This document discusses various measures of dispersion used to quantify how spread out or varied the values in a data set are. It describes absolute and relative measures of dispersion. Absolute measures are expressed in the original data units, while relative measures are dimensionless ratios. Some common measures of dispersion discussed include range, quartile deviation, mean deviation, standard deviation, and variance. The document outlines the properties, calculation methods, merits and limitations of each dispersion measure.

1. MATHEMATICAL STATISTICS - I
Prepared by
Ms. R. Umadevi
Assistant Professor
Department of Mathematics
Bon Secours College For Women
Thanjavur

2. MEASURES OF DISPERSION

3. Definition
Dispersion is a measure of variation of the items
Properties
(i) It should be capable of treating it by Algebraic or Statistical techniques
(ii) It should be easy to calculate
(iii) It should be easy to understand
(iv) It must not be affected by different samples or fluctuation of sampling
Types of Measures
(i) Absolute Measures
(ii) Relative Measures

4. Absolute Measure
Absolute measures of dispersion are expressed in same units in
which original data is presented but these measures cannot be used to
compare the variations between the two series.
Relative Measure
Relative measures are not expressed in units but it is a pure number.
It is the ration of absolute dispersion to an appropriate average such as
coefficient of standard deviation or coefficient of mean deviation.

5. Methods of Measuring Dispersion
(i) Range and Coefficient of Range
(ii) Quartile Deviation and Coefficient of Quartile Deviation
(iii) Mean Deviation and Coefficient of Mean Deviation
(iv) Standard Deviation and Coefficient of Standard Deviation

6. Range
It is the simplest of the values of Dispersion. It is merely
the difference between the largest and smallest term.
Range = L – S
Coefficient of Range =
It is also known as Ratio of range or coefficient of
Scatteredness.
S
L
S
L



7. Merits or Uses
(i) It is easiest to calculate and simplest to understand.
(ii) It is one of those measures which are rigidly defined.
(iii) It gives us the total picture of the problem even with a single glance.
(iv) It is used to check the quality of a product for quality control. Range
plays an important role in preparing R – charts, thus quality is
maintained.

8. (v) The idea about the price of Gold and Shares is also made
taking care of the range in which prices have moved for the past
some periods.
(vi) Meteorological department also makes forecasts about the
weather by keeping range of temperature in view.

9. Demerits or Limitations
(i) Range is not based on all the terms.
(ii) Due to above reason range is not a reliable measure of
dispersion.
(iii) Range does not change even the least even if all other, in
between, terms and variables are changed.

10. (iv) Range is too much affected by fluctuation of sampling.
(v) It does not tell us anything about the variability of
other data.
(vi) For open – end intervals, range is indeterminate.

11. Quartile Deviation
This measure gives a little more knowledge about the
distribution which the range does not give.
Inter Quartile Range = Q3 – Q1
Quartile Deviation =
Q1 = Size of
Q3 = Size of
2
1
3 Q
Q 
term
N
th





 
4
1
term
N
th





 
4
)
1
(
3

12. Merits or Uses
(i) It can be easily calculated and simply understood.
(ii) It does not involve much mathematical difficulties.
(iii) As it takes middle 50% terms hence it is a measure better than range.
(iv) It is not affected by extreme terms as 25% of upper and 25% of
lower terms are left out.

13. Demerits or Limitations
(i) As Q1 and Q3 are both positional measures hence are not
capable of further algebraic treatment.
(ii) Calculation are much more, but the result obtained is not of
much importance.
(iii) It is too much affected by fluctuations of sample.

14. (iv) 50% terms play no role, first and last 25% items ignored may
not give reliable result.
(v) If the values are irregular, then result is affected badly.

15. Mean Deviation
Mean Deviation of a set of observations of a series is the
arithmetic mean of all the deviations, without their algebraic
signs, taken from its central value.
Mean Deviation =
Coefficient of Mean Deviation =
N
D
f

x
n
o
i
t
a
i
v
e
D
n
a
e
M

16. Merits or Uses
(i) As in case of , every term is taken in account. Hence it is
certainly a better measure than other measures of dispersion.
(ii) Mean deviation is extensively used in other fields such as
Economics, Business, Commerce or any other field of such type.
x

17. (iii) It has least sampling fluctuations as compared to Range,
Percentile and Quartile Deviation.
(iv) When Comparison is needed this is perhaps the best
measure.
(v) Mean Deviation is rigidly defined.

18. Standard Deviation
It is defined as the square root of the mean of the squared
deviation from the actual mean. It is also called the root mean
square deviation.
S.D. =
Coefficient of S.D. =
N
x
x
  2
)
(
x


19. Variance
It is the square of standard deviation.
Variance = σ 2
Coefficient of Variance
It is used to find the relative change that is said to exist in
two or more series.
C.V = 100

x


20. Merits or Uses
(i) This is the most rigidly defined measure of dispersion and
therefore is dependable.
(ii) It is further capable of algebraic treatment.
(iii) It is based on all the terms or observations hence is more
reliable.

21. Demerits or Limitations
(i) As compared with other measures of dispersion it is more
difficult to compute and not so easy to understand.
(ii) In the case of open end intervals we have to make the
assumption of lower limit of first interval and upper limit of last
interval

22. (iii) As far as S.D. is concerned it does not compare two series itself. We
have to proceed to coefficient of S.D. or C.V. for this purpose.
(iv) The extreme terms make the impact two much, therefore in some
cases coefficient of Q.D. or of M.D. has a certain edge over it. If extreme
item differ largely, then they make a heavy change when deviations are
squared.