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# Confidence Interval

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)

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) (3)

Now let , so . Then

 (4) (5) (6)

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)

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

Confidence Limits, Alpha Value, Erf, Inverse Erf, Normal Distribution, P-Value, Significance, Standard Deviation Explore this topic in the MathWorld classroom

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## References

Kenney, J. F. and Keeping, E. S. "Confidence Limits for the Binomial Parameter" and "Confidence Interval Charts." §11.4 and 11.5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 167-169, 1962.

## Referenced on Wolfram|Alpha

Confidence Interval

## Cite this as:

Weisstein, Eric W. "Confidence Interval." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConfidenceInterval.html