How to Calculate a Z-Score

1.What a z-score is

A z-score answers one question: how many standard deviations is this value above or below the mean? Positive z means above the mean, negative z means below, and z = 0 is exactly at the mean. The number has no units of its own — pounds, test points, and millimeters all become the same scale once you divide by σ. It is not a p-value, not a percentile unless you add a distributional assumption, and not a claim that the data are healthy or unusual on their own.

2.The formula

Subtract the mean μ from the value x, then divide by the standard deviation σ. σ must be positive; a zero spread makes the ratio undefined. The same algebra works whether μ and σ come from a population, a sample, or a published norm (for example an IQ scale with μ = 100 and σ = 15). What changes is the story you attach to those two numbers, not the arithmetic.

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

3.Worked example

Take x = 115, μ = 100, and σ = 15 — the same defaults as the z-score calculator. The gap from the mean is 115 − 100 = 15. Divide by the spread: 15 / 15 = 1. So z = 1: this value sits one standard deviation above the mean. If x were 85 instead, z would be (85 − 100) / 15 = −1, one standard deviation below.

\[ z=(115-100)/15=1 \]

4.Place x on a bell curve

Move μ, σ, and x to see z update and the marker travel along a normal sketch. The shaded ±1σ band is the usual “one standard deviation from the mean” region; the marker is x, not z itself. This is a teaching widget — no full calculator chrome, no list of scores. Open the z-score calculator when you want the standalone tool.

5.Open the full calculator

The z-score calculator uses the same (x − μ) / σ equation and the same 115 / 100 / 15 example, with How it works, FAQ, and related statistics tools on one page. Browse other statistics calculators if you need standard deviation, an average, or a p-value next.