ZScores

A z-score, also known as a standard score, measures how far a data value is from the mean in units of the standard deviation. For a population with mean μ and standard deviation σ, the z-score of a value X is z = (X − μ)/σ. When working with a sample, substitute the sample mean x̄ and sample standard deviation s, giving z = (X − x̄)/s. Z-scores transform data from any distribution into a standard normal scale with mean 0 and standard deviation 1, enabling direct comparison across different measures.

Interpretation: A positive z-score indicates a value above the mean; a negative one indicates below. The magnitude |z| shows how many standard deviations the value lies from the mean. If the underlying distribution is normal, z-scores correspond to percentiles of the standard normal distribution. For example, with μ = 75 and σ = 5, a score of 85 has z = 2, roughly the 97.5th percentile.

Applications and considerations: Z-scores are used to standardize data for comparisons, detect outliers (commonly |z| > 3), and perform hypothesis tests or feature scaling in statistics and machine learning. In psychometrics and education, standardized scores and IQ scales are built around similar concepts. Caution: when the data are not normally distributed, z-scores can be misleading, and the interpretation in terms of percentiles should be done with awareness of the distribution shape. Also, z-scores rely on accurate estimates of μ and σ; with small samples, estimates may be unstable and alternative methods (such as nonparametric approaches or t-based methods) may be preferable.