How to Find a Normal Distribution Probability

For a normal distribution with a known mean and standard deviation, finds the probability of being at most a given value. It standardises with z = (x − mean) ÷ standard deviation and returns the lower-tail probability.

Explanation

For a normal distribution with mean $\mu$ and standard deviation $\sigma$ , this finds the probability of a value being at most $x$ .

First standardise:

z = \dfrac{x - \mu}{\sigma}

Then read the lower-tail probability of the standard normal distribution (mean 0, standard deviation 1), written $\Phi(z) = P(Z \le z)$ .

Example

On a test with a mean of 60 and a standard deviation of 10, what share of people score 75 or below?

z = \dfrac{75 - 60}{10} = 1.5

P(X \le 75) = \Phi(1.5) = 0.9332

93.32% score at or below 75, so scoring above it puts you in the top 6.68%.

Numbers worth remembering

In a normal distribution the probability depends only on how many standard deviations you are from the mean.

Within $\mu \pm 1\sigma$ — about 68.3%
Within $\mu \pm 2\sigma$ — about 95.4%
Within $\mu \pm 3\sigma$ — about 99.7%

The 95% used for confidence intervals corresponds precisely to $\pm 1.96\sigma$ .

Watch out

The normal distribution is a symmetric bell, so $\Phi(-z) = 1 - \Phi(z)$
Heights and exam scores are often close to normal, but incomes and wealth have a long right tail and are not. Assuming normality for everything is dangerous
The standard deviation must be positive