Mean, Median and Mode: Differences, Formulas and When to Use Each

Q: What is the difference between mean and median?

The mean is the sum of all values divided by their number; the median is the middle value of the sorted data. The mean reacts strongly to outliers, the median hardly at all.

Q: How do I find the median with an even number of values?

Sort the values and take the average of the two middle values. With 1, 3, 5, 9 the median is (3 + 5) / 2 = 4.

Q: When is the mode the best measure?

For nominal data such as major, brand preference or country, where there is no order. It is also useful to show the most common answer on a rating scale.

Q: Why is median income reported instead of mean income?

Income is strongly right-skewed: a few very high incomes raise the mean far above what a typical person earns. The median shows the income of the person in the middle.

Q: Can a data set have more than one mode?

Yes. With two equally frequent peaks the distribution is bimodal. That often signals two different subgroups in the data, for example beginners and advanced students.

Reviewed by Prof. Dr. Alexander Holtermann · Updated 16 October 2026 · 5 min read

The mean is the arithmetic average, the median the middle value, the mode the most frequent value. Use the median for skewed data or outliers.

Key takeaways

Mean: sum of all values divided by n. Median: the value in the middle of the sorted data. Mode: the most frequent value.
The mean uses every value and is sensitive to outliers; the median is robust; the mode is the only measure for nominal data.
In a symmetric distribution all three are about equal. In a right-skewed distribution the mean lies above the median, in a left-skewed distribution below it.
Rule of thumb: nominal data → mode; ordinal data or skewed metric data → median; symmetric metric data → mean.
Income, rents, reaction times and house prices are typically right-skewed, which is why official statistics usually report median income.

Definitions and formulas

All three are measures of central tendency: they summarize a distribution with one typical value.

The mean (arithmetic mean, average) adds up all values and divides by their number: M = Σx / n. For a population the symbol is μ, for a sample M or x̄.

The median (Mdn) is the value that splits the sorted data in half. With an odd n it is the middle value, with an even n the average of the two middle values. Position: (n + 1) / 2.

The mode is the value or category that occurs most often. A distribution can have no mode (all values different), one mode (unimodal) or several modes (bimodal, multimodal).

Measure	Uses	Robust to outliers?	Minimum level of measurement
Mean	every value	no	interval
Median	order of values	yes	ordinal
Mode	frequencies	yes	nominal

Worked example: monthly rent of seven students

Seven students report their monthly rent in dollars:

380, 420, 450, 450, 480, 520, 1500

The data are already sorted, which you need for the median.

Measure	Calculation	Result
Mean	(380 + 420 + 450 + 450 + 480 + 520 + 1500) / 7 = 4200 / 7	$600
Median	position (7 + 1) / 2 = 4th value	$450
Mode	450 occurs twice	$450

How skewness affects mean and median

Comparing mean and median gives you a quick check of the shape of a distribution.

Symmetric (e.g., normal distribution): mean ≈ median ≈ mode.
Right-skewed (positive skew, long tail to the right): mean > median. Typical for income, rent, waiting times, number of social media followers.
Left-skewed (negative skew, long tail to the left): mean < median. Typical for scores on an easy exam where most people score high and a few fail.

When to use mean, median or mode

Your choice depends on the level of measurement and the shape of the distribution. If in doubt, report both mean and median; readers can then judge the shape themselves.

Situation	Best measure	Why
Nominal data (major, country)	Mode	There is no order, so no middle and no sum.
Ordinal data (single Likert item, grades)	Median	Order is meaningful, distances are not.
Metric data, roughly symmetric	Mean	Uses all information; basis for t-test and ANOVA.
Metric data, skewed or with outliers	Median	Not distorted by extreme values.
Further analysis with parametric tests	Mean (plus SD)	Tests such as the t-test compare means.

Special cases: weighted mean and grouped data

A weighted mean gives some values more importance than others: M_w = Σ(w · x) / Σw. You know it from course grades.

The trimmed mean removes a fixed share of the smallest and largest values (for example 10% on each side) before averaging. It is a compromise between mean and median and is used in robust statistics.

For grouped data (only class intervals are known, such as income bands), you can estimate the mean by using the midpoint of each class, weighted by its frequency.

How to report central tendency in APA style

Always pair a measure of central tendency with a measure of spread: the mean with the standard deviation, the median with the interquartile range. Report the sample size. Use the symbols M, SD, Mdn and IQR, italicized in your document.

Common mistakes

These errors appear again and again in seminar papers and theses:

Reporting only the mean for clearly skewed variables such as income.
Computing a mean of nominal codes (for example “average major = 2.4”).
Forgetting to sort the data before finding the median.
Reporting a mode for a continuous variable with no repeated values; it is meaningless there.
Reporting a mean without any measure of spread.

Frequently asked questions

What is the difference between mean and median?