Cumulative distribution function
The cumulative distribution function for a \(\NormalDistn(\mu,\; \sigma^2)\) distribution is
\[ F(x) \;\;=\;\; \int_{-\infty}^x {\frac 1{\sqrt{2\pi}\;\sigma} e^{- \frac{\large (u-\mu)^2}{\large 2 \sigma^2}}} du \]Unfortunately this integration cannot be performed algebraically. However numerical algorithms have been written to evaluate cumulative probabilities for any specific values of \(\mu\), \(\sigma\) and \(x\). For example, Excel can find normal cumulative probabilities by typing into a spreadsheet cell
= NORM.DIST( \(x\), \(\mu\), \(\sigma\), true)
Normal probabilities from z-scores
Although probabilities for any normal distribution can be found as described above, we now describe an alternative method using z-scores. This allows probabilities about any normal random variable, \(X\), to be translated into probabilities about the standard normal distribution.
Weights of apples
The diagram below shows the distribution of weights of Fuji apples arriving at a packhouse. The distribution is normal (µ = 180 g, σ = 10 g).
Use the slider to translate apple weights, x, into z-scores.
The probability of a lower apple weight is translated into a probability about the z-score. The probability (area) is highlighted on the standard normal distribution of the z-score at the bottom of the diagram.
After translating an x-value into a z-score, \(z = \dfrac{x - \mu}{\sigma}\), we can find the cumulative probability \(P(X \le x)\) by evaluating \(P(Z \le z)\). For example, this can be done in Excel with the function
=NORM.S.DIST(z, true)
Although this offers few practical advantages when a computer is used,