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What is the Mean (Arithmetic Average)?

The mean — also called the arithmetic average — is the most widely used measure of central tendency. It answers the question: if all values were equal, what single value would represent the entire data set?

Definition & Formula

The mean is computed by summing all values and dividing by the number of values.

x̄ = (x₁ + x₂ + … + xₙ) / n

Here x̄ (pronounced "x-bar") is the sample mean, n is the count of values, and Σ denotes summation. For the full population the symbol is μ (mu) and n is replaced by N.

Step-by-Step Example

Data set: 4, 7, 13, 2, 14

Count the values: n = 5
Sum them: 4 + 7 + 13 + 2 + 14 = 40
Divide: 40 ÷ 5 = 8

The mean is 8. Note that 8 does not appear in the data set — means are often values that never occur individually.

When to Use the Mean

The mean works best when your data is:

Symmetric — values are distributed roughly evenly around the centre
Free of extreme outliers — no single very large or very small value that distorts the result
On an interval or ratio scale — temperature, weight, exam scores, prices

Common examples: class average on a test, mean daily temperature, average product rating. Try the statistics calculator to compute the mean of your own data instantly.

Sensitivity to Outliers

The mean's biggest weakness is its sensitivity to extreme values. Consider six salaries (in k€):

28, 30, 32, 35, 33, 200

Mean = (28 + 30 + 32 + 35 + 33 + 200) / 6 ≈ 59.7 k€ — a number that represents nobody. The median (32.5 k€) is far more representative. This is why income statistics typically use the median rather than the mean — see mean vs median for a detailed comparison.

Frequently asked questions

Is the mean the same as the average?

Yes — for the arithmetic mean. The word "average" can technically also refer to the median or mode, but in everyday usage and in this calculator, "mean" and "average" refer to the same thing: sum ÷ count.

Does the mean have to be a value in the data set?

No. For example, the mean of [1, 2] is 1.5, which is not in the set. The mean is a mathematical balance point, not necessarily an observed value.

What is the difference between the population mean and the sample mean?

The population mean (μ) is computed from every value in the entire group. The sample mean (x̄) is computed from a subset. The formula is the same; only the symbol and interpretation differ.

When should I use the median instead of the mean?

Use the median when your data contains outliers or is skewed (e.g. income, house prices, response times). The median is not affected by extreme values, making it more representative in those cases.

What is a weighted mean?

A weighted mean gives different values different importance (weights). It is computed as Σ(wᵢ × xᵢ) / Σwᵢ. For example, a course grade might weight the final exam at 50% and homework at 50%.

Related guides

What is the mean? What is the median? What is the mode? Standard deviation explained Mean vs median