Editing Standard deviation (section)

===Uncorrected sample standard deviation===
The formula for the ''population'' standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by ''s''<sub>''N''</sub>, is known as the ''uncorrected sample standard deviation'', or sometimes the ''standard deviation of the sample'' (considered as the entire population), and is defined as follows:<ref name=":1">{{Cite web| last=Weisstein |first=Eric W.|title=Standard Deviation |url=https://mathworld.wolfram.com/StandardDeviation.html|access-date=21 August 2020 |website=mathworld.wolfram.com |language=en}}</ref>
<math display="block">s_N = \sqrt{\frac{1}{N} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2},</math>

where <math>\{x_1, \, x_2, \, \ldots, \, x_N\}</math> are the observed values of the sample items, and <math>\bar{x}</math> is the mean value of these observations, while the denominator&nbsp;''N'' stands for the size of the sample: this is the square root of the sample variance, which is the average of the [[squared deviations]] about the sample mean.

This is a [[consistent estimator]] (it converges in probability to the population value as the number of samples goes to infinity), and is the [[maximum likelihood|maximum-likelihood estimate]] when the population is normally distributed.<ref>{{Cite web |title=Consistent estimator |url=https://www.statlect.com/glossary/consistent-estimator |access-date=2022-10-10 |website=www.statlect.com}}</ref> However, this is a [[biased estimator]], as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/''N'', and thus is most significant for small or moderate sample sizes; for <math>N > 75</math> the bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller [[mean squared error]] than the corrected sample standard deviation.