# 68–95–99.7 Rule

The 68–95–99.7 Rule is an empirical rule that applies to normal distributions.

**Context:**- It can be defined as: if [math]x[/math] is an observation from normally distributed random variable with mean value, [math]\mu[/math], and standard deviation [math]\sigma[/math] then:

- Approximately 68% of the observations ([math]x[/math] values) fall between [math]\mu - \sigma[/math] and [math]\mu+\sigma[/math]
- Approximately 95% of the [math]x[/math] values fall between [math]\mu - 2\sigma[/math] and [math]\mu+2\sigma[/math]
- Approximately 99.7% of the [math]x[/math] values fall between [math]\mu -3\sigma[/math] and [math]\mu+3\sigma[/math]

## References

### 2016

- (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/68%E2%80%9395%E2%80%9399.7_rule Retrieved 2016-07-03
- In statistics, the
**68–95–99.7 rule**is a shorthand used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of one, two and three standard deviations, respectively; more accurately, 68.27%, 95.45% and 99.73% 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 x is an observation from a normally distributed random variable, μ is the mean of the distribution, and σ is its standard deviation:

- In statistics, the

- [math]\begin{align} \Pr(\mu-\;\,\sigma \le x \le \mu+\;\,\sigma) &\approx 0.6827 \\ \Pr(\mu-2\sigma \le x \le \mu+2\sigma) &\approx 0.9545 \\ \Pr(\mu-3\sigma \le x \le \mu+3\sigma) &\approx 0.9973 \end{align}[/math]

- In the empirical sciences the so-called
**three-sigma rule of thumb**expresses a conventional heuristic that "nearly all" values are taken to lie within three standard deviations of the mean, i.e. that it is empirically useful to treat 99.7% probability as "near certainty".The usefulness of this heuristic of course depends significantly on the question under consideration, and there are other conventions, e.g. in the social sciences a result may be considered "significant" if its confidence level is of the order of a two-sigma effect (95%), while in particle physics, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a "discovery".

The "three sigma rule of thumb" is related to a result also known as the

**three-sigma rule**, which states that even for non-normally distributed variables, at least 98% of cases should fall within properly-calculated three-sigma intervals.

- In the empirical sciences the so-called

- (statisticshowto.com, 2016) ⇒ http://www.statisticshowto.com/68-95-99-7-rule/ Retrieved 2016-07-03
- De Moivre discovered the 68 95 99.7 rule with an experiment. You can do your own experiment by flipping 100 fair coins. Note:

- How many heads you would expect to see; these are “successes” in this binomial experiment.
- The standard deviation.
- The upper and lower limits for the number of heads you would get 68% of the time, 95% of the time and 99.7% of the time

### 2013

- (Gallego et al., 2013) ⇒ Gallego, G., Cuevas, C., Mohedano, R., & Garcia, N. (2013). “On the Mahalanobis distance classification criterion for multidimensional normal distributions". IEEE Transactions on Signal Processing, 61(17), 4387-4396.[DOI:10.1109/TSP.2013.2269047]
- The 68-95-99.7 rule states that 68%, 95% and 99.7% of the values drawn from a normal distribution are within 1, 2 and 3 standard deviations [math]\sigma \gt 0[/math] away from the mean [math]\mu[/math], respectively.