What is meant by Dispersion? The extent to which individual observations spread out from the central value is called dispersion. There are various measures which are used to express the degree of variation quantitatively. These measures are called measures of variation. Let’s discuss in detail measures of dispersion:-
MEASURES OF DISPERSION
There are two kinds of measures of dispersion:
(a) Absolute Measures of Dispersion
(b) Relative Measures of Dispersion
Measures of Absolute Dispersion Q3 class
(i) Range (ii) Quartile Deviation
(iii) Mean Deviation (iv) Standard Deviation and Variance
Range is the difference between the largest observation and the smallest observation in two lies between the set of data. Suppose X m is the largest observation and Xo is the smallest observation, the range denoted by R is given by: Q3 = 69
Example: Find the range of following data:
8, 10, 5, 4, 12, 15 and 9.
Solution: I’m 15 and Xo = 4 MEAN
R = Xm -Xo= 15-4 = 11 abbrev;
Absolute measures are often presented within original data and can be expressed in units. It just tells us how much spread is there between the values from their mean. Absolute measures present the dispersion in same units or square of units in which the data is recorded. A relative measure is expressed in terms of absolute dispersion relative to the average. A relative measure is free from a unit of measurement.
Quartile Deviation abbreviated Q.D. is half of the difference between third and first quartiles.
Q.D = Q3-Q1 / 2
Mean Deviation is the average of absolute deviations measured from the mean. It is abbreviated as M.D.
STANDARD DEVIATION AND VARIANCE:
Standard deviation is the square root of the mean of squared deviations taken from arithmetic mean. Standard deviation is denoted by the symbol S for sample data and c for population data. Variance is just the square of standard deviation.
S=(For ungrouped Data)
PROPERTIES OF VARIANCE:
Variance remains unchanged If we add or subtract a constant from each observation in a set of data i.e.
Var (X± a) = Var (X), where X is a variable and ” a” is a constant.
(ii) When we multiply or divide each observation in a set of data by a constant, variance is multiplied or divided by the square of the constant i.e.
x var ( i’ X!
Var (ax) = Var (X) and Var ( 7 )
(iii) Suppose we have k subgroups having a number of observations in each group respectively and their means are X k and variances are respectively. We can find the mean of all subgroups with the help of following formula:
The following measures of dispersion are used to calculate the relative dispersion:
(i) Coefficient of Range
(ii) Coefficient of Quartile Deviation
(iii) Coefficient of Mean Deviation
(iv) Coefficient of Variation(C.V.)
Coefficient of Range = x,n-xo / xm-xo
Coefficient of Q. D. Qa-Q1 / Qa-Q1