Mean, Median & Mode (Cambridge O Level Maths)

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Paul

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24 October 2023

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Mean, Median & Mode

Why do we have different types of average?

You’ll hear the phrase “on average” used a lot
- For example
  - by politicians talking about the economy
  - by sports analysts
However not all data is numerical
- For example
  - the party people voted for in the last election
- Even when data is numerical, some of the data may lead to misleading results
This is why we have 3 types of average

What are the three types of average?

1. Mean

This is what is usually meant by “average”
- it’s like an ideal world where everybody has the same
- everything is shared out equally
It is the TOTAL of all the values DIVIDED by the NUMBER OF VALUES
Problems with the mean occur when there are one or two unusually high (or low) values in the data (outliers)
- these can make the mean too high (or too low) to reflect any patterns in the data
A formula for the mean could be thought of as $Mean = \frac{Total Amount}{Number of Data}$

2. Median

This is similar to the word medium, which can mean in the middle
So the median is the middle value – but beware, the data has to be arranged into numerical order first
We would use the median instead of the mean if we did not want extreme values (outliers) affecting our data
If there are an odd number of values, there will only be one middle value
If there are an even number of values we would get two values in the middle
- In this case we take the half-way point between these two values
- This is usually obvious but, if not, add the two middle values and divide by 2
  - this is the same as finding the mean of the middle two values

3. Mode

Not all data is numerical and that is where we use mode
MOde means the Most Often
It is often used for things like “favourite …” or “… sold the most” or “… were the most popular”
Mode is sometimes referred to as modal
- you may see phrases like “modal value”
- they all mean the same thing, the value occurring most often
Be aware that the mode can be applied to numerical data
Sometimes if no value/data occurs more often than the others we say there is no mode
- If two values occur the most we may say there are two modes (bi-modal)
  - whether it is appropriate to do this will depend on what the data is about

Worked example

Briefly explain why the mean is not a suitable average to use in order to analyse the way people voted in the last general election.

Political parties/politicians have names and so the data is non-numerical

Suggest a better measure of average that can be used.

The mode average can be used for non-numerical data

Worked example

15 students were timed how long it took them to solve a maths problem. Their times, in seconds, are given below.

12	10	15	14	17
11	12	13	9	21
14	20	19	16	23

Find the mean and median times.

There are a fair amount of numbers so it may be wise to do the adding up in bits - we've used rows.

12 + 10 + 15 + 14 + 17 = 68
11 + 12 + 13 + 9 + 21 = 66
14 + 20 + 19 + 16 + 23 = 92

$\begin{array}{rcl} Mean & = & \frac{68 + 66 + 92}{15} \\ = & \frac{226}{15} \\ = & 15.066 666 . . . \end{array}$

For the median, the data needs to be in order first.

$9 10 11 12 12 13 14 14 15 16 17 19 20 21 23$

Mean = 15.1 seconds (3 s.f.)
Median time = 14 seconds

Comment on the mode of the data.

The mode (or lack of) is easiest to see from the data listed in order in the median question above.

There are two modes (bi-modal) - 12 and 14 seconds

Alternatively we could say there is no mode.

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Calculations with the Mean

What does calculations with the mean involve?

Because the mean has a formula it means you could be asked questions that use this formula backwards and in other ways
Mean = Total of values ÷ Number of values
- it is a formula involving 3 quantities
- if you know any 2, you can find the other one

What calculations with the mean might I have to do?

Typical questions ask you to either
- work backwards from a known mean or
- combine means for two data sets
As this is in the area of problem solving there may be something unusual that you haven’t seen before
- you will need to make sure you understand what the mean is, how it works and what it shows

How do I solve problems involving calculations with the mean?

Known mean, unknown data value
- This is working backwards from the mean, to an unknown data value
- Call the unknown data value $x$ , say
- Using the 'formula' for the mean, set up an equation in $x$
- Rearrange and solve the equation to find $x$ , the unknown data value
Combined means for two data sets
- This is where we know the mean for two different data sets but would like to know the overall mean
- We would need to find the overall total of values from both data sets
- Then divide by the total number of values across both data sets
- Alternatively we may know the overall mean and want to work back to the mean of one or both of the data sets, or an unknown data value
Others
- Due to the problem solving nature of such questions there will be some variation in question styles
  - The above two should give you a good idea and cover the vast majority of questions
- The best way to start tackling questions with the mean is to
  - write down the quantities you do know
  - write down those you don't know
  - use the 'formula' for the mean to link the unknown and known values

Examiner Tip

You have used the mean so often in mathematics that you do not normally think of it as a formula
- but it is - and, as with other work in using formulas,
  - write down the information you do know
  - and separately write down the information you are trying to find

Worked example

A class of 24 students have a mean height of 1.56 metres.
Two new students join the class and the mean height of the class increases to 1.58 metres.

Given that the two new students are of equal height, find their height.

Start by writing down what we do know.

No. of students originally in the class; n₁ = 24
Mean of the original 24 students; m₁ = 1.56
No. after new students; n₂ = 24 + 2 = 26
Mean after new students; m₂ = 1.58

And now write down what we don't know (but need to know to answer the question).

Height of the two new students (both equal); h metres
Total of all heights before new students; T₁Total of all heights after new students; T₂ = T₁ + h + h = T₁ + 2h

Considering the formula for the mean, and the values before the new students joined, we can work out T₁.

$\begin{array}{rcl} m_{1} & = & \frac{T_{1}}{n_{1}} \\ 1.56 & = & \frac{T_{1}}{24} \\ T_{1} & = & 1.56 \times 24 \\ T_{1} & = & 37.44 \end{array}$

Using the mean 'formula' for the overall mean we can set up, then solve, an equation for h.

$\begin{array}{rcl} m_{2} & = & \frac{T_{2}}{n_{2}} \\ Since T_{2} & = & T_{1} + 2 h = 37.44 + 2 h \\ 1.58 & = & \frac{37.44 + 2 h}{26} \\ 37.44 + 2 h & = & 1.58 \times 26 \\ 2 h & = & 41.08 - 37.44 = 3.64 \\ h & = & 1.82 \end{array}$

Both new students have a height of 1.82 metres

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Mean, Median & Mode (Cambridge O Level Maths)

Revision Note

Why do we have different types of average?

What are the three types of average?

What does calculations with the mean involve?

What calculations with the mean might I have to do?

How do I solve problems involving calculations with the mean?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Paul