Normal Distribution

Normal Distribution (Biostatistics)

1. Definition & overview

• The Normal distribution (also called the Gaussian distribution after Carl Friedrich Gauss) is a continuous, theoretical probability distribution that describes how the values of many biological and clinical measurements are spread around their average; many measured quantities in medicine — height, weight, serum sodium, lead concentration, blood pressure — follow it at least approximately (Swinscow 10e Ch.2, pp.13–14).
• It is the single most important distribution in medical statistics because the theory of the sampling distribution of the mean, the standard error, confidence intervals, and almost all of the common parametric significance tests (t-test, ANOVA, Pearson correlation, linear regression) are built on the assumption that the relevant quantity is Normally distributed (Swinscow 10e Ch.3, pp.31–34; Ch.4, pp.39–41).
• The distribution is the limiting, idealised shape of a histogram of a continuous variable as the sample becomes very large and the class intervals become very narrow — the jagged bars of a real histogram smooth into a single continuous curve (Swinscow 10e Ch.1, pp.7–9; Ch.2, pp.13–14).
• It is fully and uniquely specified by just two parameters: the mean (μ), which fixes the location (the centre) of the curve, and the standard deviation (σ), which fixes the spread (the width) of the curve. Once μ and σ are known, the entire curve is determined (Swinscow 10e Ch.2, pp.12–14).
• Notation: a variable that is Normally distributed with mean μ and variance σ² is written X ~ N(μ, σ²). The population mean and SD are denoted by Greek letters (μ, σ); the corresponding sample estimates are denoted by Roman letters (x̄, s, or SD) (Swinscow 10e Ch.2, p.12; Ch.3, pp.29–30).
• ⚠ Conceptual caution: Swinscow stresses that the Normal distribution is a mathematical model — real data only ever approximate it. The practical question in research is never "are the data exactly Normal?" (they never are) but "are they close enough to Normal for the Normal-theory methods to be valid?" (Swinscow 10e Ch.2, p.14; Ghasemi & Zahediasl 2012 [PMID 23843808]).