Standard deviation for grouped data, Applied Statistics

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Grouped data

For grouped data, the formula applied is

Where f = frequency of the variable, μ= population mean.

Example 17

A security analyst studied hundred companies and obtained the following Return on Investment (ROI) data for the year 20x3.

Returns %	0-10	10-20	20-30	30-40
No. of companies	19	32	41	8

We can find how the ROI of the company varies with the mean ROI by calculating the standard deviations for the above data.

The steps involved are:

Find mean for grouped data.
Find deviations from mean for grouped data.
Find squares of the above deviations.
Total the squared deviations taking frequency into account.
Calculate square root.

Return on investment	Mid-point	No. of companies	Deviation
%	X	f	fX	X - μ	f(X - μ )²
0-10	5	19	95	-13.8	3618.36
10-20	15	32	480	-3.8	462.08
20-30	25	41	1025	6.2	1576.04
30-40	35	8	280	16.2	2099.52
Total		100	1880		7756.00

Mean

2299_standard dviation for grouped data1.png

18.8%.

Standard Deviation	=
	=		=8.81%

Thus, the standard deviation for the return on investment is 8.8%.

In such a calculation, we always assume that all the observations in a class interval are located at the mid-point of the class. For example, the first class interval has mid-point 5 and frequency 19. Hence the assumption is that all the 19 companies have an ROI of exactly 5%.

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