Z-Score (Standard Score)
A Z-score (standard score) is a dimensionless statistical metric that quantifies the signed number of standard deviations an empirical observation lies above or below the arithmetic mean of a normal distribution.
Detailed Technical Breakdown
In parametric statistics, raw measurements across different fields have completely distinct units of measurement (e.g. test points, blood pressure in mmHg, financial yields in basis points). The Z-score normalizes these heterogeneous variables by standardizing them onto a reference Gaussian distribution with an arithmetic mean of zero (μ = 0) and standard deviation of one (σ = 1). An observation with Z = +1.5 lies 1.5 standard deviations above the population average, corresponding to approximately the 93.32nd percentile.
Key Technical Specifications
- Standard score formula: Z = (X - μ) / σ, where X is raw value, μ is mean, and σ is standard deviation.
- In a two-tailed standard normal distribution, Z = ±1.96 defines the 95% confidence interval boundary.
- Z-scores are symmetrically centered around zero; negative Z indicates a below-average observation.
- The square of a standard normal variable Z² follows a Chi-square distribution with 1 degree of freedom (χ²₁).
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