The 68-95-99.7 rule, also called the empirical rule, states that for a normal distribution, approximately 68%, 95%, and 99.7% of the probability
lies within one, two, and three standard deviations
of the mean, respectively. More precisely, if 
has a normal distribution
with mean 
and standard deviation 
, then
for 
,
where erf is the usual error
function. Taking 
,
2, and 3 gives probabilities 0.682689...(OEIS A178647), 0.954499...(OEIS A110894),
and 0.997300...(OEIS A270712), respectively.
The rule is specific to the normal distribution.
It is not a general statement about arbitrary probability
distributions.
See also
Normal Distribution,
Standard
Deviation,
Standard Normal Distribution
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References
National Institute of Standards and Technology. "What Do We Mean by 'Normal' Data?" §6.5.1 in NIST/SEMATECH e-Handbook of
Statistical Methods. https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm.Pukelsheim,
F. "The Three Sigma Rule." Amer. Statistician 48, 88-91,
1994. https://doi.org/10.1080/00031305.1994.10476030.Sloane,
N. J. A. Sequences A110894, A178647,
and A270712 in "The On-Line Encyclopedia
of Integer Sequences."
Cite this as:
Weisstein, Eric W. "68-95-99.7 Rule."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/68-95-997Rule.html
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