MEASURE OF DISPERSION

1- Control the variability : Measuring of variation helps to identify the nature and causes of variation. Such information is useful in controlling the variation.

2- Compare two or more sets of data with respect to their variability: Measures of variation help in the comparison of two or more sets of data with respect to their uniformity or consistency.

3- Facilitate the use of other statistical techniques:Measures of variation facilitate the use of other statistical techniques such as correlation and regression analysis, hypothesis testing, forecasting, quality control and so on.

SIGNIFICANCE OF MEASURING VARIATION

Characteristics of a good measure of dispersion:An ideal measure of dispersion is expected to possessthe following properties1.It should be rigidly defined2. It should be based on all the items.3. It should not be unduly affected by extreme items.4. It should lend itself for algebraic manipulation.5. It should be simple to understand and easy to calculate.Absolute and Relative Measures : There are two kinds of measures of

dispersion, namely1.Absolute measure of dispersion2.Relative measure of dispersion.

Absolute measure of dispersion indicates the amount ofvariation in a set of values in terms of units of observations.For example, when rainfalls on different days are available in mm, any

absolute measure of dispersion gives the variation inrainfall in mm. On the other hand relative measures ofdispersion are free from the units of measurements of theobservations. They are pure numbers. They are used tocompare the variation in two or more sets, which are havingdifferent units of measurements of observations.The various absolute and relative measures of dispersion are listed

below.

Absolute measure Relative measure1. Range 1.Co-efficient of Range2.Quartile deviation 2.Co-efficient of

Quartile deviation3.Mean deviation 3. Co-efficient of Mean

deviation4.Standard deviation 4.Co-efficient of variationRange and coefficient of Range: This is the simplest possible

measure of dispersion and is defined as the difference between the largest and smallest values of the variable.

In symbols, Range = L – S.Where L = Largest value. S = Smallest value.

In individual observations and discrete series, L and Sare easily identified. In continuous series, the following twomethods are followed.Method 1:L = Upper boundary of the highest classS = Lower boundary of the lowest class.Method 2:L = Mid value of the highest class.S = Mid value of the lowest class.Co-efficient of Range :Co-efficient of Range = Example1: Find the value of range and its co-efficient for the

following data.7, 9, 6, 8, 11, 10, 4

L S

L S

Solution:L=11, S = 4.Range = L – S = 11- 4 = 7Co-efficient of Range = = 0.4667Example 2:Calculate range and its co efficient from the followingdistribution. Size: 60-63 63-66 66-69 69-72 72-75 Number: 5 18 42 27 8Solution:L = Upper boundary of the highest class = 75S = Lower boundary of the lowest class = 60Range = L – S = 75 – 60 = 15

11 4

11 4

L S

L S

Co-efficient of RangeMerits:1. It is simple to understand.2. It is easy to calculate.3. In certain types of problems like quality control, weatherforecasts, share price analysis, et c., range is most widelyused. Demerits:1. It is very much affected by the extreme items. 2. It is based on only two extreme observations.3. It cannot be calculated from open-end class intervals.4. It is not suitable for mathematical treatment.5. It is a very rarely used measure.

L S

L S

75 60

75 60

75 600.1111

75 60

L S

L S

Quartile Deviation ( Q.D) :Definition: Quartile Deviation is half of the differencebetween the first and third quartiles. Hence, it is called SemiInter Quartile Range. In Symbols, among the quartiles Q1, Q2

and Q3, the range Q3 - Q1 is called inter quartile range and

, Semi inter quartile range.Co-efficient of Quartile Deviation :

Quartile Deviation and Co-efficient of Quartile Deviation :

3 1

3 1

Co-efficient of Q.D =Q Q

Q Q

3 1

2

Q Q

3 1.2

Q QQ D

3 1

3 1

Q QCoefficient of quartile Deviation

Q Q

Example 3: Find the Quartile Deviation for the following data:391, 384, 591, 407, 672, 522, 777, 733, 1490, 2488

13 3 2.75 8.25

4thN

item

And for so many problems will be discuss in the class.Merits and Demerits of Quartile DeviationMerits :1. It is Simple to understand and easy to calculate2. It is not affected by extreme values.3. It can be calculated for data with open end classes also.Demerits:1. It is not based on all the items. It is based on two positional

values Q1 and Q3 and ignores the extreme 50% of the items

2. It is not amenable to further mathematical treatment.3. It is affected by sampling fluctuations.

3 1 955.25 403.

2 2276.125

Q QQ D

Example 1Evaluate an quartile range for the following data:

Farm size Farm size (acre)(acre)

No. of FarmsNo. of Farms Farm size Farm size (acre)(acre)

No. of FarmsNo. of Farms

Below 40Below 40 394394 161 - 200161 - 200 169169

41 - 8041 - 80 461461 201 - 240201 - 240 113113

81 - 12081 - 120 391391 241 and 241 and aboveabove

148148

121 - 160121 - 160 334334

Farm size (acre)Farm size (acre) No. of FarmsNo. of Farms Cumulative Cumulative frequency (frequency (c fc f))

Below 40Below 40 394394 394394

41 – 8041 – 80 461461 855855

81 – 12081 – 120 391391 12461246

121 – 160121 – 160 334334 15801580

161 – 200161 – 200 169169 17491749

201 – 240201 – 240 113113 18621862

241 and above241 and above 148148 20102010

1

3

2010502.5 , 41 80

4 4

.(502.5 394)4

41 40 50.41461

3 3*20101507.5 , 121 160

4 43

.4

152.31

. .

th

th

nobservation This observationlies

nc f

Q L h acresf

nfind observation This observationlies

nc f

Q L h acresf

Q R

3 1 101.9Q Q acres

Mean Deviation and Coefficient of Mean Deviation: Mean Deviation:The range and quartile deviation are not based on all observations. They

are positional measures of dispersion. They do not show any scatter of the observations from an average. The mean deviation is measure of dispersion based on all items in a distribution.

Mean deviation is the arithmetic mean of the deviations of a series computed from any measure of central tendency; i.e., the mean, median or mode, all the deviations are taken as positive i.e., signs are ignored. According to Clark and Schekade,

“Average deviation is the average amount scatter of the items in a distribution from either the mean or the median, ignoring the signs of the deviations”. x x

Mean deviationn

f x xMean deviation

N

Coefficient of mean deviation: Mean deviation calculated by any measure of central tendency is an absolute measure. For the purpose of comparing variation among different series, a relative mean deviation is required. The relative mean deviation is obtained by dividing the mean deviation by the average used for calculating mean deviation.

Computation of mean deviation – Individual Series :1. Calculate the average mean, median or mode of theseries.2. Take the deviations of items from average ignoring signs and denote

these deviations by .

100

mean deviationCoefficient of mean Deviation

meanIf the result is desired in percentagethen

mean deviationcoefficient of mean deviation

mean

D

3. Compute the total of these deviations, i.e., .4. Divide this total obtained by the number of items.

Example 6: Calculate mean deviation from mean and median for the following data:

100,150,200,250,360,490,500,600,671 also calculate coefficientsof M.D.Solution:Now arrange the data in ascending order100, 150, 200, 250, 360, 490, 500, 600, 671

D

1Median = Value of ( )

2thn

item

.D

Symbolically M Dn

3321369

9

xMean x

n

= value of = value of = value of the 5th item = 360

X

100 269 260

150 219 210

200 169 160

250 119 110

360 9 0

490 121 130

500 131 140

600 231 240

671 302 311

3321 1570 1561

D X X D X Md

M.D from mean

1570174.44

9

D

n

.Co-efficient of M.D =

M D

x

M.D from median

1561173.44

9M.D 173.44

Co-efficient of M.D.= 0.48Median 360

D

n

1( )

2thn

9 1

2

th

.D

M Dn

• Q2. Find the mean absolute deviation from mean for the following frequency distribution of sales (Rs. In thousand) in a co- operative store.

• Ans= 47.05

Sales 50-100 100-150 150-200 200-250 250-300 300-350

No. of days 11 23 44 19 8 7

Variance and Standard deviation

2

2

22

2

222

i i

Ungrouped data

x xVariance

n

xx

n

d dwhere d x A

n n

2

2

22

2

222

i i

For grouped data

f x xVariance

n

f xx

n

f d f dwhere d x A

n n

Standard deviationStandard deviation is a measure of the spread

or dispersion of a set of data.It is calculated by taking the square root of

the variance and is symbolized by S.D.,

In other words

Where variance is denoted by σ2 2. .S D

Interpretation and Uses of the Standard Deviation

• Empirical Rule: • For any symmetrical, bell shaped distribution, approximately 68% of the observations will lie within 1 of the mean (); approximately 95.44% within 2 of the mean (); and approximately 99.7% within 3 of the mean ().

-3 -2 -1 0 +1 +2 +3

EXAMPLE

• The ages of the Dunn family are: 2, 18, 34, 42

What is the variance?

9624

4X

2 24 -22 48418 24 -6 3634 24 10 10042 24 18 32496 0 944

944 / 4 = 236

X x iX X 2

iX X

EXAMPLE

The hourly wages earned by a sample of five students are: 7, 5, 11, 8, 6. Find the variance.

4.75

37X

5 7.4 -2.4 5.766 7.4 -1.4 1.967 7.4 -0.4 0.168 7.4 0.6 0.3611 7.4 3.6 12.96

0 21.2

X X XX 2XX

21.2 (5) 4.24

Example-1: Find Standard Deviation of Ungroup Data

Family No.

1 2 3 4 5 6 7 8 9 10

Size (xi) 3 3 4 4 5 5 6 6 7 7

ixxxi

2xxi

Family No. 1 2 3 4 5 6 7 8 9 10 Total

3 3 4 4 5 5 6 6 7 7 50

-2 -2 -1 -1 0 0 1 1 2 2 0

4 4 1 1 0 0 1 1 4 4 20

510

50

n

xx

i

2

2 202

10 10ix x

2 1.414

Here,

EXAMPLE: Find the standard deviation

Price frequency

15--18 8

18—21 23

21—24 17

24—27 18

27—30 8

30—33 4

33—36 2

Total 80

Now we calculate standard deviationwhere =23.06

•Price frequency Midpoint

15--18 8 16.5 -6.56 43.03 344.24

18—21 23 19.5 -3.56 12.67 291.41

21—24 17 22.5 0.56 0.31 5.27

24—27 18 25.5 2.44 5.95 107.1

27—30 8 28.5 5.44 29.59 236.72

30—33 4 31.5 8.44 71.23 284.92

33—36 2 34.5 11.44 130.87 261.74

Total 80 1531.4

iX X 2iX X 2i if X X

x

21531.4

. .80

19.1425

4.3752

i if x xS D

N

21531.4i if x x

Example-2: Find Standard Deviation of Group Data

Mid point frequency

3 2

5 3

7 2

8 2

9 1

Total 10

Solution

3 2 6 -3 9 18

5 3 15 -1 1 3

7 2 14 1 1 2

8 2 16 2 4 8

9 1 9 3 9 9

Total 10 60 - - 40

2xxf ii 2xxi xxi ii xfifix

2

2 404

10i i

f x x

n

6

10

60

i

ii

f

xfx

. . 4 2S D

Assignment-1:Find standard deviation

Marks 40-50 50-60 60-70 70-80 80-90 90-100 Total

students 6 11 19 17 13 4 70

Data for Assignment-2

Marks

No. of students

0-5 3

5-10 11

10-15 9

15-20 17

20-25 3

25-30 4

Total

Data for Assignment-3

Marks

No. of students

4—8 16

8—12 11

12—16 19

16—20 17

20—24 13

24--28 14

Total 90

Coefficient of Variation

Standard deviation. *100C V

mean

The set of data for which the coefficient of variation is low is said to be more consistent or more stable.

Q1. The weekly sales of two products A and B were recordedas given below:

Product A 59 75 27 63 27 28 56

Product B 150 200 125 310 330 250 225

Find out which of the two shows greater fluctuation in sales.

1 2 1 2 1

2

2 2 2 21 1 1 2 2 2

121

Combinedstandard deviation : The combined standard deviation of two sets

of data containing n mean standard deviation

n n

n

and n observation with x and x and

and respectively is given by

d d

2

12

1 12 1 2 12 2

1 1 2 212

1 2

n

Combinedstandard deviation

,

n n

n n

where

d x x d x x

x xand x

Q1. For a group of 50 male workers, the mean and standarddeviation of their monthly wages are Rs 6,300 and Rs 900 respectively . For a group of 40 female workers, these are Rs 5400 and Rs 600 respectively. Find the standard deviation of monthly wages for the combined group of workers.

Relation Between Different Measures of Variation

2

34

55

.6

6.

55 3

standard . .4 2

Quartile deviation

Mean Absolute deviation

Quartile deviation M D

Mean Absolute deviation Q D

deviation M D or Q D