Here, you will learn what are the measures of dispersion in statistics i.e. range, mean deviation, variance and standard deviation with example.
Let’s begin –
Measures of Dispersion :
The dispersion of a statistical distribution is the measure of deviation of its values about the their average(central) value.
It gives an idea of scatteredness of the different values from the average value.
Generally these measures of dispersion are commonly used.
(i) Range
(ii) Mean deviation
(iii) Variance and standard deviation
Range :
The difference between the greatest and least values of variate of a distribution, are called range of that distribution.
If distribution is the grouped distribution, then its range is the difference between upper limit of maximum class and lower limit of a minimum class.
Also, coefficient of range = \({difference of extreme values}\over {sum of extreme values}\)
Mean deviation(M.D.) :
The mean deviation of a distribution is, the mean of absolute value of deviations of variate from their statistical average(Mean, Median, Mode).
If A is any statistical average of a distribution that mean deviation about A is defined as
Mean deviation = \({\sum_{i=1}^{n}{|x_i – A|}}\over n\) (For ungrouped distribution)
Mean deviation = \({\sum_{i=1}^{n}{f_i|x_i – A|}}\over N\) (For frequency distribution)
NOTE : Mean deviation is minimum when it taken about the median.
Example : Find the mean deviation of the numbers 3, 4, 5, 6, 7.
Solution : We have n = 5, \(\bar{x}\) = 5 here.
\(\therefore\) Mean deviation = \({\sum_{i=1}^{n}{|x_i – A|}}\over n\)
= \(1\over 5\)[|3 – 5| + |4 – 5| + |5 – 5| + |6 – 5| + |7 – 5|]
= \(1\over 5\)[2 + 1 + 0 + 1 + 2] = 1.2
Variance and Standard Deviation :
It is defined as the mean of squares of the deviation of variate from their mean. It is denoted by \(\sigma^2\) or var(x).
The positive square root of variance are called the standard deviation. It is denoted by \(\sigma\) or S.D.
Hence standard deviation = + \(\sqrt{variance}\)
Hope you learnt what are measures of dispersion in statistics, learn more concepts of statistics and practice more questions to get ahead in the competition. Good luck!
