Mean, median, mode and range
Levels 57
The objectives of this unit are to:
* calculate the mean, median and mode for a list of numbers;
* calculate the range for a set of data;
* use averages and the range to compare two sets of data;
* use dot plots to illustrate a set of data;
* calculate the mean, median and mode for a simple (ungrouped) frequency table;
* understand when it is best to use the mean, median and mode.
There are three commonly used averages in maths:
* the mode this is the value that occurs most frequently;
* the median this is the value in the middle of the data when it is listed in order of size.
* the mean this is found by adding together all the numbers and then dividing by the number of values.
Averages are used to summarise a set of data they give a typical value for the data.
As well as knowing a typical value for the data, we often also want to know how spread out (or how varied) the data is. The range measures the spread of the data. The range is found by subtracting the smallest value from the largest value, i.e.
range = largest value smallest value.
Example 1:
A school has 9 maths teachers. Their ages are:
28, 31, 62, 45, 39, 50, 31, 22, 42.
The mode (or modal value) is 31 this value occurs more frequently than the other values.
To find the median we write the data in order of size (from smallest to largest):
22, 28, 31, 31, 39, 42, 45, 50, 62.
We then find the middle value. One way of doing this is by crossing out the smallest number and the largest value; then deleting the next smallest and the next largest etc until there is a single number in the middle:
22,
28, 31, 31, 39, 42,
45, 50, 62
We see that the median is 39.
To find the mean, we first add up the numbers: 28 + 31 + 62 + 45 + 39 + 50 + 31 + 22 + 42 = 350.
We then divide this number by 9 (the number of values).
We get: mean = 350 ÷ 9 = 38.9 (to one decimal place).
The largest age is 62 and the youngest teacher is 22 years old.
So the range for the ages is 62 22 = 40.
Note 1: The mean does not have to be a whole number.
Note 2: In this example, the median, mean and mode are all different numbers (although the mean and median are close together).
It is possible to have more than one modal value, as in the following example.
Example 2:
A biased dice was thrown 15 times.
The scores obtained were:
3, 2, 1, 1, 6, 4, 6, 1, 1, 5, 2, 6, 6, 3, 3.
There are two modes here, i.e. 1 and 6. (These numbers both occurred four times each).
To find the median score, we can put the numbers onto a dot plot or we can simply list them in order of size:
1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 6.
We get the median by crossing numbers off each end until we are left with the middle number:
1, 1,
1, 1, 2,
2, 3, 3, 3,
4, 5,
6, 6, 6,
6
We see that the median number is 3.
To get the mean score, we first add up all the numbers: 3 + 2 + 1 + … = 50
Then we divide by the number of values: mean = 50 ÷ 15 = 3.3 (to one decimal place).
Finding the median if there is an even number of values
When there is an even number of values in the data, there is not a single value in the middle of the data. Instead, there is a pair of values in the middle. The median is halfway between the two values in the middle.
Example 3:
The heights (in cm) of 10 people in a class are:
135, 141, 148, 140, 132, 145, 150, 151, 139, 149.
Find the median and range for the heights.
Solution:
To find the median, we write these heights in order of size:
132, 135, 139, 140, 141, 145, 148, 149, 150, 151
If we delete numbers from each end, we get two middle numbers:
132, 135,
139, 140, 141,
145, 148, 149,
150, 151
The median height is halfway between 141 and 145, i.e. median = 143 cm.
The range in heights is 151 132 = 19 cm.
Note 1: We can find the number halfway between two numbers by adding them together and dividing by 2.
Note 2: The median gives a typical height for the class. However the range is not an average. The range tells you how spread out the data is.
Note 3: There is no mode for this data  no value occurs more often than the others.
Comparing two sets of data
We often want to compare two sets of data. We can do this by comparing average values (e.g. comparing the medians or the means) and by comparing the spread (i.e. comparing the ranges for the two sets of data).
Sometimes we also like to compare the data visually. A dot plot is a very simple way to compare two sets of data visually.
Example:
A class did a maths test. The teacher wants to compare how well the boys and the girls did in the test.
The percentage marks for the boys were:
65, 29, 42, 71, 58, 84, 66, 52, 55, 65, 40, 54, 46, 75, 36.
The percentage marks for the girls were:
60, 62, 58, 65, 51, 68, 64, 65, 69, 59, 63, 60, 64, 67.
a) Draw a dot plot to show the boys’ and the girls’ percentages.
b) Find the median mark for the boys and the girls.
c) Find the ranges for the boys’ and the girls’ percentages.
d) Compare the marks obtained by the boys and the girls.
Solution:
a) A dot plot is formed by first drawing a scale. A scale numbered from 20 to 90 will be sufficient here. We can show the marks by placing a dot above the appropriate number on the scale.
If we draw the dot plots with the same scale, one above the other, it makes it easier to compare the boys’ and the girls’ marks.
The dot plots clearly show that the boys’ marks are much more varied than the girls.
b) The median values can be found by listing the values in order of size.
The boys scores in order of size are:
29, 36, 40, 42, 46, 52, 54, 55, 58, 65, 65, 66, 71, 75, 84
The middle value is 55. So:
Median for the boys = 55%.
The girls’ values in order of size are:
51, 58, 59, 60, 60, 62, 63, 64, 64, 65, 65, 67, 68, 69.
There is an even number of girls, so there is a middle pair of values. The median is halfway between 63 and 64, so:
Median for the girls = 63.5%.
Note: The median values could have been found from the dot plots as these give the data values in order of size.
c) Boys’ range = 84 29 = 55%
Girls’ range = 69 51 = 18%
d) The median values show that the girls generally did better in the tests than the boys. The values of the range show that the boys’ scores were more spread out than the girls’ (i.e. the girls’ performances were less varied).
Note: When comparing the two sets of data, it is important that your comparisons refer to the context of the question.
The position of the median value
Instead of crossing numbers off each end of the list to find the middle value, it is sometimes easier to find the median value using the formula:
median = (n + 1)^{th} value.
So, if there are 17 values written in order of size, the median will be the (17 + 1) = 9^{th} number in the list.
If there are 22 values written in order of size, the median will be the (22 + 1) = 11.5^{th} number, i.e. halfway between the 11^{th} and 12^{th} numbers in the list.
Finding averages and the range from a frequency table
When there is a large amount of data, it is usual to summarise the data in a frequency table. It is possible to find the median, mode and mean straight from the frequency table, as shown in the following example.
For example, the Key Stage 3 Maths SATs results obtained by a class of 27 pupils were:
3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5,
5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8.
These results can be put into a frequency table:
Level 
Frequency 
3 
2 
4 
4 
5 
9 
6 
8 
7 
3 
8 
1 
The mode is the value which occurs the most frequently. Here the modal level is 5.
The median level is the middle value. As there are 27 people in the class, the median will be the (27 + 1) = 14^{th} number. We can find this from the original ordered list, but it can also be found from the frequency table.
Level 
Frequency 

3 
2 
There are 2 values to this point 
4 
4 
There are 6 values up to this position 
5 
9 
There are 15 values up to this position 
6 
8 

7 
3 

8 
1 

The 14^{th} number must occur in the level 5 row. So the median level obtained is 5.
The mean can be obtained by putting an additional column onto the end of the table:
Level 
Frequency 
Level × Frequency 
3 
2 
3 ×2 = 6 
4 
4 
4 ×4 = 16 
5 
9 
5 ×9 = 45 
6 
8 
6 ×8 = 48 
7 
3 
7 ×3 = 21 
8 
1 
8 ×1 = 8 
TOTALS 
27 
144 

↑ This total is the number of pupils in the class 
↑ This value represents the total of all the levels scored by all the students in the class. 
The mean level is 144 ÷ 27 = 5.3 (to 1 d.p.)
Steps for finding the mean from a frequency table
Step 1: Copy out the frequency table and put in a third column.
Step 2: Multiply together the numbers in the first two columns. Write the products in the third column.
Step 3: Add up the frequency column and the product column.
Step 4: To find the mean, divide the total of the product column by the total frequency.
Example
The table shows the number of people living in each property in a road:
Number of people 
Frequency 
1 
4 
2 
11 
3 
8 
4 
7 
5 
4 
Find the mean and the median number of people living in a house.
Solution:
To find the mean, we extend the table:
Number of people 
Frequency 
Product 
1 
4 
1 × 4 = 4 
2 
11 
2 × 11 = 22 
3 
8 
24 
4 
7 
28 
5 
4 
20 
Totals 
34 
98 
Mean = 98 ÷ 34 = 2.9 (to 1 d.p.)
The total frequency is 34, so the median is halfway between the 17^{th} and 18^{th} value.
As the 17^{th} and 18^{th} numbers are both 3, the median is 3.
Which is the best average to use?
When to use the mode…
· The mode is the only average that can be used for data that is not numerical. It is possible to find the modal eye colour, for example, but you couldn’t find the median or mean eye colour.
· The mode is also useful of we want to find the most popular value. For instance, if you wanted to design clothing in just once size, it would be helpful to know the modal size that way the clothing will properly fit as many people as possible.
When to use the mean …
The mean is the most commonly used average, so much so that the mean is sometimes just referred to as the “average”. The mean has the advantage that it is worked out using every piece of data. However, it is less useful in cases where the data contains extreme values (or outliers).
When to use the median…
The median is often used for numerical data that contains outliers or data that is not symmetrically distributed (i.e. skewed data).
Example
A small company employs 10 people. The annual salaries of all 10 employees are:
£18000, £18500, £19000, £19000, £21000,
£23000, £25500, £32000, £34000, £85000.
Which type of average would be most appropriate?
Solution:
The data contains one value which is much higher than all the others. This value would distort the value of the mean.
Therefore the median would be the most appropriate average to use here.