The Normal Curve and Z-Scores

The Bell Curve and Standardized Scores

The normal distribution — the familiar symmetric "bell curve" — is one of the most important patterns in statistics: unimodal (one peak), perfectly symmetric, with its mean, median, and mode all equal and located at the exact center. Many naturally occurring psychological and biological variables (height, measurement error, IQ scores) approximate this shape closely enough to make the normal distribution's mathematical properties extremely useful.

The empirical rule (sometimes called the 68-95-99.7 rule) describes how data is distributed around the mean in a normal distribution:

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

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

A z-score of 0 sits exactly at the mean; a positive z-score sits above the mean, a negative one below. Because z-scores are standardized (always in units of standard deviations), they allow direct comparison of scores from entirely DIFFERENT scales — comparing a person's height percentile to their test-score percentile, for instance, becomes possible only once both are converted to the same z-score scale.

Common pitfall: treating a z-score as though it directly gives a percentile. A z-score tells you HOW FAR a score sits from the mean, in standard deviation units — converting that distance into an exact percentile requires consulting the normal distribution's known area-under-the-curve properties (e.g. the empirical rule, or a full standard normal table), not just reading the z-score number itself.

A bell curve: a bare symmetric curve; the ±1 SD band shades in with "68%"; the ±2 SD band shades in wider with "95%", and a single raw-score marker drops onto the curve with its z-score value labeled beside it.

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