Mean, Median & Mode

What is the mode?

The mode is the value that appears the most often
- The mode of 1, 2, 2, 5, 6 is 2
There can be more than one mode
- The modes of 1, 2, 2, 5, 5, 6 are 2 and 5
The mode can also be called the modal value

What is the median?

The median is the middle value when you put values in size order
- The median of 4, 2, 3 can be found by
  - ordering the numbers: 2, 3, 4
  - and choosing the middle value, 3
If you have an even number of values, find the midpoint of the middle two values
- The median of 1, 2, 3, 4 is 2.5
  - 2.5 is the midpoint of 2 and 3
- The midpoint is the sum of the two middle values divided by 2

What is the mean?

The mean is the sum of the values divided by the number of values
- The mean of 1, 2, 6 is (1 + 2 + 6) ÷ 3 = 3
The mean can be fraction or a decimal
- It may need rounding
- You do not need to force it to be a whole number
  - You can have a mean of 7.5 people, for example!

How do I know when to use the mode, median or mean?

The mode, median and mean are different ways to measure an average
In certain situations it is better to use one average over another
For example:
- If the data has extreme values (outliers) like 1, 1, 4, 50
  The mode is 1
  The median is 2.5
  The mean is 14
  - Don't use the mean (it's badly affected by extreme values)
- If the data has more than one mode
  - Don't use the mode as it is not clear
- If the data is non-numerical, like dog, cat, cat, fish
  - You can only use the mode

Worked example

15 students were timed to see how long it took them to solve a mathematical 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 time, giving your answer to 3 significant figures.

Add up all the numbers (you can add the rows if it helps)

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

Total = 68 + 66 + 92 = 226

Divide the total by the number of values (there are 15 values)

$\begin{array}{lcl} \frac{226}{15} & = & 15.066 666 . . . \end{array}$

Write the mean to 3 significant figures
Remember to include the units

The mean time is 15.1 seconds (to 3 s.f.)

(b)

Find the median time.

Write the times in order and find the middle value

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

The median time is 14 seconds

(c)

Explain why the median is a better measure of average time than the mode.

Try to find the mode (the number that occurs the most)

There are two modes: 12 and 14

Explain why the median is better

There is no clear mode (there are two modes, 12 and 14), so the median is better

(d)

If a 16th student has a time of 95 seconds, explain why the median of all 16 students would be a better measure of average time than the mean.

The16th value of 95 is extreme (very high) compared to the other values
Means are affected by extreme values

The mean will be affected by the extreme value of 95 whereas the median will not

Calculations with the Mean

How do I solve harder problems involving the mean?

Remember what the mean is
- Mean = total of values ÷ number of values
  - It is a formula involving three quantities
  - if you know any two, you can find the other one
A question may require you to work backwards from a known mean
- It helps to rearrange the formula
- Total of values = mean × number of values
Find the total of the values before and after to help with question that involve:
- missing values
- adding in, or taking out, a value

Examiner Tip

It helps to start thinking of the mean as a formula which you can rearrange
- Total of values = mean × number of values

Worked example

A class of 24 students has a mean height of 1.56 metres.
A new student joins the class.
The mean height of the class is now 1.57 metres.

Find the height of the new student.

Rearrange the formula for mean to get 'total of heights = mean height × number of students'
Find the total of heights before

Total of heights before = 1.56 × 24 = 37.44

Find the total of heights after
Remember there are now 25 students

Total of heights after = 1.57 × 25 = 39.25

The height of the new student is the difference of the two totals above

39.25 - 37.44 = 1.81

The height of the new student is 1.81 metres

Mean, Median & Mode (CIE IGCSE Maths: Core)

Revision Note

Mean, Median & Mode

What is the mode?

What is the median?

What is the mean?

How do I know when to use the mode, median or mean?

Worked example

Calculations with the Mean

How do I solve harder problems involving the mean?

Examiner Tip

Worked example

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Author: Mark