Averages (Mean, Median, Mode and Range): O-Level / IGCSE Maths (0580)
Syllabus C9.3, E9.3 · Strand 9 Statistics
- Questions
- 6
- Total marks
- 32
- Tier mix
- 4 Core · 2 Extended
This page covers the four core summary measures for discrete data: the mean, median, mode, and range. Together they let you describe a set of numbers with just a few values: three measures of a “typical” or central value, and one measure of how spread out the data is. The objective (C9.3, E9.3) is to calculate each measure and, just as importantly, interpret what it tells you about the data.
The mean is the total of all values divided by how many there are, . The median is the middle value once the data is arranged in order; with values it sits at position , and for an even you average the two central values. The mode is the most frequently occurring value (a data set can have more than one mode, or none), and the range is the largest value minus the smallest, . When data is given in a frequency table, the same ideas apply but you weight by frequency: the mean becomes , and you use a running total of frequencies (the cumulative frequency) to locate the median class or value.
A key skill is choosing and interpreting the right measure: for example, recognising that the mean is affected by extreme values while the median is more resistant, and that the range only reflects the two end values. The worked examples below are original, written to illustrate the syllabus objective.
Question 1
Daniel sits a short mathematics quiz at the end of each week. Each quiz is marked out of 10. His marks for 9 weeks are:
(a) Write down the mode of his marks. [1]
(b) Find the median of his marks. [2]
(c) Find the range of his marks. [1]
(d) Calculate the mean of his marks. [2]
Question 2
A sports coach records the number of goals scored in each match by two of the school's hockey teams during one season.
(a) The table shows the number of goals scored by the Falcons in their 40 matches.
| Goals scored () | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Number of matches () | 6 | 11 | 9 | 8 | 4 | 2 |
Calculate the mean number of goals scored per match by the Falcons. [3]
(b) The table below shows the number of goals scored by the Kestrels. The number of matches in which they scored 3 goals is unknown and is represented by .
| Goals scored () | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Number of matches () | 3 | 6 | 12 | 5 |
The mean number of goals scored per match by the Kestrels is exactly .
Find the value of . [3]
(c) Using your results from parts (a) and (b), calculate the mean number of goals per match for all of the Falcons' and Kestrels' matches combined. [2]
Question 3
The mean of a set of test scores is marks. A ninth student's score is then included, and the mean of all scores becomes marks. What is the ninth student's score?
Question 4
The pie chart shows the favourite drink of a group of people.
(a) The sector for Tea has an angle of . Work out the number of people who chose Tea. [2]
(b) of the people chose Water. Work out the angle of the sector for Water. [2]
Question 5
A local football team played matches last season. The bar chart shows the number of goals the team scored in each match.
(a) Write down the mode of the number of goals scored. [1]
(b) Find the range of the number of goals scored. [1]
(c) Find the median number of goals scored. [2]
(d) Calculate the mean number of goals scored. Give your answer correct to 2 decimal places. [3]
Question 6
A group of students each played a reaction-time game once. The time, seconds, that each student took to respond was recorded. The cumulative frequency diagram shows the distribution of these times.
(a) Use the diagram to find an estimate for the median reaction time. [2]
(b) Use the diagram to find an estimate for the interquartile range of the reaction times. [3]