MEASURES OF DISPERSIONS

1. MEASURES OF
DISPERSIONS

2. MEASURES OF DISPERSIONS
• A quantity that measures the variability
among the data, or how the data one
dispersed about the average, known as
Measures of dispersion, scatter, or
variations.

3. • To know the average variation of different
values from the average of a series
• To know the range of values
• To compare between two or more series
expressed in different units
• To know whether the Central Tendency
truly represent the series or not

4. 2. Common Measures of
Dispersion
• The main measures of dispersion
1. Range
2. Mean deviation or the average deviation
3. The variance & the standard deviation

5. 5
The Range
• The range is defined as the difference
between the largest score in the set of
data and the smallest score in the set of
data, XL - XS
• What is the range of the following data:
4 8 1 6 6 2 9 3 6 9
• The largest score (XL) is 9; the smallest
score (XS) is 1; the range is XL - XS = 9 - 1 =
8

6. Ahad Kabir

7. 1. RANGE
• Example:
1. Find the range in the following data.
31,26,15,43,19,10,12,37
Range = xm – xo 33 = 43 – 10
2. Find the range in the following F.D. (Ungrouped)
5 = 8 – 3
Range 5 = 8 – 3
3. Find the range in the following data.
Range = 60 – 10 = 50
X 3 4 5 6 7 8
f 5 8 12 10 4 2
X 10 - 20 20 - 30 30- 40 40 – 50 50 - 60
f 5 8 12 10 4

8. MEAN (OR AVERAGE) DEVIATION
• It is defined as the “Arithmetic mean of the
absolute deviation measured either from
the mean or median.
• or for ungroup.
• or for grouped.
n
xx
DM
∑ −
=..
N
xxf∑ −
=
N
medianx∑ −
N
medianxf∑ −
=

9. Example :
Sazzad took five exams in a class and had scores of 92, 75, 95, 90,
and 98. Find the mean
deviation for his test scores.
We can say that on the average, Saddam’s test scores deviated by 6 points from
the mean.

10. Atik hasan

11. MEAN (OR AVERAGE) DEVIATION
• Exp: Calculate mean deviation from the FD (Grouped Data).
MD (x) = 33.6 / 20 = 1.68
M.D = 23.72 / 14 = 1.69
X f Class Mark
( x )
f.x I x – 6.57 I f I x – 6.57 I
2 – 4 2 3 6 3.57 7.14
4 - 6 3 5 15 1.57 4.71
6 – 8 6 7 42 0.43 2.58
8 – 10 2 9 18 2.43 4.86
10 – 12 1 11 11 4.43 4.43
Total Σf =14 Σ f.x =92 Σ f I x – 6.57 I =
23.72
=92/14=6.57ẋ

12. • It is an absolute measure.
• It’s relative measure is coefficient of M.D.
• Coefficient of M.D. =
• It is based on all the observed values.
MEAN (OR AVERAGE) DEVIATION
median
DM
or
mean
DM ....

13. THE VARIANCE AND
STANDARD DEVIATION
• It is defined as “The mean of the squares
of deviations of all the observation from
their mean.” It’s square root is called
“standard deviation”.
• Usually it is denoted by (for population of
statistics) S2
(for sample)
• = for ungrouped
2
σ
2
σ
n
xx∑ − 2
)(

14. Bakhtiare Hossain

15. • = for grouped
• It is an absolute measure;
• It is relative measure is coefficient of
variation.
•
• Shortcut method
N
xxf∑ − 2
)(2
σ
100. ×=
µ
σ
VC 100
..
.. ×=
x
DS
VC
22
2








−=
∑∑
N
x
N
x
σ
22
2
.








−=
∑∑
N
fx
N
xf
σ
THE VARIANCE AND
STANDARD DEVIATION

16. VARIANCE AND STANDARD
DEVIATION• Example:
1. Calculate Variance and SD from the FD (Ungrouped Data).
Using Short cut method
var = (564 / 20) - (98 / 20) ^ 2 = 28.2 – 24.01 = 4.09
Sd = √ σ^2 = √ 4.09 = 2.02
X f f.x X^2 f.x^2
2 3 6 4 12
4 9 36 16 144
6 5 30 36 180
8 2 16 64 128
10 1 10 100 100
Total Σf =20 Σf.x = 98 Σ f.x^2=564
22
2
.








−=
∑∑
N
fx
N
xf
σ

17. VARIANCE AND STANDARD
DEVIATION
• Exp: Calculate Variance and Standard deviation from the FD (Grouped Data).
Using Short cut method:
var = (670 /14) - (92 / 14) ^ 2 = 47.85 – 43.18 = 4.67
Sd = √ σ^2 = √ 4.67 = 2.16
X f Class Mark
( x )
f.x x^2 f.x^2
2 – 4 2 3 6 9 18
4 - 6 3 5 15 25 75
6 – 8 6 7 42 49 294
8 – 10 2 9 18 81 162
10 – 12 1 11 11 121 121
Total Σf =14 Σ f.x =92 Σ f.x^2 =670
22
2
.








−=
∑∑
N
fx
N
xf
σ

18. Imran hossain

19. Relative Measures ofRelative Measures of
DispersionDispersion
 Coefficient of Range
 Coefficient of Quartile Deviation
 Coefficient of Mean Deviation
 Coefficient of Variation (CV)

20. Relative Measures of VariationRelative Measures of Variation
Largest Smallest
Largest Smallest
Coefficient of Range
X X
X X
−
=
+
3 1
3 1
Coefficient of Quartile Deviation
Q Q
Q Q
−
=
+
Coefficient of Mean Deviation
MD
Mean
=

21. Coefficient of Variation (CV)Coefficient of Variation (CV)
Can be used to compare two or more
sets of data measured in different
units or same units but different
average size.
100%
X
S
CV ⋅







=

22. Use of Coefficient of VariationUse of Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Standard deviation = $5
but stock B is
less variable
relative to its
price
10%100%
$50
$5
100%
X
S
CVA =⋅=⋅







=
5%100%
$100
$5
100%
X
S
CVB =⋅=⋅







=
Both stocks
have the
same
standard
deviation

23. Arjun Baidya

24. Skewness
A fundamental task in many statistical analyses is to
characterize the location and variability of a data set
(Measures of central tendency vs. measures of dispersion)
Both measures tell us nothing about the shape of the
distribution
It is possible to have frequency distributions which differ
widely in their nature and composition and yet may have
same central tendency and dispersion.
Therefore, a further characterization of the data includes
skewness

25. Positive & Negative Skew
Positive skewness
There are more observations below the mean than
above it
When the mean is greater than the median
Negative skewness
There are a small number of low observations and a
large number of high ones
When the median is greater than the mean

26. Measures of Skew
Skew is a measure of symmetry in the distribution
of scores
Positive
Skew
Negative Skew
Normal
(skew = 0)

27. Measures of Skew

28. Robiul Sarkar

29. The Kurtosis is the degree of peakedness or flatness of a
unimodal (single humped) distribution,
• When the values of a variable are highly concentrated around
the mode, the peak of the curve becomes relatively high; the
curve is Leptokurtic.
• When the values of a variable have low concentration
around the mode, the peak of the curve becomes relatively
flat;curve is Platykurtic.
• A curve, which is neither very peaked nor very flat-toped, it
is taken as a basis for comparison, is called
Mesokurtic/Normal.
Measures of Kurtosis

30. Measures of Kurtosis

31. Measures of Kurtosis
1. If Coefficient of Kurtosis > 3 ----------------- Leptokurtic.
2. If Coefficient of Kurtosis = 3 ----------------- Mesokurtic.
3. If Coefficient of Kurtosis < 3 ----------------- is Platykurtic.
( )
( )
4
22
n X-X
Coefficient of Kurtosis=
X-X 
 
∑
∑