GCSE Maths · Higher · Chapter 18: Sampling and more complex diagrams · Lesson 2

Frequency polygons

Grades 4–7About 50 minutesCalculator allowed31 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Write down the midpoint of the class $40 < t \leq 50$ .
[1 mark]
Answer
A2
From chapter 16
In how many orders can 5 people stand in a line?
[1 mark]
Answer
A3
Last lesson
A school of 400 has 60 in Year 7. A stratified sample of 60 is taken. How many from Year 7?
[1 mark]
Answer
A4
From chapter 10
Write down the gradient and the $y$ -intercept of $y = 2 x - 9$ .
[1 mark]
Answer

Example, then your turn

Grades 4–6

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

Example 1

The heights of $30$ plants are in classes $0 < h \leq 6$ , $6 < h \leq 12$ , $12 < h \leq 18$ , $18 < h \leq 24$ , with frequencies $2, 8, 13, 7$ . Write down the points to plot.

Hint Plot each frequency above the middle of its class.

B1
The masses of $40$ parcels are in classes $10 < m \leq 15$ , $15 < m \leq 20$ , $20 < m \leq 25$ , $25 < m \leq 30$ , with frequencies $6, 15, 13, 6$ . Write down the points to plot.
[2 marks]
Answer

Example 2

Work out an estimate of the mean height of the plants in Example 1.

Hint Find

Σ f x

with the mid-class values, then divide by

Σ f

B2
Work out an estimate of the mean mass of the parcels in B1. Give your answer to 1 decimal place.
[3 marks]
Answer

Practice

Grades 5–6

C1
Write down the mid-class value of the class $45 < d \leq 60$ .
[1 mark]
Answer
C2
A frequency polygon has a point at $(35, 12)$ and classes $10$ seconds wide. Write down the class and its frequency.
[2 marks]
Answer

C3
The frequency polygon shows the times taken by some people to solve a puzzle. How many people were timed, and how many took more than 30 seconds?
[2 marks]
Answer $\dots\dots$ people $\dots\dots$ over 30 s
C4
The lengths, $l$ cm, of 28 leaves are grouped as $0 < l \leq 4$ : $3$ , $4 < l \leq 8$ : $9$ , $8 < l \leq 12$ : $11$ , $12 < l \leq 16$ : $5$ . Draw a frequency polygon for the data on the grid.
[2 marks]
C5
Explain why a mean worked out from a grouped table is only an estimate.
[1 mark]

Exam-style questions

Grades 6–7

D1
Waiting times of $40$ people: $0 < t \leq 20$ , $20 < t \leq 40$ , $40 < t \leq 60$ , $60 < t \leq 80$ , with frequencies $14, 18, 6, 2$ . Mo says, "The estimate of the mean is $40$ minutes, because the mean of $10$ , $30$ , $50$ and $70$ is $40$ ."
Explain why Mo is wrong, and work out the correct estimate of the mean.
[3 marks]

D2
A café records how long $40$ customers wait for a drink on a Monday and on a Saturday.
(a)
Write down the modal class for Saturday.
[1 mark]
Answer
(b)
Work out an estimate of the mean waiting time on each day.
[4 marks]
AnswerMonday $\dots\dots$ Saturday $\dots\dots$
(c)
Compare the waiting times on the two days.
[1 mark]
Total for D2: 6 marks
D3
The masses of $23$ suitcases are grouped as $0 < m \leq 10$ : $4$ , $10 < m \leq 20$ : $9$ , $20 < m \leq 30$ : $7$ , $30 < m \leq 40$ : $3$ . Kai draws this frequency polygon for the data. Write down two mistakes he has made.
[2 marks]

Extension

Grade 8 and beyond

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

E1
A frequency polygon is drawn from classes of different widths that follow on from each other, the first starting at $0$ . Its points are $(2.5, 12)$ , $(10, 18)$ , $(25, 20)$ and $(50, 10)$ . Work out the upper end of the last class.
[3 marks]
Hint 1 · What to try
Each $x$ -value is the middle of its class, so the class goes as far above it as below it.
Hint 2 · The first line
The first class is $0$ to $5$ ; the second starts at $5$ with its middle at $10$ .
Hint 3 · The full method
The classes are $0$ – $5$ , $5$ – $15$ , $15$ – $35$ and $35$ – $65$ , so the answer is $65$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw and read a frequency polygon, and spot mistakes in one.
I can work out an estimate of the mean from a grouped table.
I can compare two distributions using their averages.

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