Home › Statistics › Mean, Median & Mode
Statistics

Mean, Median & Mode Calculator

Calculate the mean, median, mode, and range of any data set instantly.

Results
Mean = Σx⊂i; / N Median = middle value (sorted); average of two middle values if N is even Mode = most frequently occurring value

These three measures of central tendency describe the center of a distribution differently. Together they give a fuller picture than any one alone.

Mean
=AVERAGE(A1:A20)
Median
=MEDIAN(A1:A20)
Mode
=MODE(A1:A20)

Three Ways to Describe the Center of Data

Mean, median, and mode each describe the "center" of a distribution from a different angle. The mean (arithmetic average) adds all values and divides by count. The median is the middle value when data is sorted. The mode is the most frequently occurring value. Each has strengths and weaknesses that make it more or less appropriate depending on the data's shape and what question you're trying to answer.

For symmetric, normally distributed data without outliers, all three measures are approximately equal, and the mean is usually preferred because it uses all the data and has the best mathematical properties for further analysis. For skewed data or data with extreme outliers, the median is a more representative measure of typical value.

When Mean Misleads: The Outlier Problem

The mean is highly sensitive to extreme values. A classic example: in a room of 10 people with annual incomes of $40,000 to $80,000, the mean income might be $55,000 — a reasonable representation. Add one billionaire with a $1 billion income, and the mean jumps to about $91 million, making it useless as a description of typical income in the room.

This is why median household income is the preferred measure in economic reporting. The distribution of income is heavily right-skewed — a small number of very high earners pull the mean up substantially while leaving most people's experience unchanged. The median correctly describes what the "typical" household actually earns.

Mode in Real-World Applications

Mode is most useful for categorical data where "average" is meaningless, and for understanding the most common value in a dataset. What is the most common shoe size sold? What color car sells most frequently? What age is most common among a company's customers? These are modal questions.

In continuous numerical data, mode is less useful because individual values rarely repeat exactly. Kernel density estimation is often used instead to find the "peak" of a continuous distribution — the mode's analog in continuous contexts.

Skewness and the Relationship Between the Three Measures

The relative positions of mean, median, and mode reveal the shape of a distribution. In a right-skewed distribution (long tail to the right), the mean is pulled rightward by extreme high values: mode ≤ median ≤ mean. In a left-skewed distribution, mode ≥ median ≥ mean. In a symmetric distribution, they're all equal (or nearly so).

This relationship provides a quick diagnostic: if the mean and median differ substantially for your dataset, your data is likely skewed and outliers may be present. Further investigation — checking for data entry errors, considering whether outliers are legitimate data points, and deciding which measure to report — becomes important.

Frequently Asked Questions

Use mean when data has no extreme outliers and is roughly symmetric. Use median when data is skewed or has outliers — it is more resistant to extreme values. Median income is a better measure than mean income because a few billionaires skew the mean dramatically.
Yes. If all values appear the same number of times there is no mode. A data set can also be bimodal (two modes) or multimodal (more than two) if multiple values tie for most frequent.
Range is the largest value minus the smallest — the simplest measure of spread. It is easy to compute but sensitive to outliers. Standard deviation is a more robust spread measure.