zscore

A z‑score, also known as a standard score, measures the number of standard deviations a data point is from the mean of a distribution. For a value x, the z‑score is calculated as (x − μ)/σ, where μ is the population mean and σ is the population standard deviation. In a sample, the sample mean and sample standard deviation are used as estimates of μ and σ.

The z‑score provides a way to compare observations from different scales or distributions by translating them onto a common standard normal scale. Positive z‑scores indicate values above the mean, while negative scores indicate values below the mean. A z‑score of zero corresponds exactly to the mean.

Z‑scores are fundamental in statistical inference. They are used to assess probabilities under the normal distribution, to detect outliers (typically values with |z| > 3), and to perform standardization before applying methods such as principal component analysis or clustering. In hypothesis testing, z‑tests compare sample means to hypothesized population means when the population variance is known or the sample size is large.

The interpretation of a z‑score assumes an approximately normal distribution; in highly skewed data the standard‑score may be misleading. Transformations or non‑parametric alternatives are preferred when normality does not hold. Related concepts include t‑scores, which replace σ with the sample standard deviation when the population variance is unknown.