How to Calculate a Weighted Mean

Calculates the weighted mean as Σ(value × weight) ÷ Σweight. It is used for grades with different credits, or classes of different sizes. List the same number of values and weights.

Explanation

A weighted mean is an average in which each value carries its own importance. A plain mean treats every value as equal, but in practice they rarely are.

\bar{x}_w = \dfrac{\sum w_i x_i}{\sum w_i} = \dfrac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}

Here $x_i$ is a value and $w_i$ is its weight. If all the weights are equal, this reduces to the ordinary mean.

Example

Take the values 80, 70, 90 with weights 2, 3, 1 — for instance a score of 80 in a two-credit course, 70 in a three-credit course and 90 in a one-credit course.

\bar{x}_w = \dfrac{80 \times 2 + 70 \times 3 + 90 \times 1}{2 + 3 + 1} = \dfrac{160 + 210 + 90}{6} = \dfrac{460}{6} = 76.67

The unweighted average would be $(80 + 70 + 90) \div 3 = 80$ . The heavily weighted 70 pulls the result down to 76.67.

Where it is used

Grades — weight each score by its credits, which is how a GPA works
Class averages — weight by class size. Averaging the average of a class of 40 with that of a class of 20 is simply wrong
Average price — weight each price by the quantity bought

The weights must not add up to zero, since that would leave nothing to divide by.