68–95–99.7 rule

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In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an

, respectively.

In mathematical notation, these facts can be expressed as follows, where Pr() is the

σ

(sigma) is its standard deviation:

$${\displaystyle {\begin{aligned}\Pr(\mu -1\sigma \leq X\leq \mu +1\sigma )&\approx 68.27\%\\\Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )&\approx 95.45\%\\\Pr(\mu -3\sigma \leq X\leq \mu +3\sigma )&\approx 99.73\%\end{aligned}}}$$

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

In the

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 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%.^[3]

Proof

We have that

$${\displaystyle {\begin{aligned}\Pr(\mu -n\sigma \leq X\leq \mu +n\sigma )=\int _{\mu -n\sigma }^{\mu +n\sigma }{\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}dx,\end{aligned}}}$$

${\displaystyle u={\frac {x-\mu }{\sigma }}}$

$${\displaystyle {\begin{aligned}{\frac {1}{\sqrt {2\pi }}}\int _{-n}^{n}e^{-{\frac {u^{2}}{2}}}du\end{aligned}},}$$

and this integral is independent of ${\displaystyle \mu }$ and ${\displaystyle \sigma }$. We only need to calculate each integral for the cases ${\displaystyle n=1,2,3}$.

$${\displaystyle {\begin{aligned}&\Pr(\mu -1\sigma \leq X\leq \mu +1\sigma )={\frac {1}{\sqrt {2\pi }}}\int _{-1}^{1}e^{-{\frac {u^{2}}{2}}}du\approx 0.6827\\&\Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )={\frac {1}{\sqrt {2\pi }}}\int _{-2}^{2}e^{-{\frac {u^{2}}{2}}}du\approx 0.9545\\&\Pr(\mu -3\sigma \leq X\leq \mu +3\sigma )={\frac {1}{\sqrt {2\pi }}}\int _{-3}^{3}e^{-{\frac {u^{2}}{2}}}du\approx 0.9973.\end{aligned}}}$$

Cumulative distribution function

These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution.

The prediction interval for any standard score z corresponds numerically to (1 − (1 − Φ_μ,σ²(z)) · 2).

For example, Φ(2) ≈ 0.9772, or Pr(X ≤ μ + 2σ) ≈ 0.9772, corresponding to a prediction interval of (1 − (1 − 0.97725)·2) = 0.9545 = 95.45%. This is not a symmetrical interval – this is merely the probability that an observation is less than μ + 2σ. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding):

$${\displaystyle \Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )=\Phi (2)-\Phi (-2)\approx 0.9772-(1-0.9772)\approx 0.9545}$$

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

Normality tests

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

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 6σ event corresponds to a chance of about two

In The Black Swan, Nassim Nicholas Taleb gives the example of risk models according to which the Black Monday crash would correspond to a 36-σ event: 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

Because of the exponentially decreasing tails of the normal distribution, odds of higher deviations decrease very quickly. From the rules for normally distributed data for a daily event:

Range	Expected fraction of population inside range	Expected fraction of population outside range	Approx. expected frequency outside range		Approx. frequency outside range for daily event
μ ± 0.5σ	0.382924922548026	6.171E-01 = 61.71 %	3 in	5	Four or five times a week
μ ± σ	0.682689492137086[4]	3.173E-01 = 31.73 %	1 in	3	Twice or thrice a week
μ ± 1.5σ	0.866385597462284	1.336E-01 = 13.36 %	2 in	15	Weekly
μ ± 2σ	0.954499736103642[5]	4.550E-02 = 4.550 %	1 in	22	Every three weeks
μ ± 2.5σ	0.987580669348448	1.242E-02 = 1.242 %	1 in	81	Quarterly
μ ± 3σ	0.997300203936740[6]	2.700E-03 = 0.270 % = 2.700 ‰	1 in	370	Yearly
μ ± 3.5σ	0.999534741841929	4.653E-04 = 0.04653 % = 465.3 ppm	1 in	2149	Every 6 years
μ ± 4σ	0.999936657516334	6.334E-05 = 63.34 ppm	1 in	15787	Every 43 years (twice in a lifetime)
μ ± 4.5σ	0.999993204653751	6.795E-06 = 6.795 ppm	1 in	147160	Every 403 years (once in the modern era)
μ ± 5σ	0.999999426696856	5.733E-07 = 0.5733 ppm = 573.3 ppb	1 in	1744278	Every 4776 years (once in recorded history)
μ ± 5.5σ	0.999999962020875	3.798E-08 = 37.98 ppb	1 in	26330254	Every 72090 years (thrice in history of modern humankind )
μ ± 6σ	0.999999998026825	1.973E-09 = 1.973 ppb	1 in	506797346	Every 1.38 million years (twice in history of humankind )
μ ± 6.5σ	0.999999999919680	8.032E-11 = 0.08032 ppb = 80.32 ppt	1 in	12450197393	Every 34 million years (twice since the extinction of dinosaurs )
μ ± 7σ	0.999999999997440	2.560E-12 = 2.560 ppt	1 in	390682215445	Every 1.07 billion years (four occurrences in history of Earth)
μ ± 7.5σ	0.999999999999936	6.382E-14 = 63.82 ppq	1 in	15669601204101	Once every 43 billion years (never in the history of the Universe, twice in the future of the Local Group before its merger)
μ ± 8σ	0.999999999999999	1.244E-15 = 1.244 ppq	1 in	803734397655348	Once every 2.2 trillion years (never in the history of the Universe, once during the life of a red dwarf)
μ ± xσ	${\displaystyle \operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}$	${\displaystyle 1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}$	1 in	${\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}}$	Every ${\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}}$ days

See also

• p-value
• Six Sigma § Sigma levels
• Standard score
• t-statistic

References