Mean, Median, Mode and Range in Depth
Mean, median, mode and range in depth extends the four basic averages and range measures to more complex situations: finding a mean from a frequency table, working out a missing data value when the mean is already known, and comparing two data sets using their averages and range. 11+ papers use these to test interpretation as well as calculation, such as saying which of two data sets is more consistent.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a frequency table, multiply each value by its frequency, add these products together, then divide by the total frequency to find the mean.
- To find the median from a frequency table, use the total frequency to work out which position the middle value falls in, then count through the frequencies in order to find it.
- To find a missing value when the mean is known, multiply the mean by the number of values to get the total, then subtract the sum of the known values.
- To compare two data sets, calculate the same measure, such as both means or both medians, for each set and state clearly which is higher or lower, in context.
- Use the range to compare consistency: a smaller range means the data is more consistent, or less spread out.
- Answer comparison questions in a full sentence referring to both data sets and the context, not just with bare numbers.
Worked example
The table shows the number of goals scored in 10 matches: 0 goals in 2 matches, 1 goal in 3 matches, 2 goals in 4 matches, 3 goals in 1 match. Work out the mean number of goals per match.
- Multiply each number of goals by its frequency: 0 x 2 = 0, 1 x 3 = 3, 2 x 4 = 8, 3 x 1 = 3.
- Add these products to find the total number of goals: 0 + 3 + 8 + 3 = 14.
- Add the frequencies to find the total number of matches: 2 + 3 + 4 + 1 = 10.
- Divide the total goals by the total matches: 14 / 10 = 1.4 goals per match.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Six numbers have a mean of 9. Work out their total.Show answer
Answer: 54
Q2A frequency table shows 1 pupil scored 5, 3 pupils scored 6, and 2 pupils scored 8 in a quiz. Work out the mean score.Show answer
Answer: 6.5 ((5 + 18 + 16) / 6 = 39 / 6)
Q3Team A's scores have a range of 12. Team B's scores have a range of 5. Which team's scores are more consistent?Show answer
Answer: Team B, because it has the smaller range
Q4Eight numbers have a mean of 7. Seven of them are 4, 6, 9, 3, 8, 10 and 5. Work out the eighth number.Show answer
Answer: 11 (total needed = 56, sum of the known seven = 45, 56 - 45 = 11)
Q5A data set of 9 values is arranged in order. Which position is the median value in?Show answer
Answer: The 5th position
Q6A frequency table shows: value 2, frequency 5; value 4, frequency 3; value 6, frequency 2. Find the mode.Show answer
Answer: 2 (it has the highest frequency, 5)
Q7Class X has a mean test score of 62 and a range of 30. Class Y has a mean test score of 62 and a range of 12. Compare the two classes' scores in one sentence.Show answer
Answer: Both classes have the same average score, but Class Y's scores are more consistent, since Class Y has the smaller range.
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
The table shows the number of pets owned by 20 pupils: 0 pets, 4 pupils; 1 pet, 6 pupils; 2 pets, 7 pupils; 3 pets, 3 pupils. Work out the mean number of pets per pupil, giving your answer to 2 decimal places.
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Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
Five numbers have a mean of 14. When a sixth number is included, the mean of all six numbers becomes 15. Work out the sixth number.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
Free printable worksheet
Want more practice on paper? Download the mean, median, mode and range in depth 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 25 of 11+ Maths Workbook 3, the whole course as one free printable PDF.
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