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 + 4   =   10   = 2.5
4 4

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

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

Which camera is best?

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

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

Sam decides to buy the Cony.

 

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.

The data:

  • 2 weeks Alex worked 1 day each week
  • 14 weeks Alex worked 2 days each week
  • 8 weeks Alex worked 5 days each week
  • 32 weeks Alex worked 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 =   294   = 5.25
56

It looks like this:

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

Example (continued):

Have:

  • the number of weeks is the weight w
  • and days (the value we want the mean of) is x

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 =   294   = 5.25
56

And that leads us to our formula:

Weighted Mean =   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 =   Σwx
Σw