Measures of Dispersion

Measures of Dispersion

The measures of any set of data show the range of the data or the amount of variation is known as variability or dispersion.
It is already mentioned earlier that if the values of data in a set are widely dispersed then the different measures of central tendency cannot represent the values of the data accurately. Likewise, it has also been mentioned that if the data are close together then the measures of central tendency may represent each value of data in the set more precisely. While working on various tasks using statistics, it is necessary to find the different measures of dispersion. Here, the following four measures of dispersion are included:

• Range
• Mean deviation
• Quartile Deviation
• Standard Deviation

Range

The difference between the highest and the lowest values of the variable of any set of data is called the range. The range tells us how far the values of data are spread out.

In any variation,
If, the Largest item = L
the smallest item = S, then
Range (R) = L - S

Coefficient of Range
A range is an absolute value. Hence, to compare variations between two distributions we generally use the relative value which is calculated using the following formula. This relative value is called the coefficient of range.
Coefficient of Range = \(\frac{L - S}{L + S}\)

Merits of range

• Rigidly identified
• Simple understand and easy to estimate
• Required minimum time for variability

Demerits of range

• Changed by variation of sampling
• Changed by extreme value
• Cannot be used for open-end distributions
• Not suitable for more mathematical treatment

Quartile Deviation

The difference between the first quartile (Q₁) and the third quartile (Q₃) is known as the interquartile range. The half of the interquartile range is called semi- interquartile range or quartile deviation.
Thus, if Q₁and Q₃ represent the first and the third quartile respectively,
(Q.D.) = \(\frac {Q_3 - Q_1}2\)

To find the quartile deviation according to this formula, we need to find Q₁and Q₃. For this, let us recall the formulas to find Q₁ and Q₃.
For individual and discrete series:
Q₁ = (\(\frac{N+1}{4}\))^th term
Q₃ = [\(\frac {3(N+1)}{4}\)]^th term
where, N = no. of terms in the series.

Similarly,
for continuous series, the quartiles are found as follows:
Q₁ = (\(\frac N4\))^th term
Q₃ = (\(\frac {3N}4\))^th term
From this the class interval in which Q₁ and Q₃ lies can be found. To find the quartiles the following formulas are used:
Q₁ = L + \(\frac {\frac N4- c.f.}f\) × i
where, L = lower limit of Q₁ class
c.f. = cumulative frequency of the class preceding the class containing Q₁
f = frequency of the class
i = class size

Similarly,
Q₃ = L + \(\frac {\frac {3N}4- c.f.}f\) × i

Coefficient of Quartile Deviation
As the quartile deviation is an absolute value, we need to find the coefficient of quartile deviation to compare this with other data.
Coefficient of Quartile Deviation = \(\frac {Q_3 - Q_1}{Q_3 + Q_1}\)

Merits of Quartile Deviation

• Rigidly described
• Simple to understand and easy to estimate
• Not changed by ultimate values
• Useful to study dispersion in open-end series

Demerits of Quartile Deviation

• Not based on all observation
• Affected by the fluctuations of sampling
• Not competent of extra mathematical practice