Editing Prediction interval (section)

=== Known mean, known variance ===
{{see also|68–95–99.7 rule}}

A prediction interval [''ℓ'',''u''] for a future observation ''X'' in a normal distribution ''N''(''μ'',''σ''<sup>2</sup>) with known [[mean]] and [[variance]] may be calculated from

:<math>\gamma=P(\ell<X<u)=P\left(\frac{\ell-\mu} \sigma < \frac{X-\mu} \sigma < \frac{u-\mu} \sigma \right)=P\left(\frac{\ell-\mu} \sigma < Z < \frac{u-\mu} \sigma \right),</math>

where <math>Z=\frac{X-\mu}{\sigma}</math>, the [[standard score]] of ''X'', is distributed as  standard normal.

Hence

:<math>\frac{\ell-\mu} \sigma = -z, \quad \frac{u-\mu} \sigma = z,</math>

or

:<math>\ell=\mu-z\sigma, \quad u=\mu+z\sigma,</math>

with ''z'' the [[quantile]] in the standard normal distribution for which:

:<math>\gamma=P(-z<Z<z).</math>
or equivalently;

:<math>\tfrac 12(1-\gamma)=P(Z>z).</math>

{|class="wikitable" align="left" style="margin-right:1em;"
! Prediction<br> interval !! z
|-
| 75% || 1.15<ref name=MedicalStatisticsA2>Table A2 in {{Harvtxt|Sterne|Kirkwood|2003|p=472}}</ref>
|-
| 90% || 1.64<ref name=MedicalStatisticsA2/>
|-
| 95% || 1.96<ref name=MedicalStatisticsA2/>
|-
| 99% || 2.58<ref name=MedicalStatisticsA2/>
|}
[[File:Standard score and prediction interval.svg|thumb|250px|right|Prediction interval (on the [[y-axis]]) given from z (the quantile of the [[standard score]], on the [[x-axis]]). The y-axis is logarithmically compressed (but the values on it are not modified).]]

The prediction interval is conventionally written  as:
:<math>\left[\mu- z\sigma,\  \mu + z\sigma \right]. </math>

For example, to calculate the 95% prediction interval for a normal distribution with a mean (''μ'') of 5 and a standard deviation (''σ'') of 1, then ''z'' is approximately 2. Therefore, the lower limit of the prediction interval is approximately 5&nbsp;‒&nbsp;(2&sdot;1) = 3, and the upper limit is approximately 5&nbsp;+&nbsp;(2&sdot;1) = 7, thus giving a prediction interval of approximately 3 to 7.

[[File:Cumulative distribution function for normal distribution, mean 0 and sd 1.svg|270px|thumb|right|Diagram showing the [[cumulative distribution function]] for the normal distribution with mean (''μ'') 0 and variance (''σ''<sup>2</sup>)&nbsp;1. In addition to the [[quantile function]], the prediction interval for any standard score can be calculated by (1&nbsp;&minus;&nbsp;(1&nbsp;&minus;&nbsp;<span style="font-size:100%;">Φ</span><sub>''μ'',''σ''<sup>2</sup></sub>(standard score))&sdot;2). For example, a standard score of ''x''&nbsp;=&nbsp;1.96 gives <span style="font-size:100%;">Φ</span><sub>''μ'',''σ''<sup>2</sup></sub>(1.96)&nbsp;=&nbsp;0.9750 corresponding to a prediction interval of (1&nbsp;&minus;&nbsp;(1&nbsp;&minus;&nbsp;0.9750)&sdot;2) =&nbsp;0.9500&nbsp;=&nbsp;95%.]]