## Confidence Interval

The probability that a measurement will fall within a given Closed Interval . For a continuous distribution,

 (1)

where is the Probability Distribution Function. Usually, the confidence interval of interest is symmetrically placed around the mean, so
 (2)

where is the Mean. For a Gaussian Distribution, the probability that a measurement falls within of the mean is
 (3)

Now let , so . Then
 (4)

where is the so-called Erf function. The variate value producing a confidence interval CI is often denoted , so
 (5)

 range CI 0.6826895 0.9544997 0.9973002 0.9999366 0.9999994

To find the standard deviation range corresponding to a given confidence interval, solve (4) for .

 (6)

 CI range 0.800 ± 1.28155 0.900 ± 1.64485 0.950 ± 1.95996 0.990 ± 2.57583 0.995 ± 2.80703 0.999 ± 3.29053

© 1996-9 Eric W. Weisstein
1999-05-26