Z-Score Calculator

A z-score (also called a standard score or standardised value) is the signed number of standard deviations a value lies from the mean of its distribution. A z of 0 means the value equals the mean; a z of +1 means one σ above the mean; −2 means two σ below. Z-scores standardise values onto a dimensionless scale so they can be compared across distributions with different means and spreads.

Use the formula z = (x − μ) / σ, where x is the raw value, μ is the mean, and σ is the standard deviation. For example, with x = 75, μ = 70, σ = 10: z = (75 − 70) / 10 = 0.5. So 75 lies half a standard deviation above the mean.

The percentile is the percentage of values in the normal distribution lying at or below your value. It comes from Φ(z), the standard-normal CDF. Some landmark values: z = 0 → 50th percentile, z = 1 → ~84th, z = 2 → ~97.7th, z = −1 → ~16th, z = −2 → ~2.3rd.

By common convention, |z| > 2 is considered unusual (fewer than 5% of values would be at least this far from the mean), and |z| > 3 is treated as an extreme outlier (less than 0.3% of values). These thresholds underpin the 68-95-99.7 rule and many statistical tests.

The z-score itself can be calculated for any data — it's just an algebraic standardisation. However, the probability and percentile lookups only make sense when the underlying distribution is approximately normal. For heavy-tailed or skewed data, use empirical percentiles, non-parametric methods, or transform the data first (log, Box-Cox, etc.) before standardising.

Rearrange the standardisation: x = μ + z × σ. So a z of 1.5 with μ = 70 and σ = 10 gives x = 70 + 1.5 × 10 = 85. This is useful for setting cut-off thresholds (e.g. "what test score corresponds to the 95th percentile?") — find the z whose Φ(z) equals 0.95 (z ≈ 1.645) and back-solve.

A z-score uses the population standard deviation σ; a t-score uses the sample standard deviation s (with degrees-of-freedom correction). Use z when σ is known (large samples, or theory-based σ); use t when only s is available, especially with small n (typically < 30). For large samples the two converge, since the t-distribution approaches the standard normal.

The cumulative probabilities use a Hart 1968 rational polynomial approximation to Φ(z), which is accurate to roughly seven significant figures for |z| ≤ 7. This is more than sufficient for typical homework, A/B testing, and research. For very deep tail probabilities (|z| ≥ 6 or so), use a dedicated statistics library with extended-precision arithmetic.

A two-tailed p-value is the probability of seeing a z at least as extreme as the observed one in either direction: 2 · (1 − Φ(|z|)). It's the standard p-value for a non-directional hypothesis test ("is x different from μ?"). The one-tailed p is half this and tests a directional alternative ("is x greater than μ?" or "less than μ?").

Yes — that's the whole point of standardisation. A student who scores 80 on a test with μ = 70, σ = 5 (z = 2.0) did better relatively than a student who scored 95 on a test with μ = 80, σ = 10 (z = 1.5), even though the raw score is lower. Z-scores give you a universal scale for "how unusual is this value within its distribution?"

Because the z-score depends on relative distance, not absolute distance. If σ is large, even big absolute deviations produce small z-scores; if σ is small, tiny absolute differences produce large z-scores. A height of 6'2" might be a z of 1.5 in the general population but a z of 0 among professional basketball players.

No. A z-score measures distance in standard deviations for a single value; the standard error is the standard deviation of a sample statistic (such as the sample mean). When you compute a z for a sample mean rather than an individual value, you divide by SE = σ / √n, not σ. That's a z-test, used for inference about means.