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Normal distribution and z-score

The normal distribution is bell-shaped and symmetric about the mean $\mu$ , with spread given by $\sigma$ . The 68–95–99.7 rule states how much probability lies within 1, 2 and 3 standard deviations. The z-score standardizes a measurement to the number of standard deviations from $\mu$ .

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

z-score (standardized deviation)

\mu \pm 2\sigma \approx 95\%

the 68–95–99.7 rule for

\pm1, \pm2, \pm3\sigma

Symbols

$\mu$	expected value
$\sigma$	standard deviation
$z$	number of standard deviations from $\mu$

Example

$\mu = 76$ , $\sigma = 4$ , $x = 72$ :

$z = (72-76)/4 = -1$ .

A negative z-score means the measurement is below average, not that something is wrong.

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Part of Statistics and Risk Analysis: Distributions.