Basic statistical tests - median

Definition

The middle value when data are ordered - the 50th percentile
- Odd n: the middle observation. Even n: the mean of the two middle observations
A measure of central tendency that is robust to outliers and to skew

	Mean	Median	Mode
Definition	Sum / n	Middle value	Commonest value
Outlier sensitivity	High	Very low	Low
Use when	Symmetric, roughly normal	Skewed, ordinal, or with outliers	Categorical
Spread reported with it	SD	IQR (25th-75th centile)	-

Skew rule: right (positive) skew -> mean > median; left skew -> mean < median
- Income, length of stay, CRP, viral load, waiting times - all right-skewed. Report the median

Median survival is the standard summary in oncology - the time at which 50% of the cohort has died, read off the Kaplan-Meier curve where it crosses 0.5
- Reportable long before the mean, which requires everyone to have had the event
Median is the default summary for length of stay, waiting time, biomarker levels and any cost data

The line inside the box represents the median of the entire dataset
The box spans the interquartile range - the 25th to 75th percentile
The whiskers typically extend to the most extreme values within 1.5 x IQR of the box
Points beyond the whiskers are plotted individually as outliers
A box plot shows the distribution shape: an off-centre median line inside the box indicates skew

Adding one extreme value shifts the mean in proportion to its magnitude, but moves the median by at most one rank position
- -> breakdown point of 50% (half the data can be corrupted before the median is arbitrary), versus 0% for the mean

Data	Summary	Two-group test	Paired
Continuous, normal	Mean (SD)	Unpaired t test	Paired t test
Continuous, skewed or ordinal	Median (IQR)	Mann-Whitney U (Wilcoxon rank-sum)	Wilcoxon signed-rank
>2 groups, normal	Mean (SD)	ANOVA	Repeated-measures ANOVA
>2 groups, non-normal	Median (IQR)	Kruskal-Wallis	Friedman
Categorical	n (%)	Chi-squared (Fisher exact if expected cell <5)	McNemar
Time to event	Median survival	Log-rank	-

Non-parametric tests rank the data rather than using its values -> robust, but with some loss of power if the data really are normal
Assess normality by inspecting a histogram or Q-Q plot, and by sample size - formal normality tests are over-powered in large samples and under-powered in small ones

Median with IQR, not median with SD - the SD belongs to the mean
State n; give the range separately where outliers matter clinically
When a paper reports a mean for an obviously skewed variable, treat it with suspicion - it usually flatters the result
Median difference between groups is not the difference of the medians in general; the Hodges-Lehmann estimator gives a proper confidence interval

A median can be identical in two groups with radically different distributions - always look at the spread
"Median survival not reached" means fewer than half the cohort has had the event, not that the treatment cured anyone
Median survival of 12 months does not mean the patient has 12 months - half live longer, and the tail can be long. This matters more in a consultation than in a paper
Do not compute a median of medians across studies in a meta-analysis
Standard error is not standard deviation: SEM = SD / sqrt(n) and shrinks with sample size. Reporting SEM makes data look tighter than it is - a common presentational trick

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