Probability Between Two Values in a Normal Distribution

Finds the probability that a normal variable falls between two bounds, as Φ((upper − mean) ÷ σ) − Φ((lower − mean) ÷ σ). Within one standard deviation of the mean, the probability is about 68.27%.

Explanation

This is the probability that a normal variable lands between $a$ and $b$ . Take the probability up to the upper bound and subtract the probability up to the lower one.

P(a \le X \le b) = \Phi\left(\dfrac{b - \mu}{\sigma}\right) - \Phi\left(\dfrac{a - \mu}{\sigma}\right)

where $\Phi$ is the lower-tail probability of the standard normal distribution.

Example

On a test with mean 60 and standard deviation 10, what share of scores fall between 50 and 70?

z_{\text{lower}} = \dfrac{50 - 60}{10} = -1 \qquad z_{\text{upper}} = \dfrac{70 - 60}{10} = 1

P(50 \le X \le 70) = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = 0.6827

68.27%. Since this is exactly one standard deviation either side of the mean, the famous 68% figure drops straight out.

Ranges worth knowing

Mean $\pm 1\sigma$ — 68.27%
Mean $\pm 1.96\sigma$ — 95.00%
Mean $\pm 2\sigma$ — 95.45%
Mean $\pm 3\sigma$ — 99.73%

The "six sigma" of manufacturing refers to a specification window six standard deviations wide, where defects become vanishingly rare.

Watch out

The lower bound must not exceed the upper bound
The standard deviation must be positive
For a continuous distribution the probability of any single point is zero, so it makes no difference whether the bounds are written with $\le$ or $<$