Zvalues

A z-value, also called a z-score or standard score, is a measure of how many standard deviations an individual observation is from the mean of a population. The standard formula is z = (X − μ) / σ, where X is the observed value, μ is the population mean, and σ is the population standard deviation. The concept can be applied to samples as z = (X − x̄) / s, using the sample mean x̄ and sample standard deviation s.

In theory, when the underlying variable is normally distributed with mean μ and standard deviation σ, the z-values follow a standard normal distribution with mean 0 and standard deviation 1. This makes z-values useful for comparing data from different scales and for calculating probabilities and percentiles.

Z-values have several common applications. In hypothesis testing, test statistics based on z, such as Z = (X̄ − μ0) / (σ/√n), underpin z-tests for means when the population standard deviation is known. P-values and confidence intervals are derived from the standard normal distribution. In regression analysis, z-values often refer to standardized coefficients or to the ratio of an estimate to its standard error, informing statistical significance.

Beyond inferential statistics, z-scores are widely used to standardize data for comparability, transforming different variables to a common scale with mean 0 and variance 1. They also serve as a simple rule-of-thumb for outlier detection, with values beyond roughly ±3 indicating potential outliers.

Important caveats include that the exact probabilities tied to z-values assume normality and known σ. When σ is unknown and the sample size is small, the t-distribution is typically more appropriate, though z-values remain a foundational concept in standardization and large-sample practice.