Editing Normal distribution (section)

==== Standard deviation and coverage ====
{{Further|Interval estimation|Coverage probability}}
[[File:Standard deviation diagram.svg|thumb|350px|For the normal distribution, the values less than one standard deviation from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.]]
About 68% of values drawn from a normal distribution are within one standard deviation ''σ'' from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations.<ref name="www.mathsisfun.com" /> This fact is known as the [[68–95–99.7 rule|68–95–99.7 (empirical) rule]], or the ''3-sigma rule''.

More precisely, the probability that a normal deviate lies in the range between <math display=inline>\mu-n\sigma</math> and <math display=inline>\mu+n\sigma</math> is given by
<math display=block>
F(\mu+n\sigma) - F(\mu-n\sigma) = \Phi(n)-\Phi(-n) = \operatorname{erf} \left(\frac{n}{\sqrt{2}}\right).
</math>
To 12 significant digits, the values for <math display=inline>n=1,2,\ldots , 6</math> are:

{| class="wikitable" style="text-align:center;margin-left:24pt"
|-
! {{tmath|n}} !! <math display=inline>p= F(\mu+n\sigma) - F(\mu-n\sigma)</math> !! <math display=inline>1-p</math>!! <math display=inline>\text{or }1\text{ in }(1-p)</math> !! [[OEIS]]
|-
|1 || {{val|0.682689492137}} || {{val|0.317310507863}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|3}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.15148718753}}
|}
|| {{OEIS2C|A178647}}
|-
|2 || {{val|0.954499736104}} || {{val|0.045500263896}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|21}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.9778945080}}
|}
|| {{OEIS2C|A110894}}
|-
|3 || {{val|0.997300203937}} || {{val|0.002699796063}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|370}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.398347345}}
|}
|| {{OEIS2C|A270712}}
|-
|4 || {{val|0.999936657516}} || {{val|0.000063342484}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|15787}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.1927673}}
|}
|-
|5 || {{val|0.999999426697}} || {{val|0.000000573303}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|1744277}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.89362}}
|}
|-
|6 || {{val|0.999999998027}} || {{val|0.000000001973}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|506797345}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.897}}
|}
|}

For large {{tmath|n}}, one can use the approximation <math display=inline>1 - p \approx \frac{e^{-n^2/2}}{n\sqrt{\pi/2}}</math>.