Cumulative distribution function
The cumulative distribution function for a \(\NormalDistn(\mu,\; \sigma^2)\) distribution is
\[ F(x) \;\;=\;\; \int_{-\infty}^x {\frac 1{\sqrt{2\pi}\;\sigma} e^{- \frac{\large (u-\mu)^2}{\large 2 \sigma^2}}} du \]This integration cannot be performed algebraically, but numerical algorithms will find cumulative probabilities for you. For example, in Excel you can use the function
= NORM.DIST( \(x\), \(\mu\), \(\sigma\), true)
Normal probabilities from z-scores
Although probabilities for any normal distribution can be found as described above, an alternative method uses z-scores. This lets us find probabilities about a normal random variable using the standard normal distribution.
In Excel, this would be evaluated as
=NORM.S.DIST(z, true)
Although this offers few practical advantages when a computer is used,