# 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: 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

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

• 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

Weeks = 2 + 14 + 8 + 32 = 56

Divide total lunches by total weeks:

Mean =  29456  = 5.25

It looks like this: 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

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