Methods of measure of dispersion

1. MEASURES OF DISPERSION
Presented by:
Sonia gupta

2. Persentation outline
 Introduction of measure of dispersion
 Defination
 Method of dispersion:
• Range
• Quartile deviation
• Mean deviation
• Standard deviation
• Lorenz curve
 Measure of skewness
2

3. What u want to know?
3

4. DEFINITION
According to Bowley “Dispersion is the
measure of the variation of the items”
According to Conar “Dispersion is a
measure of the extent to which the
individual items vary”
4

5. 5
Measures of Dispersion
Which of the
distributions of scores
has the larger
dispersion?
0
25
50
75
100
125
1 2 3 4 5 6 7 8 9 10
0
25
50
75
100
125
1 2 3 4 5 6 7 8 9 10
The upper distribution
has more dispersion
because the scores are
more spread out
That is, they are less
similar to each other

6. METHODS OF MEASURING
DISPERSION
Range
Quartile Deviation
Mean Deviation
Standard Deviation
Lorenz curve
6

7. RANGE
It is defined as the difference between the smallest
and the largest observations in a given set of data.
Formula is R = L – S
The relative measure corresponding to range, called
the coefficient of range.
Formula is Coefficient of R = L - S
L + S
7

8. Ex. Find out the range of the given
distribution: 1, 3, 5, 9, 11
Soluion:
Range = L – S
L= 11 And S= 1
Range = 11 – 1 =10
Coefficient of range = L – S
L + S
= 11 – 1 =10 = 1.2
11+1 12
8

9. QUARTILE DEVIATION
It is the second measure of dispersion, no doubt
improved version over the range. It is based on the
quartiles so while calculating this may require
upper quartile (Q3) and lower quartile (Q1) and
then is divided by 2. Hence it is half of the
deference between two quartiles it is also a semi
inter quartile range.
The formula of Quartile Deviation is
(Q D) = Q3 - Q1
2 9

10. Roll no. 1 2 3 4 5 6 7
marks 12 15 20 28 30 40 50
Solution:-
Q1=Sizeof N+1th item=Sizeof7+1=2
nd
item
4 4
Size of 2nd
item is 15. Thus Q1=15
Q3=Sizeof 3 N+1 thitem=Sizeof 3x8 thitem=6thitem
4 4
Size of 6th
item is 40. Thus Q3= 40
Q.D. =Q3-Q1/2 =40-15/2 =12.5 10
Ex:-

11. MEAN DEVIATION
Mean Deviation is also known as average
deviation. In this case deviation taken from
any average especially Mean, Median or
Mode. While taking deviation we have to
ignore negative items and consider all of
them as positive. The formula is given
below:
11

12. MEAN DEVIATION
The formula of MD is given below
MD = Σ d
N (deviation taken from mean)
MD = Σm
N (deviation taken from median)
12

13. The Mean Deviation (cont.)
EX:
13
Data
(X1)
Deviation
(X1 – X)
Absolute deviation
X - X
13 -1 1
17 +3 3
14 0 0
11 -3 3
15 +1 1
TOTAL = 70 D = 0 D = 8
MEAN = 70/5 , = 14 M.D.= 8/5, = 1.6
MEDIAN= N+1/2th item
=6/2th item , = 14
M.D.= 8/5, = 1.6

14. STANDARD DEVIATION
The concept of standard deviation was first
introduced by Karl Pearson in 1893. The
standard deviation is the most useful and
the most popular measure of dispersion.
Just as the arithmetic mean is the most of all
the averages, the standard deviation is the
best of all measures of dispersion.
14

15. STANDARD DEVIATION
The standard deviation is represented by the Greek
letter (sigma). It is always calculated from the
arithmetic mean, median and mode is not
considered. While looking at the earlier measures
of dispersion all of them suffer from one or the
other demerit i.e.
Range –it suffer from a serious drawback
considers only 2 values and neglects all the other
values of the series.
15

16. STANDARD DEVIATION
Quartile deviation considers only 50% of the item and
ignores the other 50% of items in the series.
Mean deviation no doubt an improved measure but ignores
negative signs without any basis.
Karl Pearson after observing all these things has given us a
more scientific formula for calculating or measuring
dispersion. While calculating SD we take deviations of
individual observations from their AM and then each
squares. The sum of the squares is divided by the number
of observations. The square root of this sum is knows as
standard deviation.
16

17. MERITS OF STANDARD
DEVIATION
Very popular scientific measure of dispersion
From SD we can calculate Skewness, Correlation
etc
It considers all the items of the series
The squaring of deviations make them positive
and the difficulty about algebraic signs which was
expressed in case of mean deviation is not found
here.
17

18. STANDARD DEVIATION
The formula of SD is = √∑d2
N
EX: Calculate Standard Deviation of the following series
X – 40, 44, 54, 60, 62, 64, 70, 80, 90, 96
18

19. Solution :
NO OF YOUNG ADULTS
VISIT TO THE LIBRARY
IN 10 DAYS (X)
d=X - A.M d2
40 -26 676
44 -22 484
54 -12 144
60 -6 36
62 -4 16
64 -2 4
70 4 16
80 14 196
90 24 596
96 30 900
N=10
ΣX=660
Σd2=
3048

20. Standard deviation
AM = ΣX
N
= 660 = 66 AM
10
SD = √∑d2
N
SD =√3048 = 17.46
10 20

21. 21
Measure of Skew
Skew is a measure of symmetry in the
distribution of scores
Positive
Skew
Negative Skew
Normal
(skew = 0)

22. 22
Measure of Skew
The following formula can be used to
determine skew:
( )
( )
N
N
XX
XX
s 2
3
3
∑ −
∑ −
=

23. 23
Measure of Skew
If s3
< 0, then the distribution has a negative
skew
If s3
> 0 then the distribution has a positive
skew
If s3
= 0 then the distribution is symmetrical
The more different s3
is from 0, the greater
the skew in the distribution

24. Thanku
24