Use a z-score when the population standard deviation σ is known or the sample size is large (typically n ≥ 30 so the sample SD is a reliable estimate of σ). Use a t-score when σ is unknown and the sample is small, because the t-distribution has heavier tails that account for the extra uncertainty in estimating the standard deviation. As n grows, the t-distribution converges to the standard normal, so the two tests give nearly identical results for large samples.

Look up the cumulative probability Φ(z) for your z-score in a standard normal table or use a calculator function such as NORM.S.DIST(z, TRUE) in Excel or scipy.stats.norm.cdf(z) in Python. The percentile is Φ(z) × 100. For example, z = 1.00 gives Φ(1.00) ≈ 0.8413, so the value is at the 84th percentile. Negative z-scores map to percentiles below 50.

A one-tailed test checks for a deviation in a single direction (only larger, or only smaller). At α = 0.05 the critical value is z = 1.645. A two-tailed test checks for a deviation in either direction and splits α between the two tails, giving critical values of ±1.96 at α = 0.05. Use one-tailed only when the alternative hypothesis is directional and a deviation in the opposite direction would have the same practical meaning as no effect.

By common convention, |z| ≥ 1.96 is significant at the 0.05 level (two-tailed), |z| ≥ 2.58 is significant at the 0.01 level, and |z| ≥ 3.29 corresponds to 0.001. These come directly from the standard normal CDF: P(|Z| > 1.96) ≈ 0.05. Particle physics uses a much stricter 5σ threshold (z ≈ 5) where the chance of a false positive is about 1 in 3.5 million.

Two common uses: the Altman Z-Score combines five financial ratios to predict corporate bankruptcy risk, and the Sharpe ratio is structurally a z-score that measures excess return per unit of risk, (Rp − Rf) / σp. Traders also use rolling z-scores of price or spread to flag mean-reversion entries when a series moves more than 2 standard deviations from its moving average.

Educational tests often rescale z to avoid negative numbers and decimals. T-scores use mean 50 and SD 10 (T = 50 + 10z). IQ scores use mean 100 and SD 15. Stanines use a 1-9 scale, and the SAT section score uses mean 500 and SD 100. All of these are linear transformations of z, so a z-score of +1 corresponds to T = 60, IQ = 115, or SAT = 600.