GCSE Statistics · Higher · Chapter 6: Averages

Mean, median, mode and averages from grouped data

About 80 minutesCalculator allowed79 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/stats-mean-median-mode-and-averages-from-grouped-data/

Date

Do Now

The first two are what this chapter needs. The rest bring back earlier chapters so nothing is forgotten.

A1
Needed today
The mean of 4 numbers is $21$ . One more number is added and the mean becomes $23$ . What number was added?
[2 marks]
Answer

A2
Needed today
20 phone calls are in the class $80 < t \leq 120$ (minutes). Assuming they are spread evenly through the class, estimate how many are at most 92 minutes.
[2 marks]
Answer
A3
From chapter 4
A bar chart of the mean height of plants (in cm) has a vertical axis that starts at 100. The bar for Plot A is 120 and the bar for Plot B is 180. How many times as tall as the Plot A bar does the Plot B bar look?
[2 marks]
Answer
A4
Maths skill
Work out $2 4^{2} + 1 8^{2}$ .
[2 marks]
Answer
A5
Maths skill
Work out $5.2 \div 0.02$ .
[2 marks]
Answer

Example, then your turn

Your teacher works through each example with you. Then try the one beside it.

Example 1

The annual salaries, in thousands of pounds, of the 7 staff at a small firm are 21, 23, 24, 24, 26, 28 and 95. Find the mean, median and mode. Which average best represents a typical salary? Give a reason.

Hint Look at how many salaries are below the mean.

B1
The numbers of texts sent in one hour by 9 people are 3, 4, 4, 5, 6, 6, 7, 8 and 48. Work out the mean and the median. Which of the two is the better average here? Give a reason.
[3 marks]

Example 2

40 families were asked how many children they have. The table shows the results. Calculate the mean, the median and the mode number of children.

Hint Multiply each number of children by its frequency to get the total number of children.

B2
50 households were asked how many pets they own. The table shows the results. Work out the mean and the median.
[3 marks]
Answer

Example 3

Class A has 24 students with a mean test mark of 62. Classes A and B together have 54 students with a mean mark of 59. Work out the mean mark of class B.

Hint Work with total marks, not with the means.

B3
The mean mass of 12 boxes is 4.5 kg. Three more boxes are added, and the mean mass of all 15 boxes is 4.8 kg. Work out the mean mass of the 3 boxes that were added.
[2 marks]
Answer

Example 4

A town's population grew by 20% in the first year, 50% in the second year and fell by 10% in the third year. Use the geometric mean to work out the average yearly percentage change.

Hint Change each percentage into a multiplier first.

B4
The price of a bus pass rose by 4%, 10%, 6% and 2% in four years. Use the geometric mean to work out the average yearly percentage rise. Give your answer to 2 decimal places.
[3 marks]
Answer

Example 5

The table shows the lengths, $t$ minutes, of 40 phone calls. Work out an estimate of the mean length. Explain why it is only an estimate.

Hint Use the midpoint of each class for every call in it.

B5
The table shows the masses, $m$ grams, of 60 eggs. Work out an estimate of the mean mass, to 1 decimal place. Explain why it is only an estimate.
[4 marks]

Example 6

The table shows the ages, $a$ years, of 80 people at a cinema. Estimate the median age by interpolation.

Hint The median is the 40th person. Find the class they are in, then how far into it.

B6
The table shows the journey times, $t$ minutes, of 120 commuters. Estimate the median journey time by interpolation.
[3 marks]
Answer

Practice

Each question asks for something different.

C1
Work out the median and the range of these masses, in kg: 3.2, 4.8, 2.9, 5.1, 3.6, 4.0
[2 marks]
Answer

C2
A data set has a mean of 12 and a median of 10. Every value in the data set is increased by 5. Write down the new mean and the new median.
[2 marks]
Answer
C3
The four numbers $x$ , $x + 3$ , $2 x$ and 10 have a mean of 9. Work out the value of $x$ .
[2 marks]
Answer
C4
The table shows the times, $t$ seconds, that 50 people took to solve a puzzle. Which class is the modal class? In which class does the median lie?
[2 marks]
Answer
C5
Explain why the geometric mean, and not the arithmetic mean, is used to find an average yearly growth rate.
[2 marks]
C6
A café buys 200 kg of coffee at £3.50 per kg, 300 kg at £4.20 per kg and 500 kg at £3.80 per kg. Work out the mean price per kg.
[2 marks]
Answer
C7
Tom says, "My estimate of the mean from the grouped table is exact, because I used every class."
Is Tom correct? Give a reason.
[2 marks]
C8
A hockey team recorded its goals in each match of a season, as shown in the table. The mean number of goals per match is 2.2. Work out the value of $x$ .
[3 marks]
Answer

Exam-style questions

Show your working: most of the marks are for method.

D1
Ravi says, "Class P has 20 students with a mean score of 58. Class Q has 40 students with a mean score of 66. So the mean score of all 60 students is 62."
Is Ravi correct? Show how you decide.
[3 marks]

D2
The table shows the finishing times, $t$ minutes, of 80 runners in a race.
(a)
Work out an estimate of the mean finishing time.
[3 marks]
Answer
(b)
Estimate the median finishing time by interpolation. Give your answer to 1 decimal place.
[2 marks]
Answer
(c)
The estimated mean is greater than the median. What does this tell you about the finishing times?
[1 mark]
Total for D2: 6 marks
D3
The fixed base index numbers for the price of a train ticket are shown, with 2020 = 100. In 2021 the index was 104, in 2022 it was 113, in 2023 it was 121 and in 2024 it was 125. Use the geometric mean to work out the average yearly percentage rise from 2020 to 2024. Give your answer to 1 decimal place.
[3 marks]
Answer
D4
A school combines three surveys into one satisfaction score. Students have a weight of 5, parents a weight of 3 and staff a weight of 2. The students' mean score is 6.8 and the parents' mean score is 7.4. The weighted mean of all three is 7.0. Work out the staff's mean score.
[3 marks]
Answer
D5
In a town, the mean house price is £312 000 and the median house price is £245 000. An estate agent's advert says, "The typical house here costs £312 000." Explain why the advert may be misleading.
[2 marks]
D6
In 25 netball matches, a team's mean number of goals was 18. In the next 5 matches the team scored 14, 22, 19, 16 and 25 goals. Work out the mean number of goals for all 30 matches.
[2 marks]
Answer
D7
A grouped frequency table of salaries has a last class "over £80 000". Explain why an estimate of the mean cannot be found without making an assumption, and state an assumption you could make.
[2 marks]
D8
Three whole numbers have a mode of 7, a median of 7 and a mean of 9. Work out the three numbers.
[2 marks]
Answer
D9
The mean of 20 values is 14. One value was written down as 41 by mistake; it should have been 14. Work out the correct mean.
[2 marks]
Answer

Extension

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Five positive whole numbers have a mode of 4, a median of 5, a mean of 6 and a range of 7. Work out the five numbers.
[3 marks]
Hint 1 · What to try
Put the five numbers in order. The middle one is the median.
Hint 2 · The first line
The mode is 4, so 4 appears twice, and both 4s are below the median.
Hint 3 · The full method
The total is $5 \times 6 = 30$ and the largest is the smallest plus 7.
Answer

E2
Alex says, "My investment rose by 50% one year and fell by 50% the next, so on average it did not change."
An investment of £800 does this. Work out its value after the two years and the average yearly percentage change, using the geometric mean.
[3 marks]
Hint 1 · What to try
Work out the value after each year.
Hint 2 · The first line
The two multipliers are $1.5$ and $0.5$ .
Hint 3 · The full method
The geometric mean is the square root of their product.
Answer
E3
Ten values have a mean of 12 and a median of 11. Two more values, 3 and 30, are added. Work out the new mean, and explain whether the median changes.
[3 marks]
Hint 1 · What to try
Work with the total, not with the mean.
Hint 2 · The first line
Of the two new values, one is below the median and one is above it.
Hint 3 · The full method
The middle of the data does not move when one value is added on each side.

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find and choose averages, and work with combined, weighted and geometric means.
I can estimate the mean and the median from grouped data and say why they are estimates.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers