1. Prof. Rajkumar Teotia
Institute of Advanced Management and Research (IAMR)
Address: 9th Km Stone, NH-58, Delhi-Meerut Road, Duhai,Ghaziabad (U.P)
- 201206
Ph:0120-2675904/905 Mob:9999052997 Fax: 0120-2679145
e mail: rajkumarteotia@iamrindia.com
2.
3. Dispersion measures the extent to which the items vary
from some central value. It may be noted that the measures
of dispersion measure only the degree (the amount of
variation) but not the direction of variation.
The various measures of central value give us one single
figure that represents the entire data. But the average alone
cannot adequately describe a set of observations, unless all
the observations are the same. It is necessary to describe
the variability or dispersion of the observations.
4. A good measure of dispersion should possess the following
properties
It should be simple to understand.
It should be easy to compute.
It should be rigidly defined.
It should be based on each and every item of the distribution.
It should be amenable to further algebraic treatment.
It should have sampling stability.
Extreme items should not unduly affect it.
5. Range
Inter-quartile range or Quartile Deviation
Mean deviation or Average Deviation
Standard Deviation
Lorenz curve.
6. Range is defined as the difference between the value of largest
item and the value of smallest item included in the distribution.
Measures of range may be absolute or relative. Formula for
calculating range is
Range = L – S
Where, L = Largest value
S = Smallest Value
Coefficient of range = L – S
L + S
7. MERITS OF RANGE
It should be simple to understand.
It should be easy to compute.
It should be rigidly defined
LIMITATIONS OF RANGE:-
It is based only on two items and does not cover all the items
in a distribution.
It is subject to wide fluctuations from sample to sample based
on the same population.
It fails to give any idea about the pattern of distribution.
Finally, in the case of open-ended distributions, it is not
possible to compute the range.
8. Quartile deviation is half of the difference between upper
quartile (Q3) and lower quartile (Q1). Quartile deviation
indicates the average amount by which the two quartiles differ
from the median. In symmetrical distribution the two quartiles
(Q1 and Q3) are equidistant from the median.
Symbolically, inter quartile range = Q3- Ql
9. Many times the inter quartile range is reduced in the form of
semi-inter quartile range or quartile deviation as shown below:
Semi inter quartile range or Quartile deviation = (Q3 – Ql)
2
Coefficient of Quartile Deviation = (Q3 – Ql)
(Q3 + Ql )
10. MERITS OF QUARTILE DEVIATION
The following merits are entertained by quartile deviation:
As compared to range, it is considered a superior measure
of dispersion.
In the case of open-ended distribution, it is quite suitable.
Since it is not influenced by the extreme values in a
distribution, it is particularly suitable in highly skewed or
erratic distributions.
.
11. LIMITATIONS OF QUARTILE DEVIATION
Like the range, it fails to cover all the items in a
distribution.
It is not amenable to mathematical manipulation.
It varies widely from sample to sample based on the same
population.
Since it is a positional average, it is not considered as a
measure of dispersion. It merely shows a distance on scale
and not a scatter around an average. In view of the above-mentioned
limitations, the inter quartile range or the
quartile deviation has a limited practical utility
12. Average deviation is obtained by calculating the absolute
deviations of each observation from median or mean and then
averaging these deviations by taking their arithmetic mean.
Average deviation is denoted by A.D
13. In case deviation is taken from median:-
A.D (Median) = Σ X - Median
A.D
Median
N
Coefficient of A.D (Median) =
14. In case deviation is taken from mean:-
A.D (Mean) = Σ X - Mean
N
Coefficient of A.D (Mean) = A.D
Mean
15. In case deviation is taken from median:-
A.D (Median) = Σf X - Median
N
Coefficient of A.D (Median) = A.D
Median
16. A.D (Mean) = Σf X - Mean
N
Coefficient of A.D (Mean) = A.D
Mean
17. MERITS OF MEAN DEVIATION
A major advantage of mean deviation is that it is simple to
understand and easy to calculate.
It takes into consideration each and every item in the
distribution. As a result, a change in the value of any item
will have its effect on the magnitude of mean deviation.
The values of extreme items have less effect on the value
of the mean deviation.
As deviations are taken from a central value, it is possible
to have meaningful comparisons of the formation of
different distributions.
18. LIMITATIONS OF MEAN DEVIATION
It is not capable of further algebraic treatment.
At times it may fail to give accurate results. The mean
deviation gives best results when deviations are taken from the
median instead of from the mean. But in a series, which has
wide variations in the items, median is not a satisfactory
measure.
Strictly on mathematical considerations, the method is wrong
as it ignores the algebraic signs when the deviations are taken
from the mean.
19. The standard deviation measures the absolute variation of a
distribution. The greater the amount of variation, the greater the
standard deviation, for the greater will be the magnitude of the
deviation on the values from their mean.
A small standard deviation means a high degree of uniformity
of the observations as well as homogeneity of a series, a large
standard deviation means just opposite.
Hence standard deviation is extremely useful in judging the
representativeness of the mean. It is denoted by the small
Greek letter σ (read as sigma).
24. Coefficient of variation:-
Coefficient of variation is calculated by the following formula
Where,
C.V = σ x 100
X
C.V = coefficient of variation
σ = standard deviation.
X = Arithmetic Mean.
25. Relationship between standard deviation and variance
If we square standard deviation we get variance
OR
σ = Variance
Variance = σ2
26. Just as it is possible to compute combined mean of two or
more than two groups. Similarly we can also compute
combined standard deviation of two groups is denoted by the
following formula.
σ12 = N1σ1
2 + N2σ2
2 + N1d1
2 + N2d2
2
N1 + N2
27. Where,
σ12 = combined standard deviation of two groups.
σ1 = standard deviation of first group.
σ2 = standard deviation of second group
N1 = number of items in first group.
N2 = number of items in second group.
d1 = X1 – X12 And d2 = X2 – X12
28. PROPERTIES OF THE STANDARD DEVIATION:-
The sum of squares of the deviation of all observation from arithmetic mean is
Minimum.
The standard deviation of the first n natural numbers can be obtained by the
following formula:
σ = 1 (N 2 – 1)
12
Thus standard deviation of natural numbers 1 to 10 will be
σ = 1 (10 2 – 1) = 2.87
12
29. Standard deviation is independent of change of origin but not scale.
For a symmetrical distribution, the following area relationships
hold good.
It enables us to determine as to how far individual items in a
distribution deviate from its mean. In a symmetrical, bell-shaped
curve:
(i) About 68.27 percent of the values in the population fall within: + 1
standard deviation from the mean.
(ii) About 95.45 percent of the values will fall within +2 standard
deviations from the mean.
(iii) About 99.73 percent of the values will fall within + 3 standard
deviations from the mean.
31. This measure of dispersion is graphical. It is known as the
Lorenz curve named after Dr. Max Lorenz. It is generally used
to show the extent of concentration of income and wealth. The
steps involved in plotting the Lorenz curve are:
Convert a frequency distribution into a cumulative frequency
table.
On the X – axis, start from 0 to 100 and take the percent of
variable.
On the Y – axis, start from 0 to 100 and take the percent of
variable.
32. Draw a diagonal line joining 0 with 100. This is known as line
of equal distribution. Any point on this line shows the same
percent on X as on Y
Plot the various points corresponding to X and Y and join
them. The distribution so obtained, unless it is exactly equal,
will always curve below the diagonal line. If two curves of
distribution are shown on the same Lorenz presentation, the
curve that is farthest from the diagonal line represents the
greater inequality.
33. Figure 3.1 shows two Lorenz curves by way of illustration. The straight line
AB is a line of equal distribution, whereas AEB shows complete inequality.
Curve ACB and curve ADB are the Lorenz curves
