Editing Normal distribution (section)

=== Confidence intervals ===
{{See also|Studentization|3-sigma rule}}

By [[Cochran's theorem]], for normal distributions the sample mean <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and the sample variance ''s''<sup>2</sup> are [[independence (probability theory)|independent]], which means there can be no gain in considering their [[joint distribution]]. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and ''s'' can be employed to construct the so-called ''t-statistic'':
<math display=block>
    t = \frac{\hat\mu-\mu}{s/\sqrt{n}} = \frac{\overline{x}-\mu}{\sqrt{\frac{1}{n(n-1)}\sum(x_i-\overline{x})^2}} \sim t_{n-1}
</math>
This quantity ''t'' has the [[Student's t-distribution]] with {{math|(''n'' − 1)}} degrees of freedom, and it is an [[ancillary statistic]] (independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the [[confidence interval]] for ''μ'';<ref>{{harvtxt |Krishnamoorthy |2006 |p=130 }}</ref> similarly, inverting the ''χ''<sup>2</sup> distribution of the statistic ''s''<sup>2</sup> will give us the confidence interval for ''σ''<sup>2</sup>:<ref>{{harvtxt |Krishnamoorthy |2006 |p=133 }}</ref>
<math display=block>\mu \in \left[ \hat\mu - t_{n-1,1-\alpha/2} \frac{s}{\sqrt{n}},\,
                                   \hat\mu + t_{n-1,1-\alpha/2} \frac{s}{\sqrt{n}} \right]</math>
<math display=block>\sigma^2 \in \left[ \frac{n-1}{\chi^2_{n-1,1-\alpha/2}}s^2,\,
                                        \frac{n-1}{\chi^2_{n-1,\alpha/2}}s^2\right]</math>
where ''t<sub>k,p</sub>'' and {{SubSup|χ|''k,p''|2}} are the ''p''th [[quantile]]s of the ''t''- and ''χ''<sup>2</sup>-distributions respectively. These confidence intervals are of the ''[[confidence level]]'' {{math|1 − ''α''}}, meaning that the true values ''μ'' and ''σ''<sup>2</sup> fall outside of these intervals with probability (or [[significance level]]) ''α''. In practice people usually take {{math|''α'' {{=}} 5%}}, resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''<sup>2</sup>.

Approximate formulas can be derived from the asymptotic distributions of <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and ''s''<sup>2</sup>:
<math display=block>\mu \in
              \left[ \hat\mu - \frac{|z_{\alpha/2}|}{\sqrt n}s,\,
                      \hat\mu + \frac{|z_{\alpha/2}|}{\sqrt n}s \right]</math>
<math display=block>\sigma^2 \in
                   \left[ s^2 - \sqrt{2}\frac{|z_{\alpha/2}|}{\sqrt{n}} s^2 ,\,
                          s^2 + \sqrt{2}\frac{|z_{\alpha/2}|}{\sqrt{n}} s^2 \right]</math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''<sub>''α''/2</sub> do not depend on ''n''. In particular, the most popular value of {{math|''α'' {{=}} 5%}}, results in {{math|{{!}}''z''<sub>0.025</sub>{{!}} {{=}} [[1.96]]}}.