Z-Score and Area Under the Normal Curve

A z-score tells you how many standard deviations a value is from the mean:

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

The area under the standard normal curve to the left of a z-score gives the probability $P(Z < z)$. For $z = 1.5$: $P(Z < 1.5) = 0.9332$, meaning 93.32% of values fall below 1.5 standard deviations above the mean.

The z-score transformation converts any normal distribution $N(\mu, \sigma)$ to the standard normal $N(0, 1)$, making it possible to use a single table (the z-table) for all normal distributions.

Try it in the visualization

Drag the z-score slider and watch the shaded area (probability) update. Toggle between left-tail ($P(Z < z)$), right-tail ($P(Z > z)$), and two-tail ($P(|Z| > z)$) probabilities. Enter a raw score with custom $\mu$ and $\sigma$ to see the z-score conversion.