Z-Score Calculations

What is a z-score?

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

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

A positive z-score means above the mean; negative means below.

Step-by-step: Find z for x = 85

Given: $\mu = 70$, $\sigma = 10$, $x = 85$.

$z = \frac{85 - 70}{10} = \frac{15}{10} = 1.5$

Interpretation: 85 is 1.5 standard deviations above the mean. Using a z-table: $P(Z < 1.5) = 0.9332$, meaning 85 is higher than about 93.3% of values.

Common z-scores

• $z = 0$: at the mean (50th percentile)
• $z = 1$: 84th percentile
• $z = -1$: 16th percentile
• $z = 2$: 97.7th percentile
• $z = -2$: 2.3rd percentile

Why z-scores matter

They standardize any normal distribution to $N(0,1)$, allowing comparison across different scales. A z-score of 1.5 in test scores means the same relative position as a z-score of 1.5 in heights.

Try it in the visualization

Enter $\mu$, $\sigma$, and $x$. The bell curve shows where $x$ falls. The z-score and corresponding percentile are computed. The shaded area shows $P(Z < z)$.