Zscoring

Zscoring, or standardization, is a data transformation that rescales a dataset so that its distribution has a mean of zero and a standard deviation of one. For an observation x in a dataset with mean μ and standard deviation σ, the z-score is defined as z = (x − μ)/σ. When computed from a sample, μ is typically the sample mean x̄ and σ the sample standard deviation s. If the data are approximately normally distributed, z-scores map onto the standard normal distribution N(0,1), allowing probabilities and percentiles to be read from the standard normal table.

Z-scores provide a unitless measure of how many standard deviations an observation lies from the mean, enabling comparisons across different datasets or measurement scales. They are widely used in data preprocessing for statistical analyses and machine learning, including regression, principal component analysis, and clustering, where features with different scales can bias results. They are also commonly used in outlier detection, where observations with large absolute z-scores (for example, |z| > 2 or 3) may be treated as potential anomalies. Additionally, z-scores can be transformed into percentiles using the standard normal cumulative distribution function.

Important considerations include the dependence on the estimated mean and standard deviation. Z-scores assume the data are roughly symmetric around the mean; for strongly skewed distributions, z-scores may be misleading. Outliers themselves can distort μ and σ, affecting others’ z-scores. In such cases, robust alternatives using medians and MAD (median absolute deviation) may be preferred. Overall, zscoring is a simple, widely used standardization technique with broad applicability in statistics and data analysis.