Normal Distribution

The normal distribution, also called the Gaussian distribution or bell curve, is a continuous distribution that is symmetric about its mean and fully described by its mean and standard deviation. It is the single most important distribution in biostatistics because many measurements approximate it and because sample means tend toward it, making it the basis of most standard inference.

Definition

The normal distribution is a continuous probability distribution with a symmetric, bell-shaped density determined entirely by two parameters, its mean (centre) and standard deviation (spread).

Scope

The entry covers the shape and parameters of the normal distribution, the empirical rule relating standard deviations to coverage, the standard normal distribution and z-scores, reference ranges, and the distinction between a normal distribution of individuals and a normal distribution of sample means. It is a methodological reference and does not provide clinical thresholds for individual patients.

Core questions

• What shape does the normal distribution have and what determines it?
• How much of the distribution lies within a given number of standard deviations?
• What is a z-score and how does standardisation work?
• When is it appropriate to assume normality?

Key concepts

• Mean and standard deviation
• Symmetry and bell shape
• Empirical (68-95-99.7) rule
• Standard normal distribution
• z-score and standardisation
• Reference range
• Skewness and departures from normality

Mechanisms

A normal distribution is fixed by two numbers: the mean, which locates its centre, and the standard deviation, which sets its width. Roughly 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three — the empirical rule that gives the distribution its practical usefulness. Any normal variable can be standardised by subtracting the mean and dividing by the standard deviation to give a z-score following the standard normal distribution (mean 0, standard deviation 1), which lets a single set of tables or formulas serve all normal distributions. In medical research, reference ranges for measurements such as blood values are often constructed from the central 95% of an assumed normal distribution, and many statistical tests assume either that the data or the sampling distribution of a statistic is approximately normal.

Clinical relevance

Many biological measurements are summarised and compared under an assumption of approximate normality, and reference ranges are frequently built from it, so understanding the distribution aids interpretation of laboratory and study results. This entry describes the distribution as methodology and does not set diagnostic cut-offs for individuals.

History

The bell-shaped curve emerged in the eighteenth century from de Moivre's approximation to the binomial distribution and was developed by Laplace and Gauss, the latter using it in the analysis of measurement error, which is why it is often called the Gaussian distribution. Through the nineteenth and twentieth centuries it became the default model for measured biological quantities and the cornerstone of classical statistical inference.

Debates

When does assuming normality mislead?

Many biological variables are skewed rather than symmetric, and treating them as normal can distort reference ranges and tests; whether to transform the data, use distribution-free methods, or rely on the central limit theorem for means is a recurring methodological judgement.

Key figures

• Carl Friedrich Gauss
• Pierre-Simon Laplace
• Abraham de Moivre

Seminal works

• altman-bland-1995-normal
• rosner-2015

Frequently asked questions

What is the 68-95-99.7 rule?

For a normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three; this empirical rule connects the standard deviation directly to the proportion of values covered.

Does data have to be normally distributed to use the normal distribution in inference?

Not always; many methods rely on the sampling distribution of a mean being approximately normal by the central limit theorem, which can hold even when individual measurements are not normally distributed, provided the sample is large enough.