Concept Architecture
Concept
Theoretically, the Median is a measure of central tendency representing the middle value of an ordered dataset. It divides a distribution into two equal parts, with at least half of the observations lying at or below the median and at least half lying at or above it. The median exists to provide a robust summary of central location that is less sensitive than the arithmetic mean to extreme values and skewed distributions.
Mathematically, the median is the 0.5 quantile of a probability distribution. For an ordered sample containing an odd number of observations, it is the central observation; for an even number of observations, it is conventionally calculated as the arithmetic mean of the two central observations. In continuous distributions, a population median is any value m satisfying P(X � m) � 0.5 and P(X � m) � 0.5.
In practice, the median is calculated by arranging observations in ascending order and identifying the central position. In health economics, it is used to summarise skewed variables such as healthcare costs, hospital length of stay, waiting times and resource utilisation. It is commonly reported alongside the interquartile range and may provide a more representative measure of a typical observation than the mean when distributions contain extreme values.
Purpose
Used to summarise the central location of ordered data, describe skewed distributions, reduce sensitivity to extreme observations and support comparisons of healthcare costs, outcomes and resource use.
Mathematical Formulae
Primary Formula
For ordered observations x??? � x??? � ? � x???:
Median = x????????when n is odd
Median = [x????? + x???????] / 2?when n is even
Supporting Formulae
Population median condition:
P(X � m) � 0.5
P(X � m) � 0.5
Quantile representation:
m = Q?.?
Related Mathematical Methods
- Quantile Estimation
- Interquartile Range
- Percentile Estimation
- Median Regression
- Wilcoxon Rank-Sum Test
- Sign Test
Example
A health economist analyses hospital length of stay for seven patients:
2, 3, 4, 5, 6, 8 and 21 days.
The observations are already ordered, and the sample size is:
n = 7
The median position is:
(n + 1) / 2 = (7 + 1) / 2 = 4
The fourth observation is 5 days.
Median length of stay = 5 days
Although the arithmetic mean is 7 days because of the 21-day admission, the median indicates that the central patient stayed for 5 days and is less affected by the extreme observation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MEDIAN | =MEDIAN(B2:B501) | Calculate the median healthcare cost, utility score or length of stay. |
| QUARTILE.INC | =QUARTILE.INC(B2:B501,2) | Calculate the second quartile, equivalent to the median. |
| PERCENTILE.INC | =PERCENTILE.INC(B2:B501,0.5) | Estimate the 50th percentile of a health economic variable. |
| MEDIAN with FILTER | =MEDIAN(FILTER(B2:B501,A2:A501="Treatment")) | Calculate the median for a specified treatment group. |
| COUNT | =COUNT(B2:B501) | Determine whether the number of observations is odd or even. |
VBA (Optional)
A VBA routine can calculate medians and interquartile ranges across multiple treatment groups and generate distributional summary tables.
Sources
- Hyndman RJ, Fan Y. Sample Quantiles in Statistical Packages. The American Statistician. 1996;50(4):361?365.
- Koenker R. Quantile Regression. Cambridge University Press.
- Conover WJ. Practical Nonparametric Statistics. Wiley.
- Altman DG. Practical Statistics for Medical Research. Chapman & Hall.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is the median?
A central tendency measure representing the middle value of an ordered dataset, dividing it into two equal halves.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What does the median mark in an ordered dataset?
The median marks the middle value of a dataset once the values are put in order, the point with as many observations above it as below. Because it depends only on the rank of the middle value rather than the sizes of the extremes, it is barely affected by a few very high or low figures, which makes it the better summary for skewed data such as incomes or costs. It conveys what is typical when the mean would be distorted. The central value of ordered data is what it captures. Kirkwood and Sterne (2003) describe this measure.
Source: Kirkwood & Sterne 2003
How is the median calculated?
The median is calculated by ordering the values from smallest to largest and taking the middle value; if there is an odd number of values, the median is the single middle one, and if an even number, it is the average of the two middle values. So the median is calculated from the ordered data by locating the centre, which depends only on the rank order of the values rather than their magnitudes, making it unaffected by how extreme the largest or smallest values are, in contrast to the mean, which uses every value and is therefore sensitive to outliers.
Source: Casella & Berger 2002
When is the median preferred over the mean?
The median is preferred over the mean when the data are skewed or contain outliers, since it is the middle value and resistant to extreme values, whereas the mean is pulled toward them and can misrepresent the typical value. It is also suitable for ordinal data. So the median is preferred for asymmetric distributions or data with outliers, such as incomes, costs, or survival times, where it better reflects the centre than the mean, which is why reporting the median, often with the interquartile range, is common for such data, while the mean is reserved for reasonably symmetric distributions.
Source: Casella & Berger 2002
How does the median differ from the mean?
The median is the middle value of ordered data, depending only on rank and resistant to outliers, while the mean is the arithmetic average, using all values and sensitive to extremes. In symmetric distributions they coincide, but in skewed ones they differ, with the mean pulled toward the longer tail and the median staying near the bulk of the data. So the median and mean differ in how they summarise the centre and in their robustness, with the median suited to skewed or outlier-prone data and the mean to symmetric data, which is why both are reported to convey the centre and the shape of a distribution.
Source: Casella & Berger 2002
What are the advantages of the median?
The advantages of the median include its robustness to outliers and skew, since it depends only on the middle of the ordered data and is unaffected by extreme values, making it a reliable measure of the typical value for asymmetric distributions; and its applicability to ordinal data. So the median is advantageous where the data are skewed or contain outliers, providing a stable and representative summary of the centre that the mean cannot, which is why it is widely used for such data and paired with the interquartile range, though it uses less of the information in the data than the mean and can be less efficient for symmetric distributions.
Source: Casella & Berger 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 18 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/median
- Term code
- HE-ES-SA-115
Stable URI · Machine-readable · Resolvable · CC BY 4.0