Standard Deviation Properties
The Standard deviation has some very important mathematical properties which considerably enhance its utility in the statistical work.
Combined standard deviation: - As it is possible to compute the combined mean of two or more than two groups. Similarly we can also compute the combined standard deviation of two or more groups. The combined standard deviation is represented by σ12 and is computed as follows:
σ12 = √N1σ12 + N2σ22 + N1d12 + N2d22/N1 + N2
σ12 = combined standard deviation
σ1 = standard deviation of first group
σ2 = standard deviation of second group
d1 = | X1 - X12 | & d2 = |X2 - X12 |
The above formula can be extended to find out the standard deviation of three or more groups. For e.g., the combined standard deviation of three groups would be:
σ123 = √(N1σ12 +N2σ22 +N3σ32 + N1d12 +N2d22 +N3d32)/ N1 + N2 + N3
Where, d1 | = X1 - X123 |d2 | = X2 - | X123 |; d3 = X3 |- X123.
The sum of the squares of the deviations of items in the series from their arithmetic mean is minimum. In another words, the sum of the squares of the deviations of items of any series from a value other than the arithmetic mean would always be greater. That's the reason why standard deviation is always computed from the arithmetic mean.
The standard deviation enables us to determine, with a big deal of accuracy, where the values of a frequency distribution located, with the help of Tchebycheff's theorem given by the mathematician P.L. Tchebycheff's (1821-1894), no matter what the shapes of the distribution is at least 75 percent of the values will fall within ± 2 standard deviations from the mean of the distribution, and at least 89 percent of the values will be within ± 3 standard deviations from the mean. With the help of normal curve we can measure even with great precision the number of items that fall within the specific ranges.
For a symmetrical distribution,
Mean ± 1σ covers 68.270% of the items.
Mean ± 2σ covers 95.45% of the items.
Mean ± 3σ covers 99.73% of the items.