Start by sorting your values
Five students have read 4, 5, 5, 6 and 20 pages. We use these values to compare three measures of central tendency: mean, median and mode.
Check what each value describes and that the units are the same. Here, all values are numbers of pages. Sort them in order before looking for the median. Our example is already sorted.
Also describe which group the data applies to. Results for five students do not automatically tell you how much all students at the school read.
How to calculate the mean
Add all the values and divide the sum by the number of values.
The sum in our example is 4 + 5 + 5 + 6 + 20 = 40. There are five values. The mean is therefore 40/5 = 8 pages.
None of the students needs to have read exactly eight pages. The mean describes how much each person would have read if the total number of pages were shared equally.
The value 20 raises the mean. If, in another group, you replace 20 with 10, the mean becomes 30/5 = 6, while the first four values stay the same.
Median and mode
The median is the middle value in a sorted list. In 4, 5, 5, 6, 20, there are five values. The third value is 5, so the median is 5 pages.
With an even number of values, take the mean of the two middle values. In the list 1, 2, 8, 9, the median is (2 + 8)/2 = 5. The median therefore does not have to appear among the original values.
The mode is the value that occurs most often. In our first example, it is 5. A data set can have several modes when several values share the highest frequency. If all values occur equally often, the mode does not clearly distinguish between them.
Range and choice of measure
The range shows the difference between the largest and smallest values. For 4, 5, 5, 6, 20, it is 20 − 4 = 16 pages.
The mean uses all values and is affected by large outliers. The median is less affected by how far a single value is from the centre. The mode can be useful for describing the most common value, including non-numerical data such as colours.
Choose a measure based on the question and say which one you use. Do not simply write «average» when the distinction between mean and median matters. Consider also giving a measure of spread or the original values.
Practise statistics with answers
Problem 1
Find the mean, median, mode and range of 2, 3, 3, 4, 8.
Show solution to problem 1
The sum is 20, so the mean is 4. The median and mode are 3. The range is 8 − 2 = 6.
Problem 2
Calculate the median of 12, 3, 9, 4.
Show solution to problem 2
Sort first: 3, 4, 9, 12. The median is (4 + 9)/2 = 6.5.
Problem 3
Replace 8 with 18 in the first problem. How do the mean and median change?
Show solution to problem 3
The mean becomes (2 + 3 + 3 + 4 + 18)/5 = 6. The median is still 3. Here, a high value affects the mean but not the median.
Ready to try it yourself?
Solve the problems on your own. Then open the solution for each problem and compare your steps.
Practise with answers