A confidence interval is an interval in which a measurement or trial falls corresponding to a given probability. Usually, the confidence interval of interest is symmetrically placed around the mean, so a 50% confidence interval for a symmetric probability density function would be the interval such that
(1)
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For a normal distribution, the probability that a measurement falls within standard deviations () of the mean (i.e., within the interval ) is given by
(2)
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(3)
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Now let , so . Then
(4)
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(5)
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(6)
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where is the so-called erf function. The following table summarizes the probabilities that measurements from a normal distribution fall within for with small values of .
0.6826895 | |
0.9544997 | |
0.9973002 | |
0.9999366 | |
0.9999994 |
Conversely, to find the probability- confidence interval centered about the mean for a normal distribution in units of , solve equation (5) for to obtain
(7)
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where is the inverse erf function. The following table then gives the values of such that is the probability- confidence interval for a few representative values of . These values can be returned by NormalCI[0, 1, ConfidenceLevel -> P] in the Wolfram Language package HypothesisTesting` .
0.800 | |
0.900 | |
0.950 | |
0.990 | |
0.995 | |
0.999 |