Mode, Median & Mean (Edexcel GCSE Statistics)

Revision Note

Mode, Median & Mean from Discrete Data

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

    • e.g. the party people voted for in the last election

  • And 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 people usually mean when they say “average”

    • In an ideal world where everybody had the same amount of some resource

    • the mean is the amount of that resource that each person would have

  • It is also known as the arithmetic mean

  • It is the total of all the values divided by the number of values

    • mean space equals space fraction numerator sum space of space values over denominator number space of space values end fraction

      • i.e. add up all the data values

      • then divide by how many values there are

  • 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 accurately represent any patterns in the 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!

  • Use the median instead of the mean if you don't want extreme values (outliers) affecting the average

  • If there are an odd number of values, there will only be one middle value

    • This will be the fraction numerator n plus 1 over denominator 2 end fractionth value

      • e.g. for 35 data values, n equals 35 and fraction numerator n plus 1 over denominator 2 end fraction equals fraction numerator 35 plus 1 over denominator 2 end fraction equals 18

      • So the median will be the 18th value

  • If there are an even number of values there will be two values in the middle

    • In this case we take the halfway point between these two values

    • Often the halfway point is obvious

    • If not, add the two middle values and divide by 2

      • this is the same as finding the mean of the two middle values

3. Mode

  • Think of MOde as meaning the Most Often

    • i.e. it is the value that occurs the greatest number of times

  • It is often used for things like “favourite …” or “… sold the most” or “… were the most popular”

  • Not all data is numerical and that is where the mode is especially useful

    • But be aware that the mode can be applied to numerical data

      • e.g. data about sales of clothing or shoes in different sizes

      • mode would be the best average for determining demand for the sizes

  • The mode is sometimes referred to using the word modal

    • e.g. you may see a phrase like “modal value

    • This means the same thing, the value occurring most often

  • Sometimes no value occurs more often than any other

    • In this case we say there is no mode

  • If two values occur most often we may say there are two modes

    • or say that the data set is bi-modal

    • Whether it is appropriate to do this will depend on what the data is about

How do changes in the data affect the average?

  • You should be able to determine how a change in the data can affect the mean, median or mode

    • For example adding or removing data values to the set

  • You can always recalculate the averages using the changed data set

    • This may be necessary if you need the exact values of the new averages

  • But sometimes you can use logic to decide what kind of change will occur

  • For the mode

    • If an added data value is equal to the modal value, it will not change the mode

    • If a removed data value is not equal to the modal value, it will not change the mode

    • Otherwise recalculate

  • For the mean

    • If an added data value is

      • greater than the mean, the mean will increase

      • less than the mean, the mean will decrease

    • If a removed data value is

      • greater than the mean, the mean will decrease

      • less than the mean, the mean will increase

    • (If an added or removed data value is equal to the mean, the mean will not change!)

    • Recalculate to find the exact value of a changed mean

  • For the median

    • You will need to examine the changed data set to decide if the median has changed

    • Make sure the values in the changed set are written in order!

Worked Example

(a) Briefly explain why the mean is not a suitable average to use in order to analyse people's favourite flavour of ice cream.

Ice cream flavours have names, so the data is qualitative (non-numerical)

(b) Suggest a better measure of average that can be used.

The mode can be used for non-numerical data

Worked Example

15 students were timed to see 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


(a) Find the mean and median times.

There are quite a few numbers to add up, so it helps to add the rows

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

table row Mean equals cell fraction numerator 68 plus 66 plus 92 over denominator 15 end fraction end cell row blank equals cell 226 over 15 end cell row blank equals cell 15.066 space 666 space... end cell end table

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

up diagonal strike 9 space space space space up diagonal strike 10 space space space space up diagonal strike 11 space space space space up diagonal strike 12 space space space space up diagonal strike 12 space space space space up diagonal strike 13 space space space space up diagonal strike 14 space space space space circle enclose 14 space space space space up diagonal strike 15 space space space space up diagonal strike 16 space space space space up diagonal strike 17 space space space space up diagonal strike 19 space space space space up diagonal strike 20 space space space space up diagonal strike 21 space space space space up diagonal strike 23

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

Problem-solving with the Mean

What does problem-solving with the mean involve?

  • The mean is calculated from a formula

    • You could be asked questions that require

      • using the formula 'backwards'

      • rearranging the formula

  • mean space equals space fraction numerator sum space of space values over denominator number space of space values end fraction

    • This is a formula involving 3 quantities

    • If you know any 2, you can find the other one

      • sum space of space values space equals space mean space cross times space number space of space values

      • number space of space values space equals space fraction numerator sum space of space values over denominator mean end fraction

What types of problems might I need to solve?

  • Typical questions ask you to either

    • work backwards from a known mean or

    • combine means for two data sets

  • An exam question can always ask something unusual that you haven’t seen before

    • So you can't just practice 'every type of question' for this topic

  • 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 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 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 to an unknown data value

  • Others

    • Due to the problem solving nature of such questions there will be 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 Tips and Tricks

  • After using the mean so often in mathematics

    • it's easy to forget that it's based on a formula

  • As with other work involving formulas,

    • write down the information you 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

Number of students originally in the class:  n1 = 24
Mean of the original 24 students:  m1 = 1.56
Number 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:  T1
Total 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

table row cell m subscript 1 end cell equals cell T subscript 1 over n subscript 1 end cell row cell 1.56 end cell equals cell T subscript 1 over 24 end cell row cell T subscript 1 end cell equals cell 1.56 cross times 24 end cell row cell T subscript 1 end cell equals cell 37.44 end cell end table

Using the mean formula for the overall mean, we can set up and solve an equation for h

table row cell m subscript 2 end cell equals cell T subscript 2 over n subscript 2 end cell row cell and space space T subscript 2 end cell equals cell T subscript 1 plus 2 h equals 37.44 plus 2 h end cell row cell so space space 1.58 end cell equals cell fraction numerator 37.44 plus 2 h over denominator 26 end fraction end cell row cell 37.44 plus 2 h end cell equals cell 1.58 cross times 26 end cell row cell 2 h end cell equals cell 41.08 minus 37.44 equals 3.64 end cell row h equals cell 1.82 end cell end table

Both new students have a height of 1.82 metres

Mode, Median & Mean from Tables & Charts

How can I find averages if there are lots of values?

  • In the real world there will usually be more data to deal with than just a few numbers

  • The data can be organised in a way that makes it easier understand

    • For example in a table or chart

  • We can still find the mean, median and mode

    • But we have to understand what the table or chart is telling us

  • Be careful - we are not talking here about tables for grouped data

    • In a grouped data table, the individual data values are no longer available

    • See the 'Mode & Mean from Grouped Data' and 'Linear Interpolation' revision notes

  • The tables discussed in this note give the frequencies for each data value

    • That means the entire original data set is still available

How do I find averages from a table or chart?

  • Finding the median and mode from tables/charts is fairly straightforward once you understand what the table/chart is telling you

  • Tables allow data to be summarised neatly

    • and (quite usefully!) they put the data into order

  • The instructions here show how to find averages from tables

    • For charts it is often easiest to turn the chart into a table first

      • See the Worked Example

    • But you can use the same methods directly from a chart if you feel confident doing so

Finding the mean from (discrete) data presented in tables

  • Tables tell us

    • the data value

      • e.g. the number of pets per household

    • and the frequency of that data value

      • i.e., how many times that data value occurred

      • e.g. the number of households with that number of pets

  • STEP 1
    Add a column to the table and work out "data value" × "frequency"

    • This is doing the 'adding up' part of finding the mean

    • We're just doing it one data value at a time

  • STEP 2
    Find the total of the extra column to give the overall total of the data values

  • STEP 3
    Find the mean by dividing this total by the total of the frequency column

    • i.e.  divide the total of the data values by the number of data values

Finding the median from (discrete) data presented in tables

  • The median is the middle value when the data is in order

  • The position of the median can be found by using fraction numerator n plus 1 over denominator 2 end fraction, where n is the number of data values

    • e.g. if n equals 35, then fraction numerator n plus 1 over denominator 2 end fraction equals fraction numerator 35 plus 1 over denominator 2 end fraction equals 18

      • The median is the 18th value

    • Or if n equals 48, then fraction numerator n plus 1 over denominator 2 end fraction equals fraction numerator 48 plus 1 over denominator 2 end fraction equals 24.5

      • The median is midway between the 24th and 25th data values

  • Use the table to deduce where the open parentheses fraction numerator n plus 1 over denominator 2 end fraction close parentheses to the power of th value lies

    • e.g. if the median is the 7th value

    • and the frequency of the first two rows are 4 and 9

      • then the median will be one of the 9 values in the second row of that table

Finding the mode (or modal value)

  • The mode (or modal value) is simple to identify

    • Look for the highest frequency

      • i.e. the data value that occurs the most times

    • The corresponding data value is the mode

    • Make sure you do not confuse the data value with the frequency!

Worked Example

The bar chart shows data about the shoe sizes of pupils in class 11A.

Bar Chart Shoe Size, IGCSE & GCSE Maths revision notes


(a) Find the mean shoe size for the class,

It will be easiest here to rewrite the data as a table
Add an extra column to help find the total of all the shoe size values

Shoe size ()

Frequency ()

xf

6

1

6 × 1 = 6

6.5

1

6.5 × 1 = 6.5

7

3

7 × 3 = 21

7.5

2

7.5 × 2 = 15

8

4

8 × 4 = 32

9

6

9 × 6 = 54

10

11

10 × 11 = 110

11

2

11 × 2 = 22

12

1

12 × 1 = 12

Total

31

278.5


Mean equals fraction numerator 278.5 over denominator 31 end fraction equals 8.983 space 870 space...

Mean = 8.98 (3 s.f.)

Note that the mean does not have to be an actual shoe size

(b) Find the median shoe size,

The bar chart/table has the data in order already
So we just need to find the position of the median

fraction numerator n plus 1 over denominator 2 end fraction equals fraction numerator 31 plus 1 over denominator 2 end fraction equals 32 over 2 equals 16

The median is the 16th value
There are 1 + 1 + 3 + 2 + 4 = 11 values in the first five rows of the table
There are 11 + 6 = 17 values in the first six rows of the table
Therefore the 16th value must be in the sixth row

Median shoe size is 9

(c) Suggest a reason the shop owner may wish to know the modal shoe size of the shop's customers.

The modal size will be more likely to sell than other sizes, so the shop owner should order more shoes in the modal size to stock the shop with.

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Roger B

Author: Roger B

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Author: Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.