Other Types of Mean (Edexcel GCSE Statistics): Revision Note

Exam code: 1ST0

Written by: Roger B

Reviewed by: Dan Finlay

Updated on 8 April 2024

Weighted Mean

What is a weighted mean?

A weighted mean is used when different numbers or values have different weights
- i.e. some of the values have more statistical ‘importance’ than others
To find the weighted mean from a list of values and weights
- $weighted mean = \frac{\sum_{} (value \times weight)}{\sum_{} weights}$
  - $\sum_{} (value \times weight)$ means multiply each value by its weight and add all the products together
  - $\sum_{} weights$ means add all the weights together
- This formula is not on the exam formula sheet, so you need to remember it
An exam question may tell you what weights to use
- e.g. three test papers where Paper 1 has a weight of 25, Paper 2 has a weight of 35, and Paper 3 has a weight of 40
But often the weights need to be determined from context
- The weights could be the percentages of individuals to which different values apply
- If you know the means for separate groups, the overall mean is a weighted average
  - In this case the numbers of individuals in each group are the weights
- See the Worked Example

Examiner Tips and Tricks

If the values and weights are given to you in a table
- add an extra column for working out ‘value × weight’
You may need to consider the context of a question to decide whether a weighted average is necessary

Worked Example

(a) Myfanwy sits three test papers. Paper 1 has a weight of 25, Paper 2 has a weight of 35, and Paper 3 has a weight of 40. She scores 64% on Paper 1, 60% on Paper 2, and 75% on Paper 3. Work out Myfanwy’s final mark.

Use the weighted average formula $weighted mean = \frac{\sum_{} (value \times weight)}{\sum_{} weights}$

$\frac{(64 \times 25) + (60 \times 35) + (75 \times 40)}{25 + 35 + 40} = \frac{6700}{100} = 67$

67%

(b) 50% of the workers in a company earn £460 per week. 35% of the workers earn £600 per week, and the other 15% earn £820 per week. Work out the mean weekly earnings for workers in the company.

Although the question doesn’t say it specifically, this is a weighted average question

Because different numbers of workers earn each wage, you can’t just find the mean of 460, 600 and 820

Use the weighted average formula $weighted mean = \frac{\sum_{} (value \times weight)}{\sum_{} weights}$ , with the percentages as the weights

$\frac{(460 \times 50) + (600 \times 35) + (820 \times 15)}{50 + 35 + 15} = \frac{56300}{100} = 563$

Note you could also work this out by using the decimal versions of the percentages:

$(460 \times 0.5) + (600 \times 0.35) + (820 \times 0.15) = 563$

£563

(c) Three Year 11 classes took a maths test. Class 1 has 28 students, and their mean score was 64%. Class 2 has 25 students, and their mean score was 68%. Class 3 has 27 students, and their mean score was 65%. Work out the mean score for the three classes combined, giving your answer as a percentage correct to one decimal place.

This is also a weighted average question

Because the classes have different numbers of students, you can’t just find the mean of 64, 68 and 65

Use the weighted average formula $weighted mean = \frac{\sum_{} (value \times weight)}{\sum_{} weights}$ , with the mean scores as the values and the numbers of students as the weights

$\frac{(64 \times 28) + (68 \times 25) + (65 \times 27)}{28 + 25 + 27} = \frac{5247}{80} = 65.5875$

65.6% (1 d.p.)

Geometric Mean

What is the geometric mean?

For a set of n values, the geometric mean is found by the formula
- $geometric mean = \sqrt[n]{{value}_{1} \times {value}_{2} \times . . . \times {value}_{n}}$
  - Multiply the values together
  - And take the nth root ( $\sqrt[▯]{▯}$ on your calculator)
  - (square root for two numbers, cube root for three numbers, etc.)
- e.g. the geometric mean of 2, 9 and 12 is
  - $\sqrt[3]{2 \times 9 \times 12} = \sqrt[3]{216} = 6$
An exam question will say when a geometric mean should be calculated
- But the formula is not on the exam formula sheet, so you need to remember it
The geometric mean gives a better ‘average’ for numbers that are going to be multiplied
- For example, in questions involving percentage changes
- The geometric mean of the percentage change multipliers should be found
  - Not of the percentages on their own
  - e.g. use 1.02 for an increase of 2%
  - or 0.98 for a decrease of 2%
- The geometric mean gives the average percentage change that would have the same combined result
- See the Worked Example

Examiner Tips and Tricks

Don’t calculate a geometric mean unless a question specifically asks you to
Make sure you remember the formula – it’s not on the exam formula sheet

Worked Example

A company’s profits increase by 1% in year 1, 7% in year 2, 3% in year 3, and 12% in year 4.

Calculate the geometric mean of these four percentage increases. Give your answer as a percentage, correct to 2 decimal places.

Use the geometric mean formula $geometric mean = \sqrt[n]{{value}_{1} \times {value}_{2} \times . . . \times {value}_{n}}$ with n =4

Remember that for the geometric mean of percentage changes we need to use the percentage change multipliers

1% increase means multiply by 1.01

7% increase means multiply by 1.07

3% increase means multiply by 1.03

12% increase means multiply by 1.12

$\sqrt[4]{1.01 \times 1.07 \times 1.03 \times 1.12} = 1.056671 . . .$

That multiplier corresponds to a percentage increase of 5.6671…%

Note that $1.01 \times 1.07 \times 1.03 \times 1.12 = 1.246695 . . .$

And $1.056671 . . . \times 1.056671 . . . \times 1.056671 . . . \times 1.056671 . . . = 1.246695 . . .$

So a 5.6671…% increase for four years in a row has the same result as the four individual percentage increases

Round answer to 2 decimal places

5.67% (2 d.p.)

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