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