Averages: Mean, Median, Mode and Range
An average is a single value used to represent a typical or central value in a set of data, and there are three used at KS3: the mode, the most common value, the median, the middle value once the data is placed in order, and the mean, the total of all the values divided by how many there are. The range is not an average but measures how spread out the data is, found by subtracting the smallest value from the largest, and a data set can have no mode, one mode, or more than one.
Before you start
Make sure you're comfortable with these topics first:
Method
- To find the mean, add up all the values in the data set, then divide by how many values there are.
- To find the median, first write the data in order from smallest to largest. If there is an odd number of values, the median is the middle one; if there is an even number of values, the median is the mean of the two middle values.
- To find the mode, identify the value that occurs most often. A data set can have no mode (if every value is different), one mode, or more than one mode (bimodal) if two values are tied for the highest frequency.
- To find the range, subtract the smallest value from the largest value. The range measures how SPREAD OUT the data is, and is not itself a type of average.
- To find a missing value when the mean is already known, multiply the mean by the number of values to find the total, then subtract the sum of the values that are already known.
- Always double check by re-counting how many values are in the list and re-adding the total, since these two simple slips cause most of the errors in this topic.
- When a new value is added to a data set, recalculate the mean from scratch (new total divided by new count) rather than trying to adjust the old mean directly.
Worked example
A cricket player's runs scored in her last 8 innings were: 34, 12, 58, 34, 21, 45, 34, 62. Find the mean, median, mode and range of her scores.
- Order the data: 12, 21, 34, 34, 34, 45, 58, 62.
- Mean: total = 34 + 12 + 58 + 34 + 21 + 45 + 34 + 62 = 300, divided by 8 = 37.5.
- Median: with 8 values (an even number), average the 4th and 5th values in the ordered list, both of which are 34, so the median is (34 + 34) / 2 = 34.
- Mode: 34 appears three times, more often than any other value, so the mode is 34.
- Range: largest minus smallest = 62 - 12 = 50.
Practice questions
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Q1Find the mean of these 5 delivery times, in minutes: 18, 25, 22, 30, 20.Show answer
Answer: 23 minutes (115 / 5)
Q2Find the median of these 6 quiz scores: 14, 9, 18, 11, 16, 13.Show answer
Answer: 13.5 (ordered: 9, 11, 13, 14, 16, 18; the median is the mean of the two middle values, 13 and 14)
Q3Find the mode(s) of this data set: 7, 3, 9, 3, 5, 9, 2.Show answer
Answer: 3 and 9 (this data set is bimodal - both 3 and 9 appear twice, more often than any other value)
Q4Find the range of these 5 temperatures, in degrees Celsius, recorded one January week: -4, 2, -1, 6, 0.Show answer
Answer: 10 degrees Celsius (6 - (-4) = 10)
Q5The mean of 5 numbers is 16. Four of the numbers are 12, 19, 14 and 21. Find the fifth number.Show answer
Answer: 14 (the total of all 5 numbers is 16 x 5 = 80; the four known numbers add up to 66, so the fifth number is 80 - 66 = 14)
Q6Five friends have heights, in cm, with a mean of 152 cm. A sixth friend joins the group, with a height of 164 cm. Find the new mean height of all six friends.Show answer
Answer: 154 cm (the original total is 152 x 5 = 760 cm; the new total is 760 + 164 = 924 cm; the new mean is 924 / 6 = 154 cm)
Q7State the mode of this data set: 4, 11, 7, 15, 2, 9. If there is no mode, explain why.Show answer
Answer: There is no mode, because every value in the list occurs exactly once, so there is no single value that occurs most often
Q8Explain why the range is not considered a type of average, even though it is calculated from a set of data.Show answer
Answer: The range only measures the SPREAD of the data (the difference between the largest and smallest values), not a typical or central value; an average, such as the mean, median or mode, is meant to represent a typical value in the data set, which the range does not do
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A group of 7 friends compare how many text messages they sent yesterday: 45, 62, 38, 45, 51, 70, 45. (a) Find the mode. (b) Find the median. (c) Find the mean, giving your answer to 1 decimal place. (d) Find the range.
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The mean age of a 6-person quiz team is 34. Five of the team members are aged 29, 41, 33, 27 and 38. (a) Work out the age of the sixth team member. (b) A seventh person joins the team. If the mean age of all 7 people is now 35, work out the seventh person's age.
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Five positive whole numbers have a mode of 4, a median of 6, a mean of 7, and a range of 8. Write down a possible set of five numbers that fits all four facts, and show that your set of numbers satisfies each one.
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Free printable worksheet
Want more practice on paper? Download the averages: mean, median, mode and range worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 32 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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