## 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.