Editing Standard deviation (section)

===Population standard deviation of grades of eight students===
Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the [[square root]] of the [[average]] of the squared deviations of the values subtracted from their average value. The marks of a class of eight students (that is, a [[statistical population]]) are the following eight values:
<math display="block">2,\  4,\  4,\  4,\  5,\  5,\  7,\  9.</math>

These eight data points have the [[mean]] (average) of 5:
<math display="block"> \mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = \frac{40}{8} = 5. </math>

First, calculate the deviations of each data point from the mean, and [[square (algebra)|square]] the result of each:
<math display="block">\begin{array}{lll}
    (2 - 5)^2 = (-3)^2 = 9  &&  (5 - 5)^2 = 0^2 = 0 \\
    (4 - 5)^2 = (-1)^2 = 1  &&  (5 - 5)^2 = 0^2 = 0 \\
    (4 - 5)^2 = (-1)^2 = 1  &&  (7 - 5)^2 = 2^2 = 4 \\
    (4 - 5)^2 = (-1)^2 = 1  &&  (9 - 5)^2 = 4^2 = 16. \\
\end{array}</math>

The [[variance]] is the mean of these values:
<!-- Notice:  DO NOT CHANGE the denominator below to 7. If you don't understand why it is 8, read the Talk page. -->
<!-- When you have the entire sample population, you use 'n' as the denominator. 'n-1' is only used if your data represents a SAMPLING from the entire population. -->
<math display="block">\sigma^2 = \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = \frac{32}{8} = 4.</math><!-- DO NOT CHANGE the denominator to 7. See talk page-->

and the ''population'' standard deviation is equal to the square root of the variance:
<math display="block">\sigma = \sqrt{ 4 } = 2.</math>

This <!-- 1.  the ''population'' standard deviation, 2. "the ''population'' standard deviation is equal to the square root of the [[variance]]" --> formula is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, there were 8 students randomly and independently chosen from a student population of 2&nbsp;million), then one divides by {{nowrap|7 (which is ''n'' − 1)}} instead of {{nowrap|8 (which is ''n'')}} in the denominator of the last formula, and the result is <math display="inline">s = \sqrt{32/7} \approx 2.1.</math> In that case, the result of the original formula would be called the ''sample'' standard deviation and denoted by <math display="inline">s</math> instead of <math>\sigma.</math> Dividing by <math display="inline">n-1</math> rather than by <math display="inline">n</math> gives an unbiased estimate of the variance of the larger parent population. This is known as ''[[Bessel's correction]]''.<ref>{{MathWorld|urlname=BesselsCorrection|title=Bessel's Correction}}</ref><ref>{{Cite web|title=Standard Deviation Formulas|url=https://www.mathsisfun.com/data/standard-deviation-formulas.html|access-date=21 August 2020|website=www.mathsisfun.com}}</ref> Roughly, the reason for it is that the formula for the sample variance relies on computing differences of  observations from the sample mean, and the sample mean itself was constructed to be as close as possible to the observations, so just dividing by ''n'' would underestimate the variability.