Median

The median is the middle value in a dataset when all observations are arranged in ascending or descending order. If the number of observations is odd, the median is the central value; if even, it is the average of the two central values. It divides the distribution into two equal halves and is less sensitive to extreme values than the mean.

The median is a critical measure of central tendency that provides a more representative summary than the mean when data are skewed or contain outliers. Income data, hospital lengths of stay, survival times, and health care costs are all typically right skewed, meaning that a small number of very large values pull the mean upward and away from the experience of the typical individual. In such cases, the median better reflects the "typical" value. In survival analysis, the median survival time is one of the most commonly reported outcomes. In health economics, median costs complement mean costs to provide a complete picture of the cost distribution. Understanding when to use the median versus the mean is essential for accurate data description and communication.

To find the median, all values are sorted from smallest to largest. For an odd number of observations (n), the median is the value at position (n + 1) divided by 2. For an even number, the median is the average of the values at positions n/2 and (n/2) + 1. Unlike the mean, the median uses only the ordinal positions of the data and is therefore unaffected by how extreme the largest or smallest values are.

The median has a breakdown point of 50%, meaning that almost half of the data can be replaced by arbitrarily extreme values and the median still stays within the range of the remaining, unchanged values. The mean, by contrast, has a breakdown point of 0% and is affected by even a single extreme value. This property makes the median a resistant measure of central tendency.

In skewed distributions, the relationship between the mean and median provides information about the direction of skew. In a right skewed distribution (long tail on the right), the mean usually exceeds the median. In a left skewed distribution (long tail on the left), the mean usually falls below the median; the rule can fail, particularly for discrete data or distributions with more than one peak. In a symmetric distribution, the mean and median are equal.

The median is also used as the basis for several other statistics. The interquartile range (IQR), defined as the difference between the 75th and 25th percentiles (Q3 minus Q1), is a resistant measure of spread that is typically reported alongside the median. The median absolute deviation (MAD) is another resistant measure of variability. In survival analysis, the median survival time is the time at which the Kaplan Meier survival curve crosses the 50% probability threshold.

Report the median when data are skewed, when outliers are present, when the distribution has heavy tails, or when the variable is measured on an ordinal scale. It is standard for reporting survival times, hospital lengths of stay, and income, and for describing the typical cost per patient, although economic evaluations still need mean costs because budgets depend on totals. When data are approximately normally distributed and free of outliers, the mean is preferred because it uses all the information in the data and has more desirable mathematical properties for inference. In practice, reporting both the mean and median allows readers to assess the symmetry of the distribution.

A common mistake is reporting only the mean for skewed data, giving a misleading impression of the typical value. Another error is comparing medians between groups using parametric tests (such as the t test) that assume normality and operate on means; a bootstrap confidence interval for the difference in medians, or quantile regression, compares medians directly, whereas the Mann Whitney U test compares whole distributions and tests medians only when the groups' distributions have the same shape. Some researchers report the median without any measure of dispersion, when the interquartile range should accompany it. Finally, in meta analysis, researchers sometimes convert medians to means for pooling without considering whether the conversion formula is appropriate for the data distribution, which can introduce bias.

A study of emergency department wait times at a large urban hospital records the wait times for 500 patients during one week. The mean wait time is 148 minutes, but the median is 92 minutes, reflecting a right skewed distribution where a small number of patients waited more than 6 hours. The hospital administration uses the median (92 minutes) to describe the typical patient experience and the 90th percentile (310 minutes) to show how long the slowest tenth of patients waited, rather than reporting only the mean, which is inflated by the extreme values.