View source for 68–95–99.7 rule

{{Short description|Shorthand used in statistics}}
{{More citations needed|date=September 2023}}
[[File:Empirical rule histogram.svg|thumb|upright=1.5|For an approximately [[normal distribution|normal data set]], the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%. Shown percentages are rounded theoretical probabilities intended only to approximate the empirical data derived from a normal population.]]
[[File:Standard score and prediction interval.svg|thumb|upright=1.2|right|Prediction interval (on the [[y-axis]]) given from the [[standard score]] (on the [[x-axis]]). The y-axis is logarithmically scaled (but the values on it are not modified).]]

In [[statistics]], the '''68–95–99.7 rule''', also known as the '''[[empirical measure|empirical]] rule''', and sometimes abbreviated '''3sr''' or '''3{{mvar|σ}}''', is a shorthand used to remember the percentage of values that lie within an [[interval estimate]] in a [[normal distribution]]: approximately 68%, 95%, and 99.7% of the values lie within one, two, and three [[standard deviation]]s of the [[Arithmetic mean|mean]], respectively.

In mathematical notation, these facts can be expressed as follows, where {{math|Pr()}} is the [[probability distribution|probability function]],<ref>{{cite book |last=Huber |first=Franz |date=2018 |title=A Logical Introduction to Probability and Induction |url=https://books.google.com/books?id=Z2J7DwAAQBAJ&pg=PA80 |location=New York |publisher=[[Oxford University Press]] |page=80 |isbn=9780190845414}}</ref> {{mvar|[[Χ]]}} is an observation from a normally distributed [[random variable]], {{mvar|[[μ]]}} (mu) is the mean of the distribution, and {{mvar|[[σ]]}} (sigma) is its standard deviation:
<math display="block">\begin{align}
  \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\
  \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\
  \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\%
\end{align}</math>

The usefulness of this heuristic especially depends on the question under consideration. 

{{anchor|Three-sigma rule}}In the [[empirical science]]s, the so-called  '''three-sigma rule of thumb''' (or '''3{{mvar|σ}} rule''') expresses a conventional [[heuristic]] that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% [[probability]] as near certainty.<ref>This usage of "three-sigma rule" entered common usage in the 2000s, e.g. cited in
*{{cite book |title=Schaum's Outline of Business Statistics |url=https://archive.org/details/businessstatisti0000unse |url-access=registration |publisher=McGraw Hill Professional |year=2003 |page=359 |isbn=9780071398763 |postscript=none }}
*{{cite book |first=Erik W. |last=Grafarend |title=Linear and Nonlinear Models: Fixed Effects, Random Effects, and Mixed Models |url=https://archive.org/details/linearnonlinearm00wgra |url-access=limited |publisher=Walter de Gruyter |year=2006 |page=[https://archive.org/details/linearnonlinearm00wgra/page/n573 553] |isbn=9783110162165 }}</ref>

In the [[social sciences]], a result may be considered [[Statistical significance|statistically significant]] if its [[confidence level]] is of the order of a two-sigma effect (95%), while in [[particle physics]] and [[astrophysics]], there is a convention of requiring statistical significance of a five-sigma effect (99.99994% confidence) to qualify as a [[Discovery (observation)|discovery]].<ref>{{cite arXiv |first=Louis |last=Lyons |title=DISCOVERING THE SIGIFICANCE OF 5σ |date = October 7, 2013 |class=physics.data-an |eprint=1310.1284 }}</ref>

A weaker three-sigma rule can be derived from [[Chebyshev's inequality]], stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For [[Unimodality#Unimodal probability distribution|unimodal distributions]], the probability of being within the interval is at least 95% by the [[Vysochanskij–Petunin inequality]]. There may be certain assumptions for a distribution that force this probability to be at least 98%.<ref>See:
* {{cite book |first1=D. J. |last1=Wheeler |first2=D. S. |last2=Chambers |title=Understanding Statistical Process Control |publisher=SPC Press |year=1992 |isbn=9780945320135 |url=https://books.google.com/books?id=XvMJAQAAMAAJ }}
* {{cite book |first1=Veronica |last1=Czitrom |author-link= Veronica Czitrom |first2=Patrick D. |last2=Spagon |title=Statistical Case Studies for Industrial Process Improvement |publisher=SIAM |year=1997 |page=342 |isbn=9780898713947 |url=https://books.google.com/books?id=gEpkYxwDbvgC&pg=PA342 }}
* {{cite journal |first=F. |last=Pukelsheim |title=The Three Sigma Rule |journal=American Statistician |volume=48 |year=1994 |issue=2 |pages=88–91|doi=10.2307/2684253 |jstor=2684253}}</ref>
<!--Czitrom has "more than 98.2%", but the article still lacks presentation of what the exact assumptions are, and what exactly is a "properly constructed sigma interval" for a non-normal distribution.-->

== Proof ==
We have that 
<math display="block">\begin{align}\Pr(\mu -n\sigma \leq X \leq \mu + n\sigma) = \int_{\mu-n\sigma}^{\mu + n\sigma} \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2} dx, \end{align}</math>
doing the [[Integration by substitution|change of variable]] in terms of the [[standard score]]  <math> z = \frac{x - \mu}{\sigma}</math>, we have
<math display="block">\begin{align}\frac{1}{\sqrt{2\pi}} \int_{-n}^{n} e^{-\frac{z^2}{2}}dz\end{align},</math>
and this integral is independent of <math>\mu</math> and <math>\sigma</math>. We only need to calculate each integral for the cases <math>n = 1,2,3</math>.
<math display="block">\begin{align}
\Pr(\mu -1\sigma \leq X \leq \mu + 1\sigma) &= \frac{1}{\sqrt{2\pi}} \int_{-1}^{1} e^{-\frac{z^2}{2}}dz \approx 0.6826894921 \\
\Pr(\mu -2\sigma \leq X \leq \mu + 2\sigma) &= \frac{1}{\sqrt{2\pi}}\int_{-2}^{2} e^{-\frac{z^2}{2}}dz \approx 0.9544997361 \\
\Pr(\mu -3\sigma \leq X \leq \mu + 3\sigma) &= \frac{1}{\sqrt{2\pi}}\int_{-3}^{3} e^{-\frac{z^2}{2}}dz \approx 0.9973002039.
\end{align}</math>

== Cumulative distribution function ==
{{main|Prediction interval#Known mean, known variance}}

[[File:Cumulative distribution function for normal distribution, mean 0 and sd 1.png|270px|thumb|Diagram showing the [[cumulative distribution function]] for the normal distribution with mean ({{mvar|μ}}) 0 and variance ({{math|{{var|σ}}{{sup|2}}}})&nbsp;1]]

These numerical values "68%, 95%, 99.7%" come from the [[Normal distribution#Cumulative distribution function|cumulative distribution function of the normal distribution]].

The [[prediction interval]] for any [[standard score]] ''z'' corresponds numerically to {{math|1=(1 − (1 − <span style="font-size:100%;">{{var|Φ}}</span>{{sub|{{var|μ}},{{var|σ}}{{sup|2}}}}(z)) · 2)}}.

For example, {{math|{{var|Φ}}(2) ≈ 0.9772}}, or {{math|Pr({{var|X}} ≤ {{var|μ}} + 2{{var|σ}}) ≈ 0.9772}}, corresponding to a prediction interval of {{math|1 = (1&nbsp;−&nbsp;(1&nbsp;−&nbsp;0.97725)·2) =&nbsp;0.9545 =&nbsp;95.45%}}.
This is not a symmetrical interval – this is merely the probability that an observation is less than {{math|{{var|μ}} + 2{{var|σ}}}}. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding):
<math display="block">\Pr(\mu-2\sigma \le X \le \mu+2\sigma)
 = \Phi(2) - \Phi(-2)
 \approx 0.9772 - (1 - 0.9772)
 \approx 0.9545
</math>

This is related to [[confidence interval]] as used in statistics: <math>\bar{X} \pm 2\frac{\sigma}{\sqrt{n}}</math> is approximately a 95% confidence interval when <math>\bar{X}</math> is the average of a sample of size <math>n</math>.

== Normality tests ==
{{Main|Normality test}}
The "68–95–99.7 rule" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal.  It is also used as a simple test for [[outliers]] if the population is assumed normal, and as a [[normality test]] if the population is potentially not normal.

To pass from a sample to a number of standard deviations, one first computes the [[deviation (statistics)|deviation]], either the [[Errors and residuals in statistics|error or residual]] depending on whether one knows the population mean or only estimates it.  The next step is [[standardizing]] (dividing by the population standard deviation), if the population parameters are known, or [[studentizing]] (dividing by an estimate of the standard deviation), if the parameters are unknown and only estimated.

To use as a test for outliers or a normality test, one computes the size of deviations in terms of standard deviations, and compares this to expected frequency. Given a sample set, one can compute the [[studentized residual]]s and compare these to the expected frequency: points that fall more than 3 standard deviations from the norm are likely outliers (unless the [[sample size]] is significantly large, by which point one expects a sample this extreme), and if there are many points more than 3 standard deviations from the norm, one likely has reason to question the assumed normality of the distribution. This holds ever more strongly for moves of 4 or more standard deviations.

One can compute more precisely, approximating the number of extreme moves of a given magnitude or greater by a [[Poisson distribution]], but simply, if one has multiple 4 standard deviation moves in a sample of size 1,000, one has strong reason to consider these outliers or question the assumed normality of the distribution.

For example, a 6''σ'' event corresponds to a chance of about two [[parts per billion]]. For illustration, if events are taken to occur daily, this would correspond to an event expected every 1.4 million years.  This gives a [[normality test#Back of the envelope test|simple normality test]]: if one witnesses a 6''σ'' in daily data and significantly fewer than 1 million years have passed, then a normal distribution most likely does not provide a good model for the magnitude or frequency of large deviations in this respect.

In ''[[The Black Swan: The Impact of the Highly Improbable|The Black Swan]]'', [[Nassim Nicholas Taleb]] gives the example of risk models according to which the [[Black Monday (1987)|Black Monday]] crash would correspond to a 36-''σ'' event:<!--
according to [http://www.wolframalpha.com/input/?i=erf%2836%2Fsqrt%282%29%29 wolphramalpha.com], a "36-sigma event" corresponds to an expected frequency of the order of 10<sup>−215</sup>, i.e. beyond astronomically small, perhaps one expected "event" in the history of the universe if events take place every planck time in every planck volume, or something insane like that.
this is far beyond the applicability of the "gambler's fallacy", citing gambler's fallacy here would be like calling "gambler's fallacy" if somebody complained about a certain roulette table coming up zero every time during a whole week or so.-->
the occurrence of such an event should instantly suggest that the model is flawed, i.e. that the process under consideration is not satisfactorily modeled by a normal distribution. Refined models should then be considered, e.g. by the introduction of [[stochastic volatility]].  In such discussions it is important to be aware of the problem of the [[gambler's fallacy]], which states that a single observation of a rare event does not contradict that the event is in fact rare. It is the observation of a plurality of purportedly rare events that increasingly [[Belief revision|undermines the hypothesis]] that they are rare, i.e. the validity of the assumed model. A proper modelling of this process of gradual loss of confidence in a hypothesis would involve the designation of [[prior probability]] not just to the hypothesis itself but to all possible alternative hypotheses. For this reason, [[statistical hypothesis testing]] works not so much by confirming a hypothesis considered to be likely, but by [[null hypothesis|refuting hypotheses considered unlikely]].

== Table of numerical values ==
Because of the exponentially decreasing tails of the normal distribution, odds of higher deviations decrease very quickly. From the [[Standard deviation#Rules for normally distributed data|rules for normally distributed data]] for a daily event:
{|class="wikitable" style="text-align:left"
! Range
! Expected fraction of
population inside range
! Expected fraction ofpopulation outside range! | Expected fraction of 
population outside range
! colspan="2" |Approx. expected{{br}}frequency outside range
!Approx. frequency outside range for daily event
|-
| style="text-align:left" | {{nowrap| style="text-align:left" | ''μ'' ± 0.5''σ''}}
| {{val|0.382924922548026}}
| {{val|0.6171}} = 61.71 %
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 3&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 5
| style="text-align:left" | Four or five times a week
|-
| style="text-align:left" | ''μ'' ± ''σ''
| {{val|0.682689492137086}}<ref>{{Cite OEIS|A178647}}</ref>
| {{val|0.3173}} = 31.73 %
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 3
| style="text-align:left" | Twice or thrice a week
|-
| style="text-align:left" | ''μ'' ± 1.5''σ''
| {{val|0.866385597462284}}
| {{val|0.1336}} = 13.36 %
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 2&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 15
| style="text-align:left" | Weekly
|-
| style="text-align:left" | ''μ'' ± 2''σ''
| {{val|0.954499736103642}}<ref>{{cite OEIS|A110894}}</ref>
| {{val|0.04550}} = 4.550 %
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 22
| style="text-align:left" | Every three weeks
|-
| style="text-align:left" | ''μ'' ± 2.5''σ''
| {{val|0.987580669348448}}
| {{val|0.01242}} = 1.242&nbsp;%
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 81
| style="text-align:left" | Quarterly
|-
| style="text-align:left" | ''μ'' ± 3''σ''
| {{val|0.997300203936740}}<ref>{{cite OEIS|A270712}}</ref>
| {{val|0.002700}} = 0.270&nbsp;% = 2.700&nbsp;‰
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 370
| style="text-align:left" | Yearly
|-
| style="text-align:left" | ''μ'' ± 3.5''σ''
| {{val|0.999534741841929}}
| {{val|0.0004653}} = 0.04653&nbsp;% = {{val|465.3|u=ppm}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 2149
| style="text-align:left" | Every 6&nbsp;years
|-
| style="text-align:left" | ''μ'' ± 4''σ''
| {{val|0.999936657516334}}
| {{val|6.334E-5}} = {{val|63.34|u=ppm}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|15787}}
| style="text-align:left" | Every 43&nbsp;years (twice in a lifetime)
|-
| style="text-align:left" | ''μ'' ± 4.5''σ''
| {{val|0.999993204653751}}
| {{val|6.795E-6}} = {{val|6.795|u=ppm}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|147160}}
| style="text-align:left" | Every 403&nbsp;years (once in the [[modern era]])
|-
| style="text-align:left" | ''μ'' ± 5''σ''
| {{val|0.999999426696856}}
| {{val|5.733E-7}} = {{val|0.5733|u=ppm}} = {{val|573.3|u=ppb}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|1744278}}
| style="text-align:left" | Every {{val|4776}}&nbsp;years (once in [[recorded history]])
|-
| style="text-align:left" | ''μ'' ± 5.5''σ''
| {{val|0.999999962020875}}
| {{val|3.798E-8}} = {{val|37.98|u=ppb}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|26330254}}
| style="text-align:left" | Every {{val|72090}}&nbsp;years (thrice in history of [[Homo sapiens|modern humankind]])
|-
| style="text-align:left" | ''μ'' ± 6''σ''
| {{val|0.999999998026825}}
| {{val|1.973E-9}} = {{val|1.973|u=ppb}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|506797346}}
| style="text-align:left" | Every 1.38&nbsp;million years (twice in history of [[Homo (genus)|humankind]])
|-
| style="text-align:left" | ''μ'' ± 6.5''σ''
| {{val|0.999999999919680}}
| {{val|8.032E-11}} = {{val|0.08032|u=ppb}} = {{val|80.32|u=ppt}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|12450197393}}
| style="text-align:left" | Every 34&nbsp;million years (twice since the [[extinction of dinosaurs]])
|-
| style="text-align:left" | ''μ'' ± 7''σ''
| {{val|0.999999999997440}}
| {{val|2.560E-12}} = {{val|2.560|u=ppt}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|390682215445}}
| style="text-align:left" | Every 1.07&nbsp;billion years (four occurrences in [[history of Earth]])
|-
| style="text-align:left" | ''μ'' ± 7.5''σ''
| {{val|0.999999999999936}}
| {{val|6.382E-14}} = {{val|63.82|u=ppq}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|15669601204101}}
| style="text-align:left" | Once every 43 billion years (never in the history of the [[Universe]], twice in the future of the [[Local Group]] before its merger)
|-
| style="text-align:left" | ''μ'' ± 8''σ''
| {{val|0.999999999999999}}
| {{val|1.244E-15}} = {{val|1.244|u=ppq}}
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | {{val|803734397655348}}
| style="text-align:left" | Once every 2.2 [[trillion]] years (never in the history of the [[Universe]], once during the life of a [[red dwarf]])
|-
| style="text-align:left" | ''μ'' ± {{math|''x''}}''σ''
| [[Error function|<math>\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)</math>]]
|[[Error function|<math>1-\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)</math>]]
| style="width:1px; border-left:none; border-right:none; padding-left:0; padding-right:0" | 1&nbsp;in&nbsp;
| style="text-align:left; border-left:none; border-right:none; padding-left:0; padding-right:0" | <math>\tfrac{1}{1-\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)}</math>
| style="text-align:left" | Every <math>\tfrac{1}{1-\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)}</math> days
|}

== See also ==
* [[p-value|''p''-value]]
* {{format link|Six Sigma#Sigma levels}}
* [[Standard score]]
* [[t-statistic|''t''-statistic]]

== References ==
{{reflist}}

== External links ==
* "[http://www.wolframalpha.com/input/?i=erf%28x%2Fsqrt%282%29%29 Calculate percentage proportion within ''x'' sigmas] at WolframAlpha

{{ProbDistributions|Normal distribution}}
{{Authority control}}

{{DEFAULTSORT:68-95-99.7 rule}}
[[Category:Normal distribution]]
[[Category:Statistical approximations]]
[[Category:Rules of thumb]]

[[pl:Odchylenie standardowe#Dla rozkładu normalnego]]