Research Methods · May 9, 2026

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Mean, Median, and Mode: Measures of Central Tendency

The mean, median, and mode are the three measures of central tendency — each one describes the "typical" or "central" value in a data set, but in different ways. Choosing the right measure depends on your data type, the shape of the distribution, and whether outliers are present. This guide explains each measure with formulas, worked calculations, and practical guidance on when to use each one.

What is central tendency?

Central tendency refers to the statistical concept of identifying a single value that best represents the center or typical value of an entire data set. Rather than working with every individual data point, a measure of central tendency condenses the data into one representative number.

There are three principal measures of central tendency, each appropriate for different situations:

Mean — the arithmetic average; most common for continuous, normally distributed data
Median — the middle value; preferred when data are skewed or contain outliers
Mode — the most frequent value; the only valid measure of center for nominal data

The mean (arithmetic average)

The arithmetic mean is calculated by summing all values and dividing by the total count. It is the most mathematically powerful measure of center and is used in the majority of statistical tests.

Formula

Population mean

μ = (x₁ + x₂ + … + xN) / N = Σxᵢ / N

Sample mean

x̄ = (x₁ + x₂ + … + xn) / n = Σxᵢ / n

Step-by-step calculation

Data: test scores of 6 students

Scores: 72, 85, 68, 90, 78, 82

Step 1 — Sum: 72 + 85 + 68 + 90 + 78 + 82 = 475
Step 2 — Divide: 475 / 6 = 79.2

Mean = 79.2

Properties of the mean

Uses every data point in its calculation
Minimizes the sum of squared deviations (least-squares property)
Sensitive to extreme values (outliers)
Appropriate for interval and ratio data
The foundation for variance, standard deviation, and most inferential tests

Weighted mean

When some data points are more important than others (e.g., a final exam worth 40% vs. homework worth 10%), use the weighted mean:

Weighted mean formula

x̄_w = Σ(wᵢ · xᵢ) / Σwᵢ

Example — Course grade with weights

Homework (20%): score = 88
Midterm (35%): score = 74
Final (45%): score = 82

Weighted mean = (0.20 × 88) + (0.35 × 74) + (0.45 × 82)
= 17.6 + 25.9 + 36.9 = 80.4

The median

The median is the middle value of an ordered data set. Exactly half of the values fall above it and half fall below. Because it is based on position rather than magnitude, the median is not influenced by extreme values.

How to find the median

Arrange all values in ascending (or descending) order.
If n is odd, the median is the value at position (n + 1) / 2.
If n is even, the median is the average of the values at positions n/2 and (n/2) + 1.

Example — Odd n (n = 7)

Data: 4, 7, 9, 12, 15, 18, 23
Position: (7 + 1) / 2 = 4th value
Median = 12

Example — Even n (n = 8)

Data: 3, 5, 8, 10, 14, 17, 20, 25
Positions 4 and 5: values are 10 and 14
Median = (10 + 14) / 2 = 12
Median = 12

Properties of the median

Resistant to outliers and extreme values
Appropriate for ordinal, interval, and ratio data
The preferred center measure for skewed distributions (income, house prices, response times)
Does not use the magnitude of all values, only their order

The mode

The mode is the value that occurs most frequently in a data set. Unlike the mean and median, the mode can be used with any type of data, including nominal (categorical) data.

Example — Unimodal

Data: 2, 3, 3, 4, 5, 5, 5, 6, 7
5 appears 3 times (more than any other value)
Mode = 5

Example — Bimodal

Data: 1, 2, 2, 3, 4, 4, 5
Both 2 and 4 appear twice
Modes = 2 and 4

Example — Nominal data

Survey responses: "satisfied," "neutral," "satisfied," "dissatisfied," "satisfied"
"satisfied" appears 3 times
Mode = "satisfied"
(Mean and median cannot be computed for this type of data)

Properties of the mode

The only measure of center valid for nominal data
A data set may have no mode, one mode, or multiple modes
Not affected by extreme values
Less informative than mean or median for continuous data
Useful in identifying the most common category in survey research

Effect of outliers on each measure

An outlier is a value that is substantially higher or lower than the other values in the data set. The three measures of central tendency respond to outliers very differently.

Data set — Salaries (in $1,000s) for a small company

Without CEO: 42, 45, 48, 50, 52, 55, 58
With CEO: 42, 45, 48, 50, 52, 55, 58, 850

Measure	Without CEO ($k)	With CEO ($k)	Change
Mean	50.0	150.0	+100 (dramatic)
Median	50.0	51.0	+1 (minimal)
Mode	None	None	Unchanged

Key takeaway: Adding the CEO's salary of $850,000 triples the mean, but barely moves the median. This is why income and wealth data are almost always reported using the median rather than the mean — the mean is misleading when a small number of extreme values dominate.

Mean vs. median in skewed distributions

The relationship between the mean and median reveals the shape of a distribution:

Symmetric distribution: Mean ≈ Median ≈ Mode. All three measures coincide at the center.
Positively skewed (right skew): The tail extends to the right. Extreme high values pull the mean upward. Mean > Median > Mode (roughly).
Negatively skewed (left skew): The tail extends to the left. Extreme low values pull the mean downward. Mean < Median < Mode (roughly).

Real-world examples of skewed distributions

Right skew: household income, response times, city populations, earthquake magnitudes
Left skew: age at retirement, test scores on easy tests (near ceiling), survival times for terminal illness

Reporting practice: When data are skewed, researchers typically report the median and interquartile range (IQR) instead of the mean and standard deviation. Many journals in economics and public health require this for income or cost data.

Which measure to use: a decision guide

Data type / situation	Use this measure	Why
Nominal (categorical)	Mode	Mean and median are undefined for categories
Ordinal (ranked)	Median	Rankings have order but not equal intervals
Interval / ratio, symmetric	Mean	Uses all information; basis for parametric tests
Interval / ratio, skewed	Median	Not distorted by extreme values
Outliers present	Median	Mean is pulled by outliers; median is resistant
Bimodal distribution	Both modes + context	One center value is misleading; report both peaks
Reporting income / house prices	Median	Industry and government standard due to right skew

Full worked example

A psychology researcher records the number of minutes 9 participants spent on a cognitive task: 14, 18, 14, 22, 19, 14, 16, 21, 75.

Step 1: Sort the data

Sorted values

14, 14, 14, 16, 18, 19, 21, 22, 75

Step 2: Calculate the mean

Mean

Sum = 14+14+14+16+18+19+21+22+75 = 213
x̄ = 213 / 9 = 23.7 minutes

Step 3: Find the median

Median (n = 9, odd)

Position = (9+1)/2 = 5th value
5th value in sorted list = 18
Median = 18 minutes

Step 4: Find the mode

Mode

14 appears 3 times; all other values appear once
Mode = 14 minutes

Interpretation

The 75-minute outlier pulls the mean (23.7) well above the median (18) and mode (14). The mean overstates the typical completion time. For this data set, the researcher should report the median as the primary measure of center and note the presence of one extreme outlier.

Quick summary

Feature	Mean	Median	Mode
Definition	Sum divided by n	Middle value in ordered data	Most frequent value
Formula	Σxᵢ / n	Value at position (n+1)/2	Value with highest frequency
Affected by outliers?	Yes, strongly	No (resistant)	No
Data types	Interval, ratio	Ordinal, interval, ratio	Nominal, ordinal, interval, ratio
Skewed data	Misleading	Preferred	Least informative
Number of values	Always one	Always one	Zero, one, or many
Used in	Most parametric tests	Non-parametric tests, reporting skewed data	Frequency analysis, categorical data

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