Choosing Mean and Standard Deviation versus Median and Interquartile Range

Overview

Medical reports often apply a simple rule:

• use mean and standard deviation for normal data;
• use median and interquartile range for non-normal data.

This rule is incomplete.

The choice of summary should begin with the quantity that matters clinically. The mean is the arithmetic average and relates directly to totals and mean differences. The median is the 50th percentile and describes the central ranked observation. The standard deviation describes dispersion around the mean, while the interquartile range describes the middle half of the ordered distribution.

Distribution shape matters, but it is only one consideration. Sample size, measurement scale, outliers, floor and ceiling effects, resource-use questions and the planned inferential analysis also matter.

Why this matters

Consider hospital length of stay. The distribution is usually right-skewed because most patients are discharged within several days while a smaller number require prolonged care.

If the question is:

What stay does a typical patient experience?

the median may be the most direct summary.

If the question is:

How many bed-days are required on average per admission?

the mean is essential because total bed use is the number of admissions multiplied by the arithmetic mean. Replacing the mean with a median would underestimate the contribution of long stays.

The summary should therefore answer the question rather than merely satisfy a distribution label.

What each pair describes

Mean and standard deviation

The mean uses every observed value and describes the equal-share average.

The standard deviation, abbreviated SD, describes the typical scale of deviations around the mean. It has the same unit as the measurement.

The pair is informative when:

• the mean is clinically meaningful;
• the distribution is not dominated by a few observations;
• a population mean or total resource use is the target;
• group comparison will focus on a mean difference; or
• approximately symmetric data make the pair easy to interpret.

Median and interquartile range

The median is the 50th percentile.

The interquartile range, abbreviated IQR, spans from the first to the third quartile and contains the middle 50% of observations.

The pair is informative when:

• the central ranked participant is clinically meaningful;
• the distribution is strongly skewed;
• extreme values are valid but make the mean unrepresentative of a typical observation;
• the measurement is ordinal;
• a time or cost distribution has a long tail; or
• the analysis targets quantiles rather than means.

The median and IQR are resistant to extreme values, but they do not describe tails or totals well.

Distribution shape is important, but not decisive

Approximately symmetric distributions

When a numerical distribution is approximately symmetric and unimodal without influential extremes, mean and median are often similar. Mean with SD usually provides an efficient and interpretable description.

Reporting both pairs may be unnecessary unless the comparison itself is educational or required by a reporting standard.

Right-skewed distributions

For right-skewed variables:

• the mean is commonly greater than the median;
• the SD may be large;
• high values contribute strongly to the mean;
• the median and IQR describe the central majority more robustly.

This does not make the mean invalid. The mean may remain the relevant measure for cost, burden or total resource use.

Bimodal or multimodal distributions

Neither mean–SD nor median–IQR may summarise a multimodal distribution adequately.

The investigator should:

• show the distribution;
• examine clinically meaningful subgroups;
• verify measurement and data sources; and
• avoid reducing distinct populations to one central value.

Bounded and ordinal variables

A bounded score may have floor or ceiling effects. The mean may be difficult to interpret if many observations occupy the boundary.

For a single ordinal item, median and category frequencies are generally more defensible than mean and SD because equal numerical spacing is not established.

A validated multi-item score may sometimes be analysed as approximately continuous, but this requires justification based on its construction, distribution and intended interpretation.

Do not let a normality test choose the summary

A formal test such as Shapiro–Wilk evaluates evidence against an exact normal distribution.

It does not answer:

• whether the mean is clinically meaningful;
• whether a deviation is practically important;
• whether an extreme observation is valid;
• whether a model is robust;
• whether the median answers the research question; or
• whether a transformation improves interpretation.

In a large sample, minor deviations can produce a small p-value. In a small sample, important skewness may not be detected.

Use graphs, clinical reasoning and the estimand.

Data structure

Simulated stroke-cohort summaries

simulated data

Table 1. Descriptive summaries for selected variables

Variable	Mean (SD)	Median (Q1–Q3)	Distributional feature
Age, years	67.8 (12.3)	68 (59–77)	Approximately symmetric
Admission systolic blood pressure, mmHg	158.6 (27.4)	156 (140–175)	Mild right tail
Stroke severity score	9.8 (7.6)	8 (4–14)	Discrete and right-skewed
Length of stay, days	8.4 (10.6)	5 (3–9)	Strong right skew
Time to brain imaging, minutes	68 (55)	52 (34–82)	Right-skewed
Inpatient cost, RM	12,800 (15,600)	8,200 (5,100–13,700)	Strong right skew

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

The table shows why distribution shape alone does not determine importance. Median length of stay describes the central admission, while mean length of stay supports total bed-day planning.

Worked example

Align description and inference

A report should avoid describing a variable only by its median and then performing a test or regression whose main estimand is a mean difference without explanation.

Examples of alignment include:

• mean (SD) with a mean difference and 95% confidence interval;
• median (Q1–Q3) with quantile regression estimating a median difference;
• geometric mean with a ratio of geometric means;
• ordinal category frequencies with an ordinal effect measure.

A Mann–Whitney test does not automatically estimate a median difference. Its interpretation depends on distributional assumptions and the estimand.

Similarly, a skewed outcome can be analysed with a mean-based method when the population mean is the target and the method provides valid inference.

Transformations

A logarithmic transformation can make positive right-skewed data more symmetric and convert multiplicative relationships into additive ones on the log scale.

Analysis on the log scale often leads to:

• geometric means;
• ratios of geometric means; or
• percentage differences.

It does not produce an adjusted comparison of medians automatically [1].

Transformation should be justified by the scientific model and interpretation, not used merely to obtain a non-significant normality test.

Robust inference for a population mean MMed extension

A mean can be clinically important even when observations are skewed.

Possible approaches include:

• Welch’s t-test for independent group means;
• heteroscedasticity-robust standard errors;
• bootstrap confidence intervals under an appropriate resampling design;
• generalised linear models with a suitable distribution and link; and
• influence or sensitivity analyses.

These approaches preserve a mean-based or otherwise explicitly defined estimand. Switching to a rank test may answer a different question.

Suggested reporting

Common mistakes

Knowledge check

Why may the mean be important for a strongly skewed length-of-stay distribution?

Answer and reasoning

Total bed use equals the number of admissions multiplied by the mean stay. Long stays contribute real resources and should not be ignored merely because they make the distribution skewed.

What does the median length of stay describe?

Answer and reasoning

It is the middle ordered stay: half of admissions lasted no longer than the median and half lasted at least that long, subject to ties.

Does a small normality-test p-value require median and IQR?

Answer and reasoning

No. The decision also depends on the clinical estimand, distribution, sample size, model robustness and influence of unusual values. The test alone cannot choose the summary.

Why might both mean–SD and median–IQR be reported?

Answer and reasoning

They may answer complementary questions, particularly for a skewed variable where both arithmetic burden and typical patient experience matter. The reason should be explicit.

Does log transformation make an analysis a comparison of medians?

Answer and reasoning

No. A log-scale mean difference commonly becomes a ratio of geometric means after back-transformation. It does not automatically estimate a median difference.

Key take-home messages

References

1. Bland JM, Altman DG. Transformations, means, and confidence intervals. BMJ. 1996;312(7038):1079. https://doi.org/10.1136/bmj.312.7038.1079