68–95–99.7 rule
In statistics, the 68–95–99.7 rule, also known as the empirical rule or 68–95–99.7 rule for a normal distribution and sometimes abbreviated 3sr or 3 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 deviations of the mean, respectively. In mathematical notation, these facts can be expressed as follows, where Pr() is the probability function, Χ is an observation from a normally distributed random variable, μ (mu) is the mean of the distribution, and σ (sigma) is its standard deviation: Pr ( μ − 1 σ ≤ X ≤ μ + 1 σ ) ≈ 68.27 % Pr ( μ − 2 σ ≤ X ≤ μ + 2 σ ) ≈ 95.45 % Pr ( μ − 3 σ ≤ X ≤ μ + 3 σ ) ≈ 99.73 % {\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 depends especially on the question under consideration and the manner in which the data have been collected; most particularly the heuristic depends on the data genuinely being normally distributed: Among the many bell-shaped distributions often seen in real-life data, the normal distribution has notoriously "thin tails" – an unusual concentration of probability near its center.