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Grouped data
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
Deviation
%
X
f
fX
X - μ
f(X - μ )2
5
95
-13.8
3618.36
15
480
-3.8
462.08
25
1025
6.2
1576.04
35
280
16.2
2099.52
Total
100
1880
7756.00
=
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%.
Sample Standard Deviation So far, we discussed the population standard deviation. Now, let us switch to sample standard deviation(s) that is analogous to the population stand
Mean Absolute Deviation To avoid the problem of positive and negative deviations canceling out each other, we can use the Mean Absolute Deviation which is given by
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