### Weighted Averages

#### Definition

A weighted average is a method of computing an average where some data points contribute more than others. If all the weights of the data point are equal then the weighted average is the same as the mean.

#### Worked Examples

##### Worked Example 1
###### Worked Example

Suppose your module consists of $20$% coursework, $30$% class test and $50$% exam. How would you work out your overall mark? Suppose these are your results:

Mark obtained

Weight

Coursework

$75$ out of $100$

$0.2$

Class Test

$30$ out of $60$

$0.3$

Exam

$60$ out of $75$

$0.5$

Note: the sum of the weights is $1.0$.

###### Solution

To find your overall module mark you have to find the weighted average. First work out the percentage mark of each module by dividing the number of marks obtained by the maximum number of marks in that module. Next, multiply the percentage mark by the weight of each module.

Mark obtained

Weight

Percentage mark

Percentage mark $\times$ Weight

Coursework

$75$ out of $100$

$0.2$

$\frac {75}{100}=0.75$

$0.75 \times 0.2 = 0.15$

Class Test

$30$ out of $60$

$0.3$

$\frac{30}{60}=0.5$

$0.5 \times 0.3 = 0.15$

Exam

$60$ out of $75$

$0.5$

$\frac{60}{75}= 0.8$

$0.8 \times 0.5 = 0.4$

Finally to find your mark, sum the right hand column as follows:

$0.15+0.15+0.4=0.7.$

So your overall mark for the whole module is $70$ out of $100$.

#### Worked Example 2

###### Worked Example

Given a survey of a gym, where $70$% of members are male and $30$% are female, what is the average age of all members if the mean age of the men is $30$ and the mean age of the women is $40$?

###### Solution

As there are a lot more males than female we can not use the mean value of these two numbers to find the average age of people in the gym. Once again, we use weighted averages.

$(0.7 \times 30)+(0.3 \times 40)=33.$

The mean age of a person in this gym is $33$.

#### Video Examples

###### Example 1

This is a video on the application of weighted averages produced by Alissa Grant-Walker.

###### Example 2

This is a worked example on weighted averages produced by Alissa Grant-Walker.