Distributions

Normal Distribution and Skew

Understanding the shape of a distribution is fundamental to choosing appropriate statistical analyses and interpreting data correctly. The normal distribution is the most important theoretical distribution in statistics — many psychological and biological variables approximate it, and the parametric statistical tests used in most psychological research assume normally distributed data. Skewness describes how far a distribution departs from symmetry, and has important implications for data analysis and interpretation.

The Normal Distribution

The normal distribution (also called the Gaussian distribution or bell curve) is a symmetrical, unimodal distribution in which the mean, median, and mode all coincide at the centre. It is fully described by just two parameters: the mean (which determines the location of the centre) and the standard deviation (which determines the width — a larger SD produces a flatter, wider curve; a smaller SD produces a taller, narrower curve).

A key property of the normal distribution is that fixed proportions of the data fall within defined intervals around the mean. Approximately 68% of scores fall within ±1 SD of the mean; approximately 95% fall within ±2 SD; and approximately 99.7% fall within ±3 SD. This is known as the 68-95-99.7 rule (or empirical rule). These fixed proportions are what allow researchers to calculate the probability of obtaining any particular score, and underpin hypothesis testing and the calculation of confidence intervals.

Examples of Normally Distributed Psychological Variables

Many psychological and biological variables approximate the normal distribution in large representative samples: IQ scores (mean = 100, SD = 15 by standardisation), height, reaction time in simple tasks, personality trait scores, and many physiological measures. The normal distribution emerges naturally when a measured variable is influenced by many independent factors each contributing a small effect — the Central Limit Theorem guarantees that sums or means of large samples will approach normality even when the original variable is not normally distributed.

Skewed Distributions

A skewed distribution is asymmetrical — the tail extends further in one direction than the other. Skew affects the relationship between the mean, median, and mode in a predictable way.

In a positively skewed distribution (also called right-skewed), the tail extends to the right — towards higher values. A small number of very high scores pull the mean to the right of the peak. The order is: Mode < Median < Mean. Income distribution is a classic example: most people earn modest amounts, but a small number of very high earners pull the mean far above the median. Psychological examples include reaction times with occasional very slow responses, and measures of rare behaviours.

In a negatively skewed distribution (also called left-skewed), the tail extends to the left — towards lower values. A small number of very low scores pull the mean downwards. The order is: Mean < Median < Mode. Examples include scores on an easy test where most students score near the maximum, with a few scoring very low.

Implications of Skew for Statistical Analysis

The shape of a distribution has critical implications for the choice of statistical methods. Parametric tests (t-test, Pearson's r, ANOVA) assume that data are (at least approximately) normally distributed. When data are substantially skewed, these assumptions are violated and the tests may produce invalid results. In such cases, researchers should either: use non-parametric equivalents (Mann-Whitney U, Wilcoxon, Spearman's rho) that make no distributional assumptions; transform the data (e.g. log transformation to normalise a positively skewed distribution); or report the median rather than the mean as the summary statistic. Skewness is assessed visually (from histograms) and numerically (skewness coefficient).