7. Product and Process Comparisons
7.2. Comparisons based on data from one process
7.2.6. What intervals contain a fixed percentage of the population values?

## Approximate intervals that contain most of the population values

Empirical intervals A rule of thumb is that where there is no evidence of significant skewness or clustering, two out of every three observations (67 %) should be contained within a distance of one standard deviation of the mean; 90 % to 95 % of the observations should be contained within a distance of two standard deviations of the mean; 99-100 % should be contained within a distance of three standard deviations. This rule can help identify outliers in the data.
Intervals that apply to any distribution The Bienayme-Chebyshev rule states that regardless of how the data are distributed, the percentage of observations that are contained within a distance of $$k$$ standard deviations of the mean is at least $$100(1-1/k^2)$$ %.
Exact intervals for the normal distribution The Bienayme-Chebyshev rule is conservative because it applies to any distribution. For a normal distribution, a higher percentage of the observations are contained within $$k$$ standard deviations of the mean as shown in the following table.

Percentage of observations contained between the mean and $$k$$ standard deviations

 $$k$$, No. of Standard Deviations Empircal Rule Bienayme-Chebychev Normal Distribution 1 67 % N/A 68.26 % 2 90-95 % at least 75 % 95.44 % 3 99-100 % at least 88.89 % 99.73 % 4 N/A at least 93.75 % 99.99 %