Here, erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range is therefore given by

(5)

Neither
nor erf can be expressed in terms of finite additions, subtractions,
multiplications, and root extractions, and so
must be either computed numerically or otherwise approximated.

Note that a function different from is sometimes defined as "the" normal distribution
function

(6)

(7)

(8)

(9)

(Feller 1968; Beyer 1987, p. 551), although this function is less widely encountered than the usual . The notation is due to Feller (1971).

The value of for which falls within the interval with a given probability is a related quantity called the confidence
interval.

For small values , a good approximation to is obtained from the Maclaurin
series for erf,

(10)

(OEIS A014481). For large values , a good approximation is obtained from the asymptotic
series for erf,