Normal Distribution

Definition & importance

• Definition — The Normal (Gaussian) distribution — named after Carl Friedrich Gauss — is a continuous, theoretical probability distribution describing how the values of many biological and clinical measurements scatter around their average; height, weight, serum sodium, blood pressure and lead concentration follow it at least approximately.
• Why it dominates medical statistics — The sampling distribution of the mean, the standard error, confidence intervals, and almost every common parametric test (t-test, ANOVA, Pearson correlation, linear regression) are built on the assumption that the relevant quantity is Normally distributed — making it the single most important distribution in biostatistics.
• Limiting shape of a histogram — It is the idealised shape a histogram of a continuous variable approaches as the sample grows very large and the class intervals become very narrow — the jagged bars smooth into a single continuous curve.
• Two parameters fully specify it — The curve is determined completely by just the mean (μ), which fixes its location (centre), and the standard deviation (σ), which fixes its spread (width). Written X ~ N(μ, σ²); population values use Greek letters (μ, σ), sample estimates use Roman letters (x̄, s).
• A model, not a fact — The Normal distribution is a mathematical model — real data only ever approximate it. The practical research question is never "are the data exactly Normal?" (they never are) but "are they close enough for Normal-theory methods to be valid?" [PMID 23843808]