Sample Standard Deviation
Suppose we have a bowl of ten apples with masses in grams.200, 220, 225, 244, 250, 252, 253, 257, 260, 262
The standard deviation formula is
\[\sigma =\sqrt{\frac{\sum x^2 - ( \sum x)^2/n}{n}}\]
.The standard deviation of the masses of these apples is
\[\sigma = \sqrt{\frac{200^2+220^2+225^2+244^2+250^2+252^2+253^2+257^2+260^2+262^2 - \\ (200+220+225+244+250+252+253+257+260+262)^2/10}{10}} =19.43 \]
to 2 decimal places.Suppose now we want as estimate of the standard deviation of all the apples in the country. Obviously we cannot weight them all. What we can do is take a sample of the apples in the country at one time and find the 'sample standard deviation'
\[s= \sqrt{\frac{n}{n-1}} \sigma=\sqrt{\frac{\sum x^2 - ( \sum x)^2/n}{n-1}}\]
Treating the above apple as a sample, we have
\[s=\sqrt{\frac{200^2+220^2+225^2+244^2+250^2+252^2+253^2+257^2+260^2+262^2 - \\ (200+220+225+244+250+252+253+257+260+262)^2/10}{10-1}}=20.47 \]
to two decimal places.