The Normal Curve and Z-Scores

The Bell Curve and Empirical Rule

The normal distribution is the familiar symmetric bell curve: unimodal with its mean, median, and mode all equal at the exact center. Naturally occurring variables like height and IQ approximate this shape.

The empirical rule describes how data spreads around the mean in a normal distribution:

Use this rule to quickly estimate where individual values fall.

Computing and Using Z-Scores

To compare a specific score to a distribution, compute a z-score: how many standard deviations a raw score sits above or below the mean.

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

Where $x$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation. A z-score of 0 is at the mean; positive is above, negative is below. Standardization allows comparing completely different scales.

Common pitfall: treating a z-score as a percentile. A z-score measures distance in SD units; converting it to a percentile requires the empirical rule or a standard normal table.