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Measures of Dispersion

Prof G R C Nair

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Need

Central value alone do not give a truepicture of the distribution

Dispersion measures how the values arescattered around the central value.

A low value shows they are all clusteredclose to the central value.

A high value shows they are all scatteredaway from the central value

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Range

Range = Largest value Smallest

value = (L- S) Simple, Easy , Quick

Not Accurate/reliable measure, not

based on all data, influenced by extremevalues

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Given any set of numerical observations,order them according to magnitude.

The P th percentile in the ordered set isthat value below which lie P % (P percent)of the observations in the set.

The position of the P th percentile is givenby (n+1)P/100, where n is the total numberof observations in the set.

Percentiles & Quartiles

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A large department storecollects data on sales made byeach of its salespeople. Thenumber of sales made on a givenday by each of 2020 salespeople is

shown on the next slide. Also, thedata has been sorted inmagnitude.

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Sales Sorted Sales

9 66 912 1010 1213 1315 1416 1414 1514 1616 1617 1616 1724 17

21 1822 1818 1919 2018 2120 2217 24

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To find the 90th percentile, determine thedata point in position (n+1)P/100 = (20+1)

(90/100) = 18.9. Thus, the percentile is located at the 18.9thposition.

The 18th observation is 21, and the 19th

observation is also 22. The 90th percentile is a point lying 0.9 ofthe way from 21 to 22 and is thus 21.9.

Percentiles

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Quartiles are the percentage points that breakdown the ordered data set into quarters. The first quartile is the 25th percentile. It isthe point below which lie 1/4 of the data.

The second quartile is the 50th percentile. Itis the point below which lie 1/2 of the data. Thisis also called the median.

The third quartile is the 75th percentile. It is

the point below which lie 3/4 of the data.

The interquartile range is the differencebetween the third and the first quartiles.

Quartiles

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Quartile Deviation

Quartile Deviation is (Q3-Q1)/2 For grouped data,Q1= L1+(0.25N-C)h/fQ3= L3+(0.75N-C)h/f, where,

L1=Lower boundary of first quartile class L3=Lower boundary of third quartile class N= total frequency; h= Class width

C=Cum frequency up to the lower limit ofthe concerned quartile class

f = frequency of the concerned quartileclass

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Sales Sorted Sales

9 66 912 1010 1213 1315 14

16 1414 1514 1616 1617 1616 1724 17

21 1822 1818 1919 2018 2120 22

17 24

First QuartileFirst Quartile

MedianMedian

Third QuartileThird Quartile

(n+1)P/100(n+1)P/100

(20+1) 1/4=5.25

(20+1) 1/2=10.5

(20+1) 3/4=15.75

13+(.25)(1)=13.25

16+(.5)(0) = 16

18+(.75)(1)=18.75

QuartilesQuartiles

Example

Position

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SortedSales Sales Rank9 6 16 9 212 10 310 12 413 13 515 14 616 14 714 15 814 16 916 16 1017 16 1116 17 1224 17 1321 18 1422 18 1518 19 1619 20 1718 21 1820 22 1917 24 20

First Quartile

Third Quartile

Q1 = 13 + (.25)(1) = 13.25

Q3 = 18+ (.75)(1) = 18.75

Minimum

Maximum

Range Maximum-Minimum24 - 6 = 18

InterquartileRange

Q3 - Q1 =18.75 - 13.25 = 5.5

Range and Interquartile Range

QuartileDeviation

(Q3 - Q1)/2 = 2.75

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ean deviation

Mean deviation !7 |x Q|/N

For grouped data,7 f |x Q|/N

Simple, easy, considers all data,

But less reliable as it ignores sign, notconducive for mathematical treatment

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Variance

The variance is the average of

the squared deviations from thepopulation mean.

All values are used in the calculation.Not influenced by extreme values.The units are awkward, the square ofthe original units.

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The ages of afamily are: 2, 18, 34, 42

What is thevariance?

24

4

96!!

7!

N

XQ

2364

944

4

2442...242)( 222

2

!!

!

7!

N

X Q

W

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Standard Deviation

Standard deviation is the squareroot of the variance.

Find the Standard deviation for thelast problem

36.152362

!!! WW

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Short cut Formula

Std deviation

Where, d = (m - A) / i

i = Class interval m = mid value of class

A = Assumed Mean

Wfd 2

= ix [ ]-{7 7fd

2

N N }

1/2

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Example HWans -hidden

A factory produces bulbs, whose length of lifewas found to be as given in the table below.Find the mean life and the std deviation bynormal and by short cut method.

Life No of lamps 500-700 5 700-900 11 900-1100 26

1100-1300 10 1300-1500 8 Mean =1016.67 Assumed mean = 1000

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X d d2 f fd fd2

600 -2 4 5 -10 20 800 -1 1 11 -11 11

1000 0 0 26 0 0

1200 1 1 10 10 10

1400 2 4 8 16 32

7 60 5 73

Std deviation = 200{( 73/60) - (5/60)2}1/2

= 219.8

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Coefficient ofVariation

The coefficient ofvariation is theratio of thestandard deviationto the arithmeticmean, expressedas a percentage:

CV

W

!

(100%)

Q

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Combined Values

Combined Mean of 2 groups

Q = Q1N1+ Q2 N2

N1+N2Combined Variance

W 2 = N1 W12 + N2 W22 + N1d12 + N2d22

N1+N2

where, d1= QQd2 = QQ

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Chebyshevs Rule

Irrespective of the shape of thedistribution curve, at least 75 % ofvalues will fall between +/- 2W and89% within +/- 3W from the mean.

% age data with in +/- k times s of

the mean will be at least (1-1/k2)x100