weighted.mean() — Weighted average

An average where some numbers count more than others.

weighted.mean(x, w)

Key idea

• A weighted mean is an average where each value counts by an assigned weight (its importance).
• It equals the sum of (value x weight) divided by the sum of the weights. (e.g. Sum(value x weight)/Sum(weight))
• Heavier weights pull the result toward those values. (e.g. scores 80,90 weights 1,3 -> 87.5)
• If all weights are equal, it becomes the ordinary mean (plain average). (e.g. weights 1,1 -> plain average)
• It is useful for grades where exams count more than quizzes. (e.g. exam weight 3, quiz weight 1)

Definition

$${\bar{x}}_w=\frac{{\sum }_i{w}_i{x}_i}{{\sum }_i{w}_i}$$

$x_i$

each value

$w_i$

its weight (importance)

$\sum$

add up

$${\bar{x}}_w=\frac{0.2(72)+0.3(85)+0.4(90)+0.1(61)}{0.2+0.3+0.4+0.1}=82.0$$

Where you'd use it

• Your grade is 60 on quizzes (worth 20%) and 90 on the exam (worth 80%). weighted.mean gives 0.2 × 60 + 0.8 × 90 = 84, much closer to the exam because it counts more.
• You bought milk at $2 (10 bottles) and $3 (2 bottles). Weighting price by quantity, the average you actually paid is weighted.mean ≈ $2.17, not the simple $2.50.
• A product has ratings of 5 stars (from 100 people) and 2 stars (from 10 people). Weighting by votes gives ≈ 4.7 stars, reflecting that far more people loved it.

Frequently asked questions

Do the weights need to sum to 1?

No — they’re normalized automatically; they just need to be non-negative and aligned with the values.