Concept/Probability and Statistics/No. 0599
Mean vs. Median
The mean and median are measures of center in descriptive statistics. The mean is the sum of all values divided by their count; the median is the middle of ordered values. Extreme values can shift the mean substantially while leaving the median unchanged.
- Evidence
- Well established
- Read
- 6 min
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- 6 connections
01You've seen this when…
- at work
A recruiter points to the company’s average salary. You ask whether executive pay is included, and the number suddenly seems less useful.
- in life
You look up local home prices before making an offer. A handful of waterfront mansions push the mean far above the prices of the houses you’re actually considering.
- out in the world
A clinic reports a median appointment wait of eight days. Patients waiting six weeks wonder why their experience barely registers in the headline.
02The idea
A pay announcement and your coworkers’ paychecks can both be accurate. They may simply describe different versions of the middle.
- The mean balances the values. Add every value and divide by how many there are. Each dollar counts, whether it belongs to an assistant or the chief executive. A sufficiently large salary can pull the mean above what nearly everyone earns.
- The median marks the middle position. Put the values in order. With an odd number of observations, take the middle one. With an even number, the usual convention is to average the two middle values. At least half the observations are at or below the median, and at least half are at or above it.
The mean uses the size of every observation. The median mainly uses their order and the central value or pair of values. Making an already-largest salary ten times bigger changes the mean but leaves the median alone.
Neither is the universally correct average. They answer different questions. Calling either one typical without explaining the question is where trouble starts.
03Why it matters
The distinction changes what you can reasonably infer from a headline number.
- A total needs the mean. Mean salary multiplied by employee count gives total payroll. Median salary describes the middle of the pay distribution. For estimating spending across many comparable cases, the mean helps estimate expected cost, including expensive cases that occur.
- A middle-of-the-pack experience often needs the median. If you want to know where someone falls within a group, the median gives a useful dividing line. When a few large values dominate the mean, the median is often a better starting point for describing ordinary amounts of money (pay or home prices) and waiting times. Severe outcomes need more than either measure. A hospital can have a reassuring median wait while some patients wait dangerously long. For safety decisions, examine the long waits themselves to see the risks a central number can hide.
The median’s resistance to extreme values is one reason it appears in robust statistics. That resistance can also be a drawback. A rare, enormous repair bill may barely affect the median while still threatening your budget.
Both measures also depend on whom you counted. Switching from mean to median cannot fix selection bias, such as a salary survey that leaves out unemployed graduates.
04A worked example
Consider an invented six-person firm. Its annual salaries, in order, are $40,000, $45,000, $50,000, $55,000, $60,000 and $350,000.
Total payroll is $600,000. Divide by six and the mean is $100,000. The two middle salaries are $50,000 and $55,000, so the median is $52,500.
What it looks like A company offering six-figure pay to an ordinary employee. An applicant sees the mean in a recruitment brochure and uses it to set salary expectations.
What’s actually going on Five of the six employees earn less than the advertised mean. The applicant has used a correct calculation to draw a misleading inference about ordinary pay. The median gives a more useful dividing point, though nobody actually earns exactly $52,500. If the highest salary rises to $650,000, the mean jumps to $150,000 while the median stays unchanged. No one else’s pay improves.
What would have helped Naming the measure and showing both figures, alongside the salary range for the advertised role. For the finance team, the mean remains useful: multiplying it by six recovers payroll. For the applicant, the role’s pay band matters more than either company-wide average. Combining executives and assistants hides meaningful differences within the group.
05Where people trip up
- Treating average as a complete description. Ask which measure was used, which observations were included and what each observation represents. Average household income and average individual income answer different questions.
- Assuming the median is what most people experience. It is a dividing point, not the most common value. People can be widely scattered around it, and an even-sized dataset can have a median that nobody actually experiences.
- Calling an extreme value an error just because it moves the mean. A multimillion-dollar insurance claim may be unusual and legitimate. Correct recording mistakes, and retain legitimate expensive cases even when they make the average less appealing.
- Taking an unweighted mean of group means. A branch with ten employees and a branch with ninety employees need weights proportional to their sizes when calculating mean pay per employee. Weight each branch mean by its employee count. Group medians, meanwhile, generally cannot be combined into an overall median without more information.
- Reading the gap as a complete picture of the distribution. A long upper tail often pushes the mean above the median; a long lower tail can pull it below. Two central numbers describe the center. To see more of the distribution’s bumps, clusters and extremes, add a plot or a measure of spread, such as standard deviation.
- Using either measure to summarize compounded growth. An investment that gains 50% and then loses 50% ends down 25%. The arithmetic mean and median of those two returns are both zero. For compounded performance, distinguish arithmetic from geometric growth.
06Roots
In 1722, six years after the mathematician Roger Cotes died, a collection of his work appeared. One problem it addressed was familiar to astronomers: several observations of the same object gave slightly different positions. Cotes described combining observations through a center-of-gravity construction, with more reliable observations carrying more weight. Picture readings placed along a line, pulling toward a balance point.
This work helped turn arithmetic averaging, an old calculation, into a systematic way to handle measurement error. Astronomers wanted one estimate from many imperfect readings, and averaging could make their combined result more dependable.
The middle-of-the-list approach also developed through work on estimation. Pierre-Simon Laplace gave the median a role in his eighteenth-century studies of unknown quantities and error. Judging errors by their absolute size leads naturally to a median; judging them by their squared size leads to a mean. In the nineteenth century, Francis Galton helped popularize medians and ranked positions through his studies of human variation.
The practical question broadened. Scientists expanded their use of these measures from combining repeated measurements of one star’s position to describing different people and outcomes. A balance point and a middle rank remained useful, though they could describe different things.
07How solid is this?
The definitions, sensitivity to extreme values and relationship between the mean and totals are mathematical facts. Which measure is more useful depends on the question, the distribution and how the data were collected.
08Connections
- Part ofRobust Statistics, Expected Value
- See also Selection Bias, Heterogeneity, Arithmetic vs. Geometric Growth, Standard Deviation
09Origin and sources
Classical descriptive statistics, with no single inventor. Averaging developed through measurement and error theory; Laplace studied median-based estimation in the eighteenth century, and Galton helped popularize ranked summaries in the nineteenth.
- [1]Illowsky, B., & Dean, S. (2013). Introductory Statistics. OpenStax, section 2.5, Measures of the Center of the Data.
- [2]NIST/SEMATECH. e-Handbook of Statistical Methods, section 1.3.5.1, Measures of Location.
- [3]Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Harvard University Press.
Suggest an edit· Updated 2026-10-02