KS3 Maths · Year 8 · Chapter 4: Using data · Lesson 4

Cumulative frequency diagrams

Year 8About 70 minutesNo calculator45 marks

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Date

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Quick questions to start.

A1
Last lesson
$4$ values are in a class with mid-value $10$ and $6$ values are in a class with mid-value $20$ . Estimate the mean.
[1 mark]
Answer
A2
Needed today
Write down the running totals of $5$ , $8$ , $12$ and $9$ .
[1 mark]
Answer
A3
Needed today
Work out $\frac{3}{4}$ of $120$ .
[1 mark]
Answer
A4
From chapter 2
Factorise $6 x + 15$ .
[1 mark]
Answer

Example, then your turn

Year 8

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

Example 1

The table shows the times of $60$ swimmers. Work out the cumulative frequencies, and list the points you would plot.

Hint Keep a running total. Each total goes with the TOP of its class.

B1
The table shows the heights of $50$ sunflowers. Work out the cumulative frequency for each class, and list the points you would plot.
[3 marks]

Example 2

The graph shows the times $80$ people took to finish a puzzle. Estimate the median and the interquartile range.

Hint The median is read across from

\frac{1}{2}

80

; the quartiles from

\frac{1}{4}

and

\frac{3}{4}

80

B2
The graph shows the lengths of $60$ leaves. Estimate the median and the interquartile range.
[3 marks]
Answermedian $\dots$ cm IQR $\dots$ cm

Example 3

Use the graph in Example 2. Estimate how many people took more than $36$ minutes.

Hint Read UP from

36

to find how many took

36

minutes or less. Take that from

80

B3
Use the graph in B2. Estimate the percentage of the leaves that are longer than $15$ cm.
[2 marks]
Answer

Example 4

Two classes did the same test. Class P: median $24$ , interquartile range $16$ . Class Q: median $28$ , interquartile range $9$ . Compare the marks of the two classes.

Hint Make one comparison of the medians (the average) and one of the IQRs (the spread). Say what each means.

B4
Two shops record how long customers wait. Shop X: median $6$ minutes, IQR $5$ minutes. Shop Y: median $9$ minutes, IQR $2$ minutes. Make two comparisons of the waiting times.
[2 marks]

Practice

Year 8

C1
$40$ students were measured. Nobody was shorter than $150$ cm. The cumulative frequencies were: $5$ up to $160$ cm, $16$ up to $170$ cm, $36$ up to $180$ cm and $40$ up to $190$ cm. Draw the cumulative frequency graph on the grid, joining the points with straight lines. Use it to estimate the median height.
[4 marks]
Answermedian $\dots\dots$ cm
C2
The table shows the cumulative frequencies (Cum. freq.) of the times of $50$ people. Work out how many people are in the class $20 < t \leq 30$ , and how many took more than $10$ minutes but no more than $30$ minutes.
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

C4
Most of Ben's times to walk to school are between $15$ and $25$ minutes, but one day he took $70$ minutes. Explain why the interquartile range is a better measure of the spread of his times than the range.
[2 marks]

Exam-style questions

GCSE Foundation

D1
The graph shows the times $120$ people spent in a museum. Leo says, "The interquartile range is $90 - 30 = 60$ minutes."
Write down the median. Then show that Leo is wrong by working out the interquartile range, and say what Leo did wrong.
[5 marks]
Answermedian $\dots$ min IQR $\dots$ min

D2
$80$ students sat Paper A and Paper B. The graph shows the cumulative frequencies of their marks.
(a)
Write down the median mark for each paper.
[2 marks]
AnswerA $\dots$ B $\dots$
(b)
Work out the interquartile range for each paper.
[3 marks]
AnswerA $\dots$ B $\dots$
(c)
Sophia says, "Paper A was easier than Paper B." Is she right? Give a reason for your answer.
[2 marks]
Total for D2: 7 marks
D3
The graph shows the times of $200$ runners in a race.
(a)
Estimate how many runners took more than $50$ minutes.
[2 marks]
Answer
(b)
Ali says, "Half the runners took less than $40$ minutes." Is Ali right? Give a reason for your answer.
[2 marks]
Total for D3: 4 marks
D4
The table shows the ages of $120$ people. Which class contains the median? Show your working.
[2 marks]
Answer

Extension

Stretch

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

E1
For $100$ bags, the cumulative frequencies up to $10$ , $20$ , $30$ , $40$ and $50$ kg are $12$ , $30$ , $x$ , $90$ and $100$ . With the points joined by straight lines, the median is $25$ kg. Find $x$ , the number of bags in $20 < m \leq 30$ , and the upper quartile.
[5 marks]
Hint 1 · What to try
The median is read across from $50$ .
Hint 2 · The first line
$25$ kg is halfway from $20$ to $30$ kg, so $50$ is halfway from $30$ to $x$ .
Hint 3 · The full method
Read across from $75$ , between $(30, 70)$ and $(40, 90)$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw a cumulative frequency graph.
I can read the median, quartiles and IQR from a graph.
I can compare two groups with the median and IQR.

Answers and mark scheme

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