Editing Standard deviation (section)

===Confidence interval of a sampled standard deviation===
{{see also|Margin of error|Variance#Distribution of the sample variance|Student's t-distribution#Robust parametric modeling}}
The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by the [[confidence interval]] or CI. 

To show how a larger sample will make the confidence interval narrower, consider the following examples: 
A small population of {{math|{{var|N}} {{=}} 2}} has only one degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45&nbsp;×&nbsp;SD to 31.9&nbsp;×&nbsp;SD; [[Confidence interval#Statistical theory|the factors here are as follows]]:

<math display="block">\Pr\left(q_\frac{\alpha}{2} < k \frac{s^2}{\sigma^2} < q_{1 - \frac{\alpha}{2}}\right) = 1 - \alpha,</math>

where <math>q_p</math> is the {{mvar|p}}-th quantile of the chi-square distribution with {{mvar|k}} degrees of freedom, and {{math|1 − {{var|α}}}} is the confidence level.  This is equivalent to the following:

<math display="block">\Pr\left(k\frac{s^2}{q_{1 - \frac{\alpha}{2}}} < \sigma^2 < k\frac{s^2}{q_{\frac{\alpha}{2}}}\right) = 1 - \alpha.</math>

With {{math|{{var|k}} {{=}} 1}}, {{math|{{var|q}}{{sub|0.025}} {{=}} 0.000982}} and {{math|{{var|q}}{{sub|0.975}} {{=}} 5.024}}. The reciprocals of the square roots of these two numbers give us the factors 0.45 and 31.9 given above.

A larger population of {{math|{{var|N}} {{=}} 10}} has 9 degrees of freedom for estimating the standard deviation. The same computations as above give us in this case a 95% CI running from 0.69&nbsp;×&nbsp;SD to 1.83&nbsp;×&nbsp;SD. So even with a sample population of 10, the actual SD can still be almost a factor 2 higher than the sampled SD. For a sample population {{math|{{var|N}} {{=}} 100}}, this is down to 0.88&nbsp;×&nbsp;SD to 1.16&nbsp;×&nbsp;SD. To be more certain that the sampled SD is close to the actual SD we need to sample a large number of points.

These same formulae can be used to obtain confidence intervals on the variance of residuals from a [[least squares]] fit under standard normal theory, where {{mvar|k}} is now the number of [[Degrees of freedom (statistics)|degrees of freedom]] for error.