Mean

The arithmetic average — the sum of all values divided by how many there are.

The mean as fair share — pool everything and redistribute equally

(3 + 7 + 2 + 8 + 5) ÷ 5 = 25 ÷ 5 = 5 — level all bars to that height and each reaches the red line

Definition

The mean (or arithmetic average) of a dataset is the sum of all values divided by how many values there are.

\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}

It answers the question: if all the values were made equal, what would each one be?

Finding the mean

Five students scored: 72, 85, 90, 68, 95.

\bar{x} = \frac{72 + 85 + 90 + 68 + 95}{5} = \frac{410}{5} = 82

The mean score is 82.

The mean can be misleading

If one value is very different from the rest — an outlier — the mean gets pulled toward it. Five people earn $30,000 per year; one earns $1,000,000. The mean salary is about $197,000, which doesn't represent anyone's actual income. In cases like this, the median is often more informative.

Try it

Find the mean of: 4, 7, 7, 10, 12.

Solution

$\bar{x} = \dfrac{4 + 7 + 7 + 10 + 12}{5} = \dfrac{40}{5} = 8$

Related concepts

Median Normal Distribution Sampling Variance

Uses this

K-Means Clustering Law of Large Numbers Linear Regression

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