Weighted Mean

Also called Weighted Average

A mean where some values contribute more than others.

Mean

When we do a simple mean (or average), we give equal weight to each number.

Here is the mean of 1, 2, 3 and 4:

weighted average equal

Add up the numbers, divide by how many numbers:

Mean = 1 + 2 + 3 + 44 = 104 = 2.5

Weights

We could think that each of those numbers has a "weight" of ¼ (because there are 4 numbers):

Mean = ¼ × 1 + ¼ × 2 + ¼ × 3 + ¼ × 4
= 0.25 + 0.5 + 0.75 + 1 = 2.5

Same answer.

Now let's change the weight of 3 to 0.7, and the weights of the other numbers to 0.1 so the total of the weights is still 1:

weighted average more weight

Mean = 0.1 × 1 + 0.1 × 2 + 0.7 × 3 + 0.1 × 4
= 0.1 + 0.2 + 2.1 + 0.4 = 2.8

This weighted mean is now a little higher ("pulled" there by the weight of 3).

When some values get more weight than others,
the central point (the mean) can change:

Decisions

Weighted means can help with decisions where some things are more important than others:

camera

Example: Sam wants to buy a new camera, and decides on the following rating system:

Image Quality 50%
Battery Life 30%
Zoom Range 20%

The Sonu camera gets 8 (out of 10) for Image Quality, 6 for Battery Life and 7 for Zoom Range

The Conan camera gets 9 for Image Quality, 4 for Battery Life and 6 for Zoom Range

Which camera is best?

Sonu: 0.5 × 8 + 0.3 × 6 + 0.2 × 7 = 4 + 1.8 + 1.4 = 7.2

Conan: 0.5 × 9 + 0.3 × 4 + 0.2 × 6 = 4.5 + 1.2 + 1.2 = 6.9

Sam decides to buy the Sonu.

What if the Weights Don't Add to 1?

When the weights don't add to 1, divide by the sum of weights.

lunch

Example: Alex usually eats lunch 7 times a week, but some weeks only gets 1, 2, or 5 lunches.

Alex had lunch:

on 2 weeks: only one lunch for the whole week
on 14 weeks: 2 lunches each week
on 8 weeks: 5 lunches each week
on 32 weeks: 7 lunches each week

What is the mean number of lunches Alex has each week?

Use "Weeks" as the weighting:

Weeks × Lunches = 2 × 1 + 14 × 2 + 8 × 5 + 32 × 7
= 2 + 28 + 40 + 224 = 294

Also add up the weeks:

Weeks = 2 + 14 + 8 + 32 = 56

Divide total lunches by total weeks:

Mean = 29456 = 5.25

It looks like this:

weighted average 5.25

But it is often better to use a table to make sure you have all the numbers correct:

Example (continued):

Let's use:

w for the number of weeks (the weight)
x for lunches (the value we want the mean of)

Multiply w by x, sum up w and sum up wx:

Weight w	Lunches x	wx
2	1	2
14	2	28
8	5	40
32	7	224
Σw = 56		Σwx = 294

The symbol Σ (Sigma) means "Sum Up"

Divide Σwx by Σw:

Mean = 29456 = 5.25

(Same answer as before.)

And that leads us to our formula:

Weighted Mean = ΣwxΣw

In other words: multiply each weight w by its matching value x, sum that all up, and divide by the sum of weights.

Summary

Weighted Mean: A mean where some values contribute more than others.
When the weights add to 1: just multiply each weight by the matching value and sum it all up
Otherwise, multiply each weight w by its matching value x, sum that all up, and divide by the sum of weights:

Weighted Mean = ΣwxΣw