68–95–99.7 rule

We have that \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} doing the change of variable in terms of the standard score z = \frac{x - \mu}{\sigma}, we have \begin{align}\frac{1}{\sqrt{2\pi}} \int_{-n}^{n} e^{-\frac{z^2}{2}}dz\end{align}, and this integral is independent of \mu and \sigma. We only need to calculate each integral for the cases n = 1,2,3. \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} == Cumulative distribution function ==

for the normal distribution with mean () 0 and variance () 1 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 . For example, , or , corresponding to a prediction interval of . This is not a symmetrical interval – this is merely the probability that an observation is less than . To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): \Pr(\mu-2\sigma \le X \le \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: \bar{X} \pm 2\frac{\sigma}{\sqrt{n}} is approximately a 95% confidence interval when \bar{X} is the average of a sample of size 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 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, either the 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 residuals 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 6σ 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 simple normality test: if one witnesses a 6σ 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. Black Monday—October 19, 1987—was an extreme tail event in global financial markets, marked by the Dow Jones Industrial Average falling 22.6% in a single day, the largest one‑day percentage drop in its history. This dramatic decline reflected a rare confluence of factors, including overvaluation concerns, negative macroeconomic news, and the amplifying effects of computerized portfolio‑insurance trading strategies. The event was severe, sudden, and globally contagious, but it does not fit within the assumptions of a normal (Gaussian) distribution, which is why traditional statistical models fail to describe it meaningfully. Some commentators have described Black Monday as a “36‑standard‑deviation event,” but this characterization is mathematically and conceptually flawed: 1) A 36σ event under a normal distribution is effectively impossible—its probability is so small that it would not be expected to occur even once in the lifetime of the universe. 2) The claim arises from misapplying the normal distribution to financial returns, which are well‑known to exhibit fat tails, volatility clustering, and structural breaks—features incompatible with Gaussian assumptions. 3) Because markets do not follow a normal distribution, calculating sigma‑equivalents for extreme events produces nonsensical results that exaggerate the improbability rather than explain the phenomenon. More appropriate models are those that incorporate fat tails, volatility clustering, and discontinuous jumps, such as the Student‑t distribution, Lévy‑stable distributions, GARCH‑type volatility models, and Extreme Value Theory (EVT) for tail behavior. These frameworks acknowledge that large price movements occur far more frequently than Gaussian models predict and that market volatility can shift abruptly during stress periods. As a result, they provide a more realistic foundation for understanding extreme events like Black Monday, without resorting to misleading notions such as “36‑sigma” outcomes. == Table of numerical values ==

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: == See also ==