standaardscores

Standardscores, or standard scores, are numerical values that express how far a data point lies from the mean of a distribution in units of its standard deviation. They are commonly called z-scores. The standard score of an observation X is z = (X − μ) / σ, where μ is the mean and σ the standard deviation of the population. When working with a sample, z = (X − x̄) / s, with x̄ the sample mean and s the sample standard deviation.

A z-score indicates the position relative to the distribution’s center: z = 0 places the value at the mean, positive z-values lie above the mean, negative below. The magnitude of the score shows how many standard deviations the observation is from the mean, facilitating comparisons across different scales or tests.

Standard scores are widely used to normalize different measurements for comparison, standardize inputs for statistical models, and identify outliers (for example, scores beyond ±3 are often treated as unusual). If the underlying distribution is approximately normal, z-scores follow a standard normal distribution with mean 0 and standard deviation 1, making it possible to translate scores into percentiles using a standard normal table.

Limitations include dependence on the mean and standard deviation of the chosen population or sample, sensitivity to outliers, and reduced interpretability for highly skewed distributions. Despite these caveats, standardscores provide a common framework for comparing and combining data from different sources.