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

7.2.5.1.

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 tandard deviations of the mean is at least (1 - 1/k2)100%.
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%

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