What is a Z-Score?
A Z-score (also called a standard score) measures how many standard deviations a data point is from the mean of its distribution. A Z-score of 0 means the value equals the mean; a Z-score of 1 means it is one standard deviation above the mean; a Z-score of –2 means it is two standard deviations below.
How to calculate a Z-Score?
To find the Z-score, subtract the mean (μ) from the observed value (x) and divide by the standard deviation (σ):
Z = (x − μ) / σ
For example, if a test score is 85, the class mean is 70, and the standard deviation is 10, the Z-score is (85 − 70) / 10 = 1.5.
What is the Z-score formula?
The standard Z-score formula is Z = (x − μ) / σ, where x is the individual value, μ is the population mean, and σ is the population standard deviation. For sample data, replace μ with the sample mean x̄ and σ with the sample standard deviation s.
How to convert a Z-score to probability?
The cumulative probability P(X ≤ z) is the area under the standard normal curve to the left of the Z-score. It is calculated using the standard normal CDF: Φ(z) = 0.5 × (1 + erf(z / √2)). For example, z = 1.96 corresponds to a cumulative probability of about 0.975, meaning 97.5% of values fall below it.
What are some Z-score examples?
- A student scores 115 on an IQ test (mean = 100, σ = 15). Z = (115 − 100) / 15 ≈ 1.0 — the score is one standard deviation above average, at the ~84th percentile.
- A product weighs 480 g (mean = 500 g, σ = 10 g). Z = (480 − 500) / 10 = −2.0 — it is two standard deviations below average, at the ~2.3rd percentile.
- z = 0 always gives a cumulative probability of exactly 0.5 (50th percentile).
When are Z-scores useful?
Z-scores are used wherever standardised comparison matters: grading on a curve, comparing scores across tests with different scales, quality control (Six Sigma defect rates), finance (normalising returns across assets), and hypothesis testing (z-tests). They are most reliable when the underlying data follows a normal distribution. Z-scores can also support practical geometry tasks — for example, standardising the measurements you collect when using the cylinder calculator or the sphere calculator.