What is Weighted average? — Mathematics

Definition 61.7 Primary & Middle School Mathematics · Chapter 61 — Proportionality, Speed and Averages

When values $v_1, v_2, \dots$ come with counts (or weights) $n_1, n_2, \dots$ , their weighted average is

\bar v = \frac{n_1 v_1 + n_2 v_2 + \dots}{n_1 + n_2 + \dots} :

total of the values, divided by total of the weights.

Examples

Example 61.8

A test was taken by two groups: group 1, $12$ students, average $11$ ; group 2, $18$ students, average $14$ . Average of the whole class:

\bar v = \frac{12 \times 11 + 18 \times 14}{12 + 18} = \frac{132 + 252}{30} = \frac{384}{30} = 12.8 .

Not $\frac{11 + 14}{2} = 12.5$ : the larger group pulls the average towards its own — the weights matter.

Example 61.9 (Average speed over two legs)

A cyclist rides $30$ km at $30$ km/h, then $30$ km at $15$ km/h. The average speed over the whole trip is not $\frac{30 + 15}{2} = 22.5$ km/h. Compute with the definition:

times: $\frac{30}{30} = 1$ h, then $\frac{30}{15} = 2$ h; total $3$ h;
total distance $60$ km, so $v = \frac{60}{3} = 20$ km/h.

The slow leg lasts longer, so it weighs more — an average of speeds must be weighted by time.

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