Editing Standard deviation (section)

===Rules for normally distributed data===
[[File:Standard deviation diagram.svg|thumb|Dark blue is one standard deviation on either side of the mean. For the normal distribution, this accounts for 68.27 percent of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45 percent; three standard deviations (light, medium, and dark blue) account for 99.73 percent; and four standard deviations account for 99.994 percent. The two points of the curve that are one standard deviation from the mean are also the [[inflection point]]s.]]

The [[central limit theorem]] states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a [[probability density function]] of

<math display="block">f\left(x, \mu, \sigma^2\right) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}</math>

where {{mvar|μ}} is the [[expected value]] of the random variables, {{mvar|σ}} equals their distribution's standard deviation divided by {{math|{{var|n}}{{sup|{{frac|1|2}}}}}}, and {{mvar|n}} is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the [[normalizing constant]].

If a data distribution is approximately normal, then the proportion of data values within {{mvar|z}} standard deviations of the mean is defined by:

<math display="block">\text{Proportion} = \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)</math>

where <math>\textstyle\operatorname{erf}</math> is the [[error function]]. The proportion that is less than or equal to a number, {{mvar|x}}, is given by the [[cumulative distribution function]]:<ref>{{cite web |url= http://mathworld.wolfram.com/DistributionFunction.html |author= Eric W. Weisstein |title= Distribution Function |work=MathWorld |publisher=Wolfram |access-date= 30 September 2014}}</ref>

<math display="block">\text{Proportion} \le x = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{x - \mu}{\sigma\sqrt{2}}\right)\right] = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)\right].</math>

If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, {{math|{{var|μ}} ± {{var|σ}}}}, where {{mvar|μ}} is the arithmetic mean), about 95 percent are within two standard deviations ({{math|{{var|μ}} ± 2{{var|σ}}}}), and about 99.7 percent lie within three standard deviations ({{math|{{var|μ}} ± 3{{var|σ}}}}). This is known as the ''[[68–95–99.7 rule]]'', or ''the empirical rule''.

For various values of {{mvar|z}}, the percentage of values expected to lie in and outside the symmetric interval, {{math|{{var|CI}} {{=}} (−{{var|z}}{{var|σ}}, {{var|z}}{{var|σ}})}}, are as follows:
[[File:Confidence interval by Standard deviation.svg|thumb|Percentage within(''z'')]]
[[File:Standard deviation by Confidence interval.svg|thumb|''z''(Percentage within)]]
{{anchor|Table}}
{| class="wikitable" style="font-size:&nbsp;"
|-
! rowspan=2 | Confidence <br />interval
! Proportion within
! colspan=2 | Proportion without
|-
! Percentage
! Percentage
! Fraction
|-
| {{val|0.318639}}{{mvar|σ}}
| 25%
| 75%
| 3 / 4
|-
| {{val|0.674490}}{{mvar|σ}}
| {{val|50}}%
| {{val|50}}%
| 1&nbsp;/&nbsp;{{val|2}}
|-
| {{val|{{#expr:0.97792452561403 round 6}}}}{{mvar|σ}}
| 66.6667%
| 33.3333%
| 1&nbsp;/&nbsp;3
|-
| {{val|0.994458}}{{mvar|σ}}
| 68%
| 32%
| 1&nbsp;/&nbsp;3.125
|-
| 1{{mvar|σ}}
| {{val|68.2689492}}%
| {{val|31.7310508}}%
| 1&nbsp;/&nbsp;{{val|3.1514872}}
|-
| {{val|1.281552}}{{mvar|σ}}
| 80%
| 20%
| 1&nbsp;/&nbsp;5
|-
| {{val|1.644854}}{{mvar|σ}}
| 90%
| 10%
| 1&nbsp;/&nbsp;10
|-
| {{val|1.959964}}{{mvar|σ}}
| 95%
| 5%
| 1&nbsp;/&nbsp;20
|-
| 2{{mvar|σ}}
| {{val|95.4499736}}%
| {{val|4.5500264}}%
| 1&nbsp;/&nbsp;{{val|21.977895}}
|-
| {{val|2.575829}}{{mvar|σ}}
| 99%
| 1%
| 1&nbsp;/&nbsp;100
|-
| 3{{mvar|σ}}
| {{val|99.7300204}}%
| {{val|0.2699796}}%
| 1&nbsp;/&nbsp;370.398
|-
| {{val|3.290527}}{{mvar|σ}}
| 99.9%
| 0.1%
| 1&nbsp;/&nbsp;{{val|1000}}
|-
| {{val|3.890592}}{{mvar|σ}}
| 99.99%
| 0.01%
| 1&nbsp;/&nbsp;{{val|10000}}
|-
| 4{{mvar|σ}}
| {{val|99.993666}}%
| {{val|0.006334}}%
| 1&nbsp;/&nbsp;{{val|15787}}
|-
| {{val|4.417173}}{{mvar|σ}}
| 99.999%
| 0.001%
| 1&nbsp;/&nbsp;{{val|100000}}
|-
| {{val|4.5}}{{mvar|σ}}
| {{gaps|99.999|320|465|3751%}}
| {{gaps|0.000|679|534|6249%}}
| 1&nbsp;/&nbsp;{{val|147159.5358}}<br />6.8&nbsp;/&nbsp;{{val|1000000}}
|-
| {{val|4.891638}}{{mvar|σ}}
| {{val|99.9999}}%
| {{val|0.0001}}%
| 1&nbsp;/&nbsp;{{val|1000000}}
|-
| 5{{mvar|σ}}
| {{val|99.9999426697}}%
| {{val|0.0000573303}}%
| 1&nbsp;/&nbsp;{{val|1744278}}
|-
| {{val|5.326724}}{{mvar|σ}}
| {{val|99.99999}}%
| {{val|0.00001}}%
| 1&nbsp;/&nbsp;{{val|10000000}}
|-
| {{val|5.730729}}{{mvar|σ}}
| {{val|99.999999}}%
| {{val|0.000001}}%
| 1&nbsp;/&nbsp;{{val|100000000}}
|-
| [[Six Sigma#Sigma levels|{{val|6}}{{mvar|σ}}]]
| {{val|99.9999998027}}%
| {{val|0.0000001973}}%
| 1&nbsp;/&nbsp;{{val|506797346}}
|-
| {{val|6.109410}}{{mvar|σ}}
| {{val|99.9999999}}%
| {{val|0.0000001}}%
| 1&nbsp;/&nbsp;{{val|1000000000}}
|-
| {{val|6.466951}}{{mvar|σ}}
| {{val|99.99999999}}%
| {{val|0.00000001}}%
| 1&nbsp;/&nbsp;{{val|10000000000}}
|-
| {{val|6.806502}}{{mvar|σ}}
| {{val|99.999999999}}%
| {{val|0.000000001}}%
| 1&nbsp;/&nbsp;{{val|100000000000}}
|-
| 7{{mvar|σ}}
| {{gaps|99.999|999|999|7440%}}
| {{val|0.000000000256}}%
| 1&nbsp;/&nbsp;{{val|390682215445}}
|}