# 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 = \frac{1 + 2 + 3 + 4}{4} = \frac{10}{4} = 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**:

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 |

## 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 Sany 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?

Sany: 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 Sany.

## 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 works 7 days a week, but sometimes just 1, 2, or 5 days.

Alex worked:

- 2 weeks: 1 day each week
- 14 weeks: 2 days each week
- 8 weeks: 5 days each week
- 32 weeks: 7 days each week

What is the mean number of days Alex works per week?

Use "Weeks" as the weighting:

Weeks × Days = 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:

Mean = \frac{294}{56} = 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 days (the value we want the mean of)

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

Weight w |
Days x |
wx |
---|---|---|

2 | 1 | 2 |

14 | 2 | 28 |

8 | 5 | 40 |

32 | 7 | 224 |

Σw = 56 | Σwx = 294 |

Note: Σ (Sigma) means "Sum Up"

Divide **Σwx** by **Σx**:

Mean = \frac{294}{56} = 5.25

(Same answer.)

And that leads us to our formula:

Weighted Mean = \frac{Σwx}{Σw}

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 = \frac{Σwx}{Σw}