Statistics How To

68 95 and 99.7 Rule in Statistics

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What is the 68 95 and 99.7 Rule?

When you use a standard normal model in statistics:

  • About 68% of values fall within one standard deviation of the mean.
  • About 95% of the values fall within two standard deviations from the mean.
  • Almost all of the values — about 99.7% — fall within three standard deviations from the mean.

This fact is what is called the 68 95 and 99.7 rule, sometimes called the Empirical Rule. That’s because the rule originally came from observations (“empirical” means “based on observation”). The 68 95 and 99.7 rule was first coined by Abraham de Moivre in 1733–75 years before the normal distribution model was published.

68 95 and 99.7 rule

Standard normal distribution showing standard deviations. Image credit: University of Virginia.

The normal curve is the most common type of data distribution. In a normal distribution, all of the measurements are computed as distances from the mean. These measurements are reported in standard deviations.

The normal curve is a symmetric distribution, so the middle 68.2% can be divided in two. Zero to 1 standard deviations from the mean contains 34.1% of the data. The opposite side is the same (0 to -1 standard deviations). Together, this area adds up to about 68% of the data.

In a normal distribution, the percentages of scores you can expect to find for any standard deviations from the mean are the same.

In a normal distribution, the percentages of scores you can expect to find for any standard deviations from the mean are the same.

When to use the 68 95 and 99.7 Rule

You can use the rule when you are told your data is normal, nearly normal, or if you have a unimodal distribution that is symmetric. If a question mentions a normal or nearly normal distribution, and you’re given standard deviations; That almost certainly means you can use the rule to approximate how many of your scores will fall within a certain number of standard deviations.

Who was Abraham de Moivre?

Abraham de Moivre was a French mathematician born in Vitry-le-Fran├žois. He was involved in the development of the theory of probability. Perhaps his biggest contribution to statistics was the 1756 edition of The Doctrine of Chances, containing his work on the approximation to the binomial distribution by the normal distribution in the case of a large number of trials.

Portrait of Abraham de Moivre, c.1736

Portrait of Abraham de Moivre, c.1736

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