13 Research methods, statistics, critical review and evidence-based practice13.8 Descriptive statistics
13.8.9.1 Recognising data distributions
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Written by Resident Doctor Committee · Published
New in the 2027 syllabus
The College added this topic for exams from January 2027.
Shape of a distribution
Look at the shape of the data before deciding how to analyse them.
A normal distribution is continuous, symmetric and bell-shaped. The mean, median and mode all sit at the centre.
Skewness describes how lopsided a distribution is. Positive skew means a longer tail on the right. Negative skew means a longer tail on the left.
Kurtosis describes how heavy the tails are, which reflects how often extreme values turn up. Do not read it simply as the height of the central peak.
Feature | Normal distribution | Possible departure |
|---|---|---|
Symmetry | Both tails have the same shape | Skewness |
Peaks | One | Several peaks, suggesting a mixture of groups |
Kurtosis | 3, or excess kurtosis 0 | Heavier or lighter tails |
Bounds | Extends without limit in both directions | Floor or ceiling effects |
Two traps follow from this.
Zero skewness does not prove normality. A symmetric distribution can still have unusual tails or more than one peak.
Excess kurtosis is ordinary kurtosis minus 3. Check which convention a paper uses before you interpret a reported value.
Examine plots
Plotting the data is the first step. Each display shows some features well and hides others.
Display | What to look at | Limitation |
|---|---|---|
Histogram | Shape, tails, gaps and peaks | Its appearance changes with the bin width |
Box plot | Median, spread and possible outliers | Can hide more than one peak |
Quantile-quantile plot | The ordered data values plotted against the values a normal distribution would predict (theoretical normal quantiles) | Small samples give irregular patterns by chance |
A normal quantile-quantile plot is roughly a straight line when the data are compatible with a normal distribution. A consistent curve away from the line suggests a departure.
Separate groups can each have a simple distribution, yet their pooled values may show several peaks (a multimodal distribution).
Formal tests and model assumptions
The Shapiro-Wilk test takes normality as its null hypothesis. A small p value suggests the data depart from normality.
Failing to reject normality does not show that the data are normal, especially in a small sample. In a large sample, the test can pick up deviations too small to affect the planned analysis.
Regression assumptions usually concern the errors, or residuals, conditional on the predictors. They do not require every predictor to be normally distributed.
The central limit theorem applies to the sampling distribution of a mean (the distribution of means you would get from repeated samples), under suitable conditions. It does not make the original observations normally distributed.