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 socalled 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 