Understanding Data Distributions

Overview

A distribution describes how the values of a variable are arranged. It includes more than whether values are “normal” or “not normal”.

A complete description considers:

• centre;
• spread;
• symmetry or skewness;
• number of modes;
• tails;
• gaps;
• unusual observations;
• discreteness;
• lower or upper boundaries; and
• differences across clinically relevant groups.

Understanding distributions helps investigators select meaningful summaries, identify data problems and assess model assumptions. It also prevents the common mistake of allowing one normality-test p-value to determine the entire analysis.

Why this matters

Two variables can have the same mean and standard deviation but different distributions.

One may be approximately symmetric. Another may contain two distinct subgroups. A third may include one influential observation. Anscombe showed that datasets with nearly identical conventional summaries can have strikingly different graphical structures [1].

Medical variables often have non-normal features:

• hospital stay cannot be negative and commonly has a long right tail;
• symptom scores have lower and upper limits;
• admission counts are discrete and may contain many zeros;
• biomarkers may be multiplicative and right-skewed;
• age may show several modes because of the populations sampled; and
• blood pressure may appear bimodal if two distinct clinical groups are combined.

A graph and clinical context are therefore essential.

Components of a distribution

Centre

Centre describes where values tend to lie.

Common measures are:

• mean;
• median; and
• mode.

Their relationship can suggest distributional shape but should not be used as a rigid diagnostic rule.

Spread

Spread describes how far observations vary.

Common measures include:

• standard deviation;
• interquartile range;
• range;
• median absolute deviation; and
• selected percentiles.

A distribution may have the same centre but very different spread.

Symmetry and skewness

A distribution is symmetric when its lower and upper sides are broadly mirror images around the centre.

A right-skewed distribution has a longer tail towards higher values. The mean is often greater than the median because high observations pull the mean upward.

A left-skewed distribution has a longer tail towards lower values. The mean is often lower than the median.

These are tendencies, not mathematical certainties in every finite dataset.

Modality

A mode is a local concentration or peak.

A distribution may be:

• unimodal;
• bimodal; or
• multimodal.

Multiple modes may indicate:

• distinct patient subgroups;
• different measurement processes;
• site effects;
• rounding;
• treatment groups pooled together; or
• data errors.

Tails

Tails are the less frequent values extending from the main body of a distribution.

Heavy tails contain more extreme observations than a normal distribution would predict. Tail behaviour matters for means, standard deviations, confidence intervals and model influence.

Boundaries and discreteness

Some variables have natural limits:

• oxygen saturation: bounded above;
• questionnaire scores: bounded below and above;
• counts: non-negative integers;
• proportions: from 0 to 1;
• durations: non-negative.

A theoretical normal distribution extends without limit in both directions, so it may be unsuitable for modelling variables near strong boundaries.

Graphs for understanding distributions

Dot plot

A dot plot displays individual values and is especially useful for small or moderate samples.

It reveals:

• exact observations;
• clusters;
• gaps;
• repeated values; and
• unusual points.

Histogram

A histogram divides the numerical axis into intervals and displays the number or density of observations in each interval.

Its appearance depends on bin width and starting point. Very wide bins hide features; very narrow bins create noise. Inspecting more than one reasonable bin width may be useful.

Density plot

A density plot provides a smoothed representation of distribution shape.

The smoothing bandwidth affects the result. It should not be interpreted as showing actual observed frequencies at every point.

Box plot

A box plot displays median, quartiles and whiskers, with possible outlier flags.

It is compact for group comparison but can conceal modality and sample size. Values beyond whiskers are not automatically erroneous.

Quantile–quantile plot

A quantile–quantile plot, commonly called a Q–Q plot, compares observed quantiles with those expected under a theoretical distribution such as the normal distribution.

Approximate alignment suggests compatibility in the plotted range. Systematic curvature or tail departures indicate differences. A Q–Q plot does not prove the distribution is normal.

Table 1. What common displays reveal

The normal distribution

The normal distribution is a symmetric, bell-shaped theoretical probability distribution defined by its mean and standard deviation.

For an exact normal distribution:

• approximately 68% of values lie within one standard deviation of the mean;
• approximately 95% lie within 1.96 standard deviations;
• approximately 99.7% lie within three standard deviations.

The normal distribution is useful because:

• many measurement distributions are approximately symmetric;
• means and linear-model residuals often have tractable properties;
• the central limit theorem can make sampling distributions of some estimators approximately normal; and
• many statistical procedures are robust to modest departures under suitable conditions.

However, individual clinical data do not need to be exactly normal for every parametric method. The relevant assumption may concern residuals, sampling distributions or model specification rather than the raw outcome alone [2].

Raw data versus residual distribution

For a two-group comparison, a normality assumption may concern outcomes within each group or residuals after fitting the model—not the pooled variable.

Pooling two groups with different means can produce an apparently bimodal distribution even when each group is individually compatible with the planned model.

Formal normality tests

Tests such as Shapiro–Wilk evaluate evidence against a specified normal distribution.

They have limitations:

• in small samples, important non-normality may not be detected;
• in large samples, trivial departures may produce very small p-values;
• they do not identify the clinical consequence of departure;
• they do not assess independence, linearity or influential observations; and
• they do not select an alternative method automatically.

Normality assessment should combine:

• clinical knowledge;
• distribution plots;
• group-specific displays;
• Q–Q plots;
• sample size;
• model diagnostics;
• robustness considerations; and
• the estimand.

Data structure and simulated summaries

Four numerical distributions

simulated data

Table 2. Distributional summaries in the hypertension survey

Variable	Mean (SD)	Median (Q1–Q3)	Shape
Systolic blood pressure, mmHg	139.2 (18.4)	136 (126–149)	Broadly unimodal with high tail
Time since diagnosis, years	8.7 (8.9)	6.0 (2.5–12.0)	Right-skewed
Number of medicines	2.2 (1.1)	2 (1–3)	Discrete, bounded below by zero
Triglycerides, mmol/L	2.05 (1.64)	1.55 (1.05–2.35)	Strongly right-skewed

Note: Q1–Q3 gives the first and third quartiles; SD = standard deviation.

The table does not replace graphs. It signals which features should be inspected.

Worked example

Distribution of data versus sampling distribution MMed extension

The data distribution describes participant observations.

A sampling distribution describes how an estimator, such as a sample mean, would vary across repeated samples under a design and model.

The central limit theorem can make the sampling distribution of a mean approximately normal even when individual observations are not normal, particularly as sample size increases and no single observation dominates.

This does not make all parametric analyses automatically valid. Independence, variance, clustering, model form and influence still matter.

Suggested reporting

Common mistakes

Knowledge check

What features should be considered besides centre and spread?

Answer and reasoning

Consider symmetry or skewness, modality, tails, boundaries, gaps, unusual observations, discreteness and clinically relevant subgroup patterns.

Why can a pooled histogram be misleading when two treatment groups have different means?

Answer and reasoning

Pooling can create a broad or bimodal shape even if the distribution within each group is compatible with the planned model. Assumptions should be assessed at the level relevant to the analysis.

What does a small Shapiro–Wilk p-value establish?

Answer and reasoning

It provides evidence that the data differ from an exact normal distribution under the test. It does not identify the clinical importance of the departure, select the correct analysis or assess other assumptions.

Why should a box plot be supplemented by another display?

Answer and reasoning

A box plot shows median and quartiles compactly but can hide multimodality, gaps, individual values and sample size. A dot plot, histogram or density plot can reveal these features.

Can the sampling distribution of a mean be approximately normal when participant data are skewed?

Answer and reasoning

Yes, under suitable conditions and sufficient sample size. This is a property of the estimator’s sampling distribution, not evidence that the individual observations are normally distributed.

Key take-home messages

References

1. Anscombe FJ. Graphs in statistical analysis. Am Stat. 1973;27(1):17–21. https://doi.org/10.1080/00031305.1973.10478966
2. Bland JM, Altman DG. The normal distribution. BMJ. 1995;310(6975):298. https://doi.org/10.1136/bmj.310.6975.298