central tendency measures lectures 3 4 5

8/3/2019 Central Tendency Measures Lectures 3 4 5

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Summarizing and Describing

Numerical Data


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Lectures 3+4+5 Topics

Measures ofCentral Tendency

Mean, Median, Mode

Measures ofVariationThe Range, Variance and

Standard Deviation

ShapeSymmetric, Skewed, Skewness, Kurtosis


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Summary Measures

Central Tendency

MeanMedian

Mode

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range


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Measures of Central Tendency

Central Tendency

Mean Median Mode

n

xn

ii

!1


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Sum of the observations

Number of observationsMean =

This is the most popular and usefulThis is the most popular and usefulmeasure of central locationmeasure of central location

The ArithmeticThe Arithmetic

MeanMean


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n

xx i

n1i!!

Sample mean Population mean

N

x iN

1i!!Q

Sample size Population size

n

xx i

n1i!!


MeanMean


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!

!

! !

10

...

10

102110

1 xxxxx ii

Example 1The reported time spent on the Internet of10 adults are 0, 7,12, 5,

33,14, 8,0, 9,22hours. Find the mean time spent on the Internet.

00 77 222211.0hours11.0hours

Example 2

Suppose the telephone bills representthe population of measurements ( 200). The population meanis

!

!

!Q !

200

x...xx

200

x 20021i200

1i 42.1942.19 38.4538.45 45.7745.7743.5943.59


MeanMean

The arithmetic

mean


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Weighted mean for data groupedWeighted mean for data grouped

by categories or variantsby categories or variants

i

ii

k

i

ffxx

! !1


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When many of the measurements have the same value, the

measurement can be summarized in a frequency table. Suppose

the number ofchildrenin a sample of16 families were recorded

as follows:

NUMBER OF CHILDREN 0 1 2 3

NUMBER OF FAMILIES 3 4 7 2

16 families

5.116

)3(2)2(7)1(4)0(3

16

....

16

16162211

16

1 !

!

!

! !fxfxfxfx

x iii


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The Median

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

Important Measure of Central Tendency

In an ordered array, the median is the

middle number.If n is odd, the median is the middle number.If n is even, the median is the average of the 2

middle numbers.

Not Affected by Extreme Values


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Odd number of observations

0, 0, 5, 7, 8 9, 12, 14, 220, 0, 5, 7, 8, 9, 12, 14, 22, 330, 0, 5, 7, 8, 9, 12, 14, 22, 33

Even number of observations

Example 4.3

Find the median of the time spent on the internet

for the adults of example 1

TheThe MedianMedian of a set of observations is theof a set of observations is thevalue that falls in the middle when thevalue that falls in the middle when theobservations are arranged in order ofobservations are arranged in order ofmagnitude or ranked increasinglymagnitude or ranked increasingly

The MedianThe Median

Suppose only 9 adults were sampled

(exclude, say, the longest time (33))

Comment

8


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The Mode

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

A Measure of Central Tendency

Value that Occurs Most Often

Not Affected by Extreme Values

There May Not be a Mode

There May be Several Modes

Used for Either Numerical or Categorical Data

0 1 2 3 4 5 6

No Mode


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TheThe ModeMode of a set of observations is theof a set of observations is thevariable value that occurs most frequently.variable value that occurs most frequently.

Set of data may have one mode (or modalSet of data may have one mode (or modalclass), or two or more modes.class), or two or more modes.

The modal classFor large data sets

the modal class is

much more relevant

than a single-value

mode.

The ModeThe Mode


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Approximating DescriptiveApproximating Descriptive

Measures for groupedMeasures for groupedData by CLASSESData by CLASSESApproximating descriptive measures forApproximating descriptive measures for

grouped data may be needed in twogrouped data may be needed in twocases:cases:

when approximated values.suffices the needs,when approximated values.suffices the needs,

when only secondary grouped data arewhen only secondary grouped data areavailable.available.

i

k

i

ii

k

i

f

fxx

1

1

!

!

!

x midpoint

f frequency


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Class Class Frequency Midpointi limits fi xi xi fi1 2-5 3 3.5 10.5

2 5-8 6 6.5 39.0

3 8-11 8 9.5 76.0

. . . . .

6 17-20 2 18.5 37.0

n =sample size= 30=f1++fn 312.0

Example 3Example 3Approximate the mean (calculate the mean) ofApproximate the mean (calculate the mean) ofthe telephone call durations problem asthe telephone call durations problem asrepresented by the frequency distributionrepresented by the frequency distribution

5 8 11 14 17 20 More6.5

26.10

:valueReal

!x


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Median and ModeMedian and Mode

MedianMedian

Me

1-Me

1i

i

0n

n-1)(21

Kx

!

!

in

Me


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Median and ModeMedian and Mode

ModeMode

21

1

0 Kx

((

(!Mo


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If a distribution is symmetrical, theIf a distribution is symmetrical, themean, median and mode coincidemean, median and mode coincide

If a distribution is non symmetrical, andIf a distribution is non symmetrical, andskewed to the left or to the right, theskewed to the left or to the right, thethree measures differ.three measures differ.

A positively skeweddistribution(skewed to the right)

Mean

Median

Mode MeanMedian

Mode

A negatively skeweddistribution(skewed to the left)

Relationship among Mean, Median,Relationship among Mean, Median,and Modeand Mode


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Summary Measures

Central Tendency

MeanMedian

Mode

n

xn

ii!1

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of

Variation

Range

1n

xxs

2

i2

!


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Measures of Variation

Variation

Variance Standard Deviation Coefficient of

VariationPopulation

Variance

Sample

Variance

Population

Standard

Deviation

Sample

Standard

Deviation

Range

100%

! X

S

CV


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Measure of Variation

Difference Between Largest & Smallest

Observations:

Absolute Range =

Relative Range =

Ignores How Data Are Distributed:

The Range

SmallestrgestLa xx

7 8 9 10 11 12

Range = 12 - 7 = 5

7 8 9 10 11 12

Range = 12 - 7 = 5

meanxxSmallestLa

/)( rgest


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DeviationDeviation

Individual deviation from the mean =Individual deviation from the mean =

Overall deviation = 0, becauseOverall deviation = 0, because

Summing squared deviationsSumming squared deviations

ororabsolute values of the deviationsabsolute values of the deviations

meanxi

! 0XXi

2

XXi

|| xxi


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Important Measure of Variation

Shows Variation About the Mean

Computed as an arithmetic mean of

squared deviations or as a square mean ofindividual deviations

For the Population:

For the Sample:

Variance

N

Xi !

2

2 QW

1

2

2

!

n

XXs

i

For the Population: use N in the

denominator.

For the Sample : use n - 1

in the denominator.


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Most Important Measure of Variation

Shows Variation About the Mean:

For the Population:

For the Sample:

Standard Deviation

N

Xi !

2Q

W

1

2

!

nXXs i

For the Population: use N in the

denominator.

For the Sample : use n - 1

in the denominator.


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Sample Standard Deviation

1

2

!

n

XXi

Data: 10 12 14 15 17 18 18 24

s =

n = 8 Mean =16

18

16241618161716151614161216102222222

)()()()()()()(

= 4.2426

s

:Xi


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Comparing Standard Deviations

1

2

n

XXis = = 4.2426

N

Xi

!

2Q

W = 3.9686

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

Data : 10 12 14 15 17 18 18 24:Xi

N= 8 Mean =16


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Comparing Standard Deviations

Mean = 15.5

s = 3.33811 12 13 14 15 16 17 18 19 20 21

11 12 13 14 15 16 17 18 19 20 21

Data B - AGE

Data A - AGE

Mean = 15.5

s = .9258

11 12 13 14 15 16 17 18 19 20 21

Mean = 15.5

s = 4.57

Data C - AGE


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Coefficient of VariationCoefficient of Variation

Measure ofMeasure of Relative VariationRelative Variation

Always aAlways a % or coefficient% or coefficient

Shows Variation Relative to MeanShows Variation Relative to Mean

Used toUsed to Compare 2 or More GroupsCompare 2 or More Groups

Formula ( for Sample):Formula ( for Sample):

100%

!

X

SCV


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Comparing Coefficient of VariationComparing Coefficient of Variation

Stock A:Stock A: Average Price last year =Average Price last year = $50$50 Standard Deviation (sd)Standard Deviation (sd) == $5$5 Stock B:Stock B: Average Price last yearAverage Price last year == $100$100 (sd) =(sd) = $5$5

100%

!

X

SCV

Coefficient of Variation:

Stock A: CV = 10%Stock B: CV = 5%

Both average prices are

representatives


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ShapeShape

Describes How Data Are DistributedDescribes How Data Are Distributedbetween smallest and largest valuesbetween smallest and largest values

Measures of Shape:Measures of Shape: Symmetric or skewedSymmetric or skewed

Right-Skewed or

Positively SkewedLeft-Skewed or

Positive Skew-ness Symmetric

Mean = Median = ModeMean Median Mode Median MeanMod

e


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Box plotBox plot graphical presentation ofgraphical presentation of

CTMCTM


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Central tendencyCentral tendency

measures summarymeasures summary Discussed Measures ofDiscussed Measures of Central TendencyCentral Tendency Mean, Median, ModeMean, Median, Mode Addressed Measures ofAddressed Measures ofVariationVariation The RangeThe Range,, Variance,Variance, Standard Deviation, Coefficient ofStandard Deviation, Coefficient ofVariationVariation DeterminedDetermined ShapeShape of Distributionsof Distributions Symmetric or SkeweSymmetric or SkeweddCoefficient of skewnessCoefficient of skewness

Mean= Median =ModeMean Median Mode Mode Median Mean

central tendency measures lectures 3 4 5

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