zvärde

Zvärde, commonly called the z-score, is a statistical measure that expresses how many standard deviations an observation is from the mean of a distribution, typically the standard normal distribution with mean 0 and standard deviation 1. It is used to standardize values so comparisons can be made across different scales and distributions.

Calculation and variants: For a population with mean μ and standard deviation σ, the zvärde of a value X is z = (X − μ) / σ. If σ is unknown and a sample standard deviation s is used, the corresponding statistic is often approximated by z ≈ (X − x̄) / s, though in inferential procedures the t-statistic is preferred when σ is unknown and the sample size is small. For large samples, the distinction between z and t becomes less critical.

Distribution and interpretation: Zvärden are interpreted relative to the standard normal distribution N(0, 1). The probability that a standard normal variable falls within a given z-interval is obtained from the cumulative distribution function Φ. In hypothesis testing, z-values are converted to p-values; for a two-tailed test, p = 2(1 − Φ(|z|)).

Applications: Zvärde is widely used to standardize data, enabling comparison across different units or scales. It underpins many statistical methods, including confidence intervals for known σ, hypothesis testing with known σ, and outlier detection (commonly using thresholds such as |z| > 3). It also provides a basis for comparing observations from different distributions by converting them to the standard normal scale.

See also: standard normal distribution, z-score, p-value, t-statistic.