Editing Normal distribution (section)

==== Quantile function ====
{{Further|Quantile function#Normal distribution}}

The [[quantile function]] of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the [[probit function]], and can be expressed in terms of the inverse [[error function]]:
<math display=block>
\Phi^{-1}(p) = \sqrt2\operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1).
</math>
For a normal random variable with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, the quantile function is
<math display=block>
F^{-1}(p) = \mu + \sigma\Phi^{-1}(p)
= \mu + \sigma\sqrt 2 \operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1).
</math>
The [[quantile]] <math display=inline>\Phi^{-1}(p)</math> of the standard normal distribution is commonly denoted as {{tmath|z_p}}. These values are used in [[hypothesis testing]], construction of [[confidence interval]]s and [[Q–Q plot]]s. A normal random variable {{tmath|X}} will exceed <math display=inline>\mu + z_p\sigma</math> with probability <math display=inline>1-p</math>, and will lie outside the interval <math display=inline>\mu \pm z_p\sigma</math> with probability {{tmath|2(1-p)}}. In particular, the quantile <math display=inline>z_{0.975}</math> is [[1.96]]; therefore a normal random variable will lie outside the interval <math display=inline>\mu \pm 1.96\sigma</math> in only 5% of cases.

The following table gives the quantile <math display=inline>z_p</math> such that {{tmath|X}} will lie in the range <math display=inline>\mu \pm z_p\sigma</math> with a specified probability {{tmath|p}}. These values are useful to determine [[tolerance interval]] for [[Sample mean and sample covariance#Sample mean|sample averages]] and other statistical [[estimator]]s with normal (or [[asymptotic]]ally normal) distributions.<ref>{{Cite book |last=Vaart |first=A. W. van der |url=http://dx.doi.org/10.1017/cbo9780511802256 |title=Asymptotic Statistics |date=1998-10-13 |publisher=Cambridge University Press |doi=10.1017/cbo9780511802256 |isbn=978-0-511-80225-6}}</ref> The following table shows <math display=inline>\sqrt 2 \operatorname{erf}^{-1}(p)=\Phi^{-1}\left(\frac{p+1}{2}\right)</math>, not <math display=inline>\Phi^{-1}(p)</math> as defined above.

{| class="wikitable" style="text-align:left;margin-left:24pt;border:none;"
! {{tmath|p}} !! <math display=inline>z_p</math>
| rowspan="8" style="border:none;"|&nbsp;
! {{tmath|p}} !! <math display=inline>z_p</math>
|-
| 0.80  || {{val|1.281551565545}} || 0.999       || {{val|3.290526731492}}
|-
| 0.90  || {{val|1.644853626951}} || 0.9999      || {{val|3.890591886413}}
|-
| 0.95  || {{val|1.959963984540}} || 0.99999     || {{val|4.417173413469}}
|-
| 0.98  || {{val|2.326347874041}} || 0.999999    || {{val|4.891638475699}}
|-
| 0.99  || {{val|2.575829303549}} || 0.9999999   || {{val|5.326723886384}}
|-
| 0.995 || {{val|2.807033768344}} || 0.99999999  || {{val|5.730728868236}}
|-
| 0.998 || {{val|3.090232306168}} || 0.999999999 || {{val|6.109410204869}}
|}

For small {{tmath|p}}, the quantile function has the useful [[asymptotic expansion]]
<math display=inline>\Phi^{-1}(p)=-\sqrt{\ln\frac{1}{p^2}-\ln\ln\frac{1}{p^2}-\ln(2\pi)}+\mathcal{o}(1).</math>{{citation needed|date=February 2023}}