the midbetween them (in ord

to reflect the locationboth of them).

If the third data point appearsexactly in the middle of thecurrent range, the centrallocation should not change

(because it is currentlyresiding in the middle).



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n

xx i

n

1i=∑=

– This is the most popular and useful measure of central location

Sum of the measurements

Number of measurements

Mean =

Sample mean Population mean

N

xi

N

1i=∑=µ

Sample size Population size

nxx i

n

1i=∑=

l ArithmeticArithmetic

meanmean



7

=+++++

=∑

= =

66

654321

6

1x x x x x x x

xii

• Example 4.1

mean of the sample of six measurements 7, 3, 9, -2, 4, 6 is

77 33 99 44 664.54.5

• Example 4.2

ppose the telephone bills of example 2.1 represent populatmeasurements. The population mean is

=+++

=∑

=µ =

200

x...xx

200

x 20021i2001i 42.1942.19 15.3015.30 53.2153.21

43.5943.59

2−



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26,26,28,29,30,32,60,31

Odd number of observations

26,26,28,29,30,32,60

Example 4.4

Seven employee salaries were recorded(in 1000s) : 28, 60, 26, 32, 30, 26, 29.Find the median salary.

– The median of a set of measurements is the

value that falls in the middle when the

measurements are arranged in order of

magnitude.

Suppose one employee’s salary of $31,was added to the group recorded beforFind the median salary.

Even number of observation

26,26,28,29, 30,32,60,3126,26,28,29, 30,32,60,31

There are two middle values!First, sort the salaries.

Then, locate the valuein the middle

First, sort the salaries. Then, locate the values in the middle26,26,28,29, 30,32,60,3129.5,

l The medianThe median



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– The mode of a set of measurements is the value

that occurs most frequently.

– Set of data may have one mode (or modal class),

or two or more modes.

The modal classFor large data setsthe modal class ismuch more relevant

than the a single-value mode.

l The modeThe mode



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• Example 4.6

professor of statistics wants to report the results of a midtexam, taken by 100 students. The data appear in file XM04-Find the mean, median, and mode, and describe the informahey provide.

Marks

Mean 73.98

Standard Error 2.1502163Median 81

Mode 84

Standard Deviation 21.502163Sample Variance 462.34303Kurtosis 0.3936606Skewness -1.073098Range 89Minimum 11Maximum 100

Sum 7398Count 100

The mean provides information

about the over-all performance levelof the class. The Median indicates that half of theclass received a grade below 81%,and half of the class received a grade

above 81%. The mode must be used when data isqualitative. If marks are classified byletter grade, the frequency of eachgrade can be calculated.Then, the modbecomes a logical measure to comput

Excel Results



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Relationship among Mean, Median,Relationship among Mean, Median,

and Modeand Mode

If a distribution is symmetrical, the

mean, median and mode coincide

If a distribution is non symmetrical, and

skewed

to the left or to the right, the three

measuresdiffer.

A positively skewed distribution(“skewed to the right”)

MeanMedian

Mode



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`̀

If a distribution is symmetrical, the mean,

median and mode coincide

If a distribution is non symmetrical, and

skewed to the left or to the right, the three

measures differ.

A positively skewed distribution(“skewed to the right”)

MeanMedian

ModeMeanMedianMode

A negatively skewed distribu(“skewed to the left”)



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Measures of Variation

Variation

Variance Standard Deviation Coefficient of

VariationPopulation

Variance

Sample

Variance

Population

Standard

Deviation

Sample

Standard

Deviation

Range

Interquartile Range

100%

= X

S

CV



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Measures of variabilityMeasures of variability(Looking beyond the average)(Looking beyond the average)

Measures of central location fail to tell the

whole story about the distribution.

A question of interest still remains unanswered:

How typical is the average value of allthe measurements in the data set?

How much spread out are the measurementsabout the average value?

or



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Observe two hypothetical data sets

The average value provides

a good representation of thevalues in the data set.

Low variability data set

High variability data set

he same average value does notrovide as good presentation of the

alues in the data set as before.

This is the previous

data set. It is nowchanging to...



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– The range of a set of measurements is the difference

between the largest and smallest measurements.

– Its major advantage is the ease with which it can becomputed.

– Its major shortcoming is its failure to provide

information on the dispersion of the values between

the two end points.

? ? ?

But, how do all the measurements spread out?

Smallestmeasurement

Largestmeasurement

The range cannot assist in answering this question

Range

l The rangeThe range



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– This measure of dispersion reflects the values of

all the measurements.

– The variance of a population of N

measurements

x1, x2,…,x N having a mean µ is defined as

– The variance of a sample of n measurements

x1, x2, …,xn having a mean is defined as

N

)x( 2i

N1i2 µ−∑

=σ =

x

1n

)xx(

s

2i

n1i2

−

−∑

==

l The varianceThe variance



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onsider two small populations:opulation A: 8, 9, 10, 11, 12

opulation B: 4, 7, 10, 13, 16

1098

74 10

11 12

13 16

8-10= -2

9-10= -1

11-10= +1

12-10= +2

4-10 = - 6

7-10 = -3

13-10 = +3

16-10 = +6

Sum = 0

Sum = 0

The mean of bothpopulations is 10...

…but measurements in Bare much more dispersedthen those in A.

Thus, a measure of dispersionis needed that agrees with thisobservation.

us start by calculatingsum of deviations

A

B

The sum of deviationis zero in both cases,therefore, anothermeasure is needed.



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1098

74 10

11 12

13 16

8-10= -2

9-10= -1

11-10= +1

12-10= +2

4-10 = - 6

7-10 = -3

13-10 = +3

16-10 = +6

Sum = 0

Sum = 0

A

B

The sum of deviationis zero in both cases,therefore, anothermeasure is needed.

The sum of squared deviationsis used in calculating the variance.

See example next.



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Let us calculate the variance of the two populations

185

)1016()1013()1010()107()104( 222222B

=−+−+−+−+−

=σ

25

)1012()1011()1010()109()108(22222

2A

=−+−+−+−+−

=σ

hy is the variance defined ashe average squared deviation?

hy not use the sum of squareddeviations as a measure of dispersion instead?

After all, the sum of squareddeviations increases inmagnitude when the dispersion

of a data set increases!!



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– Example 4.8

Find the mean and the variance of the followingsample of measurements (in years).

3.4, 2.5, 4.1, 1.2, 2.8, 3.7

– Solution

=

∑−∑−=−

−∑= =

==

n

)x(x

1n1

1n

)xx(s

2

i

n

1i2i

n

1i

2

i

n

1i2

95.26

7.17

6

7.38.22.11.45.24.3

6

xx

i6

1i ==+++++

=∑

= =

A shortcut formula

=[3.42+2.52+…+3.72]-[(17.7)2/6] = 1.075 (years)2



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Sample Standard Deviation

( )

1

2

−∑ −

=n

X X i

For the Sample : use n - 1

in the denominator.

Data: 10 12 14 15 17 18 18 24

s =

n = 8 Mean =16

18

16241618161716151614161216102222222

−

− )()()()()()()(

= 4.2426

s

: X i



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Interpreting StandardInterpreting Standard

DeviationDeviation

The standard deviation can be used to

– compare the variability of several distributions

– make a statement about the general shape of adistribution.

The empirical rule: If a sample of measurements

has a mound-shaped distribution, the interval

tsmeasurementheof 68%elyapproximatcontains)sx,sx( +−tsmeasuremetheof 95%elyapproximatcontains)s2x,s2x( +−

tsmeasurementheof allvirtuallycontains)s3x,s3x( +−



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Comparing Standard Deviations

( )1

2

−∑ −

n

X X i s = = 4.2426

( )

N

Xi−

=

2µ

σ = 3.9686

Value for the Standard Deviation is larger for data considered as a Sample.

Data : 10 12 14 15 17 18 18 24: X i

N= 8 Mean =16



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Comparing Standard Deviations

Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B

Data A

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5

s = 4.57

Data C



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Measures of AssociationMeasures of Association

Two numerical measures are presented, for

the description of linear relationship

between two variables depicted in thescatter diagram.

– Covariance - is there any pattern to the way two

variables move together?

– Correlation coefficient - how strong is the linear

relationship between two variables



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N

)y)((x Y)COV(X,covariancePopulation

yixiµ−µ−∑

==

µx (µy) is the population mean of the variable X (Y)

N is the population size. n is the sample size.

1-n

)y)((x

Y)cov(X,covarianceSampleyixi

µ−µ−∑

==

l TheThe

covariancecovariance



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– This coefficient answers the question: How strong

is the association between X and Y.

yx

) Y,X(COV

ncorrelatioof tcoefficienPopulation

σσ

=ρ

yxss

) Y,Xcov(r

ncorrelatioof tcoefficienSample

=

l The coefficient of correlationThe coefficient of correlation



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COV(X,Y)=0

ρ or r =

+1

0

-1

Strong positive linear relationship

No linear relationship

Strong negative linear relationship

or

COV(X,Y)>0

COV(X,Y)<0

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2 Mean Median Mode Variance