1. MEASURE OF DISPERSION

2. 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

3. Characteristics of a good measure of dispersion:
An ideal measure of dispersion is expected to possess
the following properties
1.It should be rigidly defined
2. 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, namely
1.Absolute measure of dispersion
2.Relative measure of dispersion.

4. Absolute measure of dispersion indicates the amount of
variation 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 in
rainfall in mm. On the other hand relative measures of
dispersion are free from the units of measurements of the
observations. They are pure numbers. They are used to
compare the variation in two or more sets, which are having
different units of measurements of observations.
The various absolute and relative measures of dispersion are listed
below.

5. Absolute measure Relative measure
1. Range 1.Co-efficient of Range
2.Quartile deviation 2.Co-efficient of
Quartile deviation
3.Mean deviation 3. Co-efficient of Mean
deviation
4.Standard deviation 4.Co-efficient of variation
Range 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.

6. In individual observations and discrete series, L and S
are easily identified. In continuous series, the following two
methods are followed.
Method 1:
L = Upper boundary of the highest class
S = 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

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

8. Co-efficient of Range
Merits:
1. It is simple to understand.
2. It is easy to calculate.
3. In certain types of problems like quality control, weather
forecasts, share price analysis, et c., range is most widely
used.
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 60
0.1111
75 60
L S
L S

9. Quartile Deviation ( Q.D) :
Definition: Quartile Deviation is half of the difference
between the first and third quartiles. Hence, it is called Semi
Inter 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
C o-efficient of Q .D =
Q Q
Q Q
3 1
2
Q Q
3 1
.
2
Q Q
Q D
3 1
3 1
Q Q
Coefficient of quartile Deviation
Q Q

10. Example 3: Find the Quartile Deviation for the following data:
391, 384, 591, 407, 672, 522, 777, 733, 1490, 2488
1
3 3 2.75 8.25
4
thN
item

11. And for so many problems will be discuss in the class.
Merits and Demerits of Quartile Deviation
Merits :
1. It is Simple to understand and easy to calculate
2. 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 2
276.125
Q Q
Q D

12. Example 1
Evaluate an quartile range for the following data:
Farm size
(acre)
No. of Farms Farm size
(acre)
No. of Farms
Below 40 394 161 - 200 169
41 - 80 461 201 - 240 113
81 - 120 391 241 and
above
148
121 - 160 334

13. Farm size (acre) No. of Farms Cumulative
frequency (c f)
Below 40 394 394
41 – 80 461 855
81 – 120 391 1246
121 – 160 334 1580
161 – 200 169 1749
201 – 240 113 1862
241 and above 148 2010

14. 1
3
2010
502.5 , 41 80
4 4
.
(502.5 394)4
41 40 50.41
461
3 3* 2010
1507.5 , 121 160
4 4
3
.
4
152.31
. .
th
th
n
observation This observation lies
n
c f
Q L h acres
f
n
find observation This observation lies
n
c f
Q L h acres
f
Q R 3 1
101.9Q Q acres

15. 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
M ean deviation
n
f x x
M ean deviation
N

16. 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 the
series.
2. Take the deviations of items from average ignoring signs and denote
these deviations by .
100
m ean deviation
C oefficient of m ean D eviation
m ean
If the result is desired in percentage then
m ean deviation
coefficient of m ean deviation
m ean
D

17. 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 coefficients
of M.D.
Solution:
Now arrange the data in ascending order
100, 150, 200, 250, 360, 490, 500, 600, 671
D
1
M edian = Value of ( )
2
thn
item
.
D
Symbolically M D
n
3321
369
9
x
Mean x
n

18. = 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 M d
M .D from mean
1570
174.44
9
D
n
.
Co-efficient of M .D =
M D
x
M .D from median
1561
173.44
9
M .D 173.44
Co-efficient of M .D.= 0.48
M edian 360
D
n
1
( )
2
thn
9 1
2
th
.
D
M D
n

19. • 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

20. Variance and Standard deviation
2
2
2
2
2
2
2
2
i i
Ungrouped data
x x
Variance
n
x
x
n
d d
where d x A
n n

21. 2
2
2
2
2
2
2
2
i i
For grouped data
f x x
Variance
n
f x
x
n
f d f d
where d x A
n n

22. Standard deviation
Standard 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

23. 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

24. EXAMPLE
• The ages of the Dunn family are:
2, 18, 34, 42
What is the variance?
96
24
4
X
2 24 -22 484
18 24 -6 36
34 24 10 100
42 24 18 324
96 0 944
944 / 4 = 236
X x i
X X
2
i
X X

25. EXAMPLE
The hourly wages earned by a sample of five students
are: 7, 5, 11, 8, 6. Find the variance.
4.7
5
37
X
5 7.4 -2.4 5.76
6 7.4 -1.4 1.96
7 7.4 -0.4 0.16
8 7.4 0.6 0.36
11 7.4 3.6 12.96
0 21.2
X X XX
2
XX
21.2 (5) 4.24

26. 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

27. i
x
xxi
2
xxi
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
5
10
50
n
x
x
i
2
2 20
2
10 10
i
x x
2 1.414
Here,

28. 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

29. Now we calculate standard deviation
where =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
i
X X
2
i
X X
2
i i
f X X
x

30. 2
1531.4
. .
80
19.1425
4.3752
i i
f x x
S D
N
2
1531.4i i
f x x

31. Example-2: Find Standard Deviation of
Group Data
Mid point frequency
3 2
5 3
7 2
8 2
9 1
Total 10

32. 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
2
xxf ii
2
xxixxiii
xfi
fi
x
2
2 40
4
10
i i
f x x
n
6
10
60
i
ii
f
xf
x
. . 4 2S D

33. 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

34. 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

35. 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

36. 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 recorded
as 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.

37. 1 2 1 2 1
2
2 2 2 2
1 1 1 2 2 2
12
1
Combined standard 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 2
12
1 2
n
Combined standard deviation
,
n n
n n
where
d x x d x x
x x
and x

38. Q1. For a group of 50 male workers, the mean and standard
deviation 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.

39. Relation Between Different Measures
of Variation
2
3
4
5
5
.
6
6
.
5
5 3
standard . .
4 2
Quartile deviation
M ean Absolute deviation
Quartile deviation M D
M ean Absolute deviation Q D
deviation M D or Q D