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Standard Deviation and Variance
13. Laguerre-Samuelson inequality


Using standard deviation, we can estimate where the elements of a dataset are located. Suppose we have n elements, \overline{x} is the arithmetic mean, and SD is the standard deviation. By the Laguerre-Samuelson inequality we know that elements from this dataset are located in the interval:

[ \overline{x}-SD\sqrt{n-1}; \overline{x}+SD\sqrt{n-1} ]

Let's see what this means for the particular set sizes where the arithmetic mean is zero (\overline{x}=0) and standard deviation equals one (SD = 1) or two (SD = 2).

n Interval for SD = 1 Interval for SD = 2
10 [-3.00;3.00] [-6.00;6.00]
50 [-7.00;7.00] [-14.00;14.00]
100 [-9.95;9.95] [-19.90;19.90]
500 [-22.34;22.34] [-44.68;44.68]
1000 [-31.61;31.61] [-63.21;63.21]