Histograms (GCSE Statistics) - Worksheets, Questions and Revision

15 original exam-style questions - 11 pages of questions with a full mark scheme - free printable PDF.

Download PDF Jump to mark scheme (page 12)Read the revision guide

Next in this strand

Revision Library

revisionlibrary.co.uk

GCSE · Statistics

S25 Histograms

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

For each grouped frequency table described below, write down whether the class widths are equal or unequal.

(i)Class boundaries (time, minutes): 0, 15, 30, 45, 60(1)

(ii)Class boundaries (mass, kg): 0, 5, 15, 40(1)

(Total for Question 1 is 2 marks)

24 pupils were timed, to the nearest whole minute, on how long they took to complete a puzzle. The histogram shows the results.

(a)Write down the frequency of pupils whose time was 6 ≤ t < 7 minutes.(1)

(b)Write down the frequency density for the class 6 ≤ t < 7.(1)

(c)Write down the modal class.(1)

(d)Work out the number of pupils whose time was at least 7 minutes.(2)

(Total for Question 2 is 5 marks)

For each statement about histograms, write down whether it is True or False.

(i)Frequency density is calculated as frequency divided by class width.(1)

(ii)In a histogram, all the bars must have the same width.(1)

(iii)The vertical axis of a histogram should be labelled 'Frequency density', not 'Frequency'.(1)

(Total for Question 3 is 3 marks)

Use the formula frequency density = frequency / class width for each part.

(a)A class has frequency 18 and class width 3. Work out the frequency density for this class.(2)

(b)A different class has frequency 52 and class width 8. Work out the frequency density for this class.(2)

(Total for Question 4 is 4 marks)

Use the formula frequency density = frequency / class width for each part.

(a)A class has frequency density 4 and class width 7. Work out the frequency for this class.(2)

(b)A different class has frequency 54 and frequency density 9. Work out the class width.(2)

(Total for Question 5 is 4 marks)

51 adults were surveyed about the number of hours, h, they spent on household chores in a week. The times were grouped using unequal class widths, since chore time varies a lot between people.

Hours, h	Frequency	Class width	Frequency density
0 ≤ h < 5	9	5	1.8
5 ≤ h < 10	24	5	?
10 ≤ h < 20	15	10	?
20 ≤ h < 35	3	15	?

(a)Work out the frequency density for the class 5 ≤ h < 10.(2)

(b)Work out the frequency density for the class 10 ≤ h < 20.(2)

(c)Work out the frequency density for the class 20 ≤ h < 35.(2)

(Total for Question 6 is 6 marks)

A company recorded the distance, d (km), travelled to work by 60 of its employees. The histogram shows the results.

(a)Write down the frequency density for the class 5 ≤ d < 10.(1)

(b)Write down the modal class.(1)

(c)Work out the number of employees who travel between 10 km and 20 km to work.(2)

(d)Work out the number of employees who travel less than 10 km to work.(2)

(Total for Question 7 is 6 marks)

In Question 7, the distances travelled to work by 60 employees were grouped so that the class 5 ≤ d < 10 has frequency density 3.0 (frequency 15), and the class 10 ≤ d < 20 has frequency density 2.0 (frequency 20). A colleague says: 'The class 10 ≤ d < 20 must be the modal class, because it has the most employees.' Explain why the colleague is wrong.

(Total for Question 8 is 2 marks)

A student is given a grouped frequency table with class widths 2, 5, 3 and 10 hours, and frequencies 2, 18, 9 and 40. She draws a histogram by plotting bar heights equal to the frequencies (2, 18, 9 and 40) directly, and labels the vertical axis 'Frequency'.

(Total for Question 9 is 2 marks)

The weight, w (kg), of 80 parcels handled by a delivery depot one morning was recorded. The histogram shows the results.

(Total for Question 10 is 3 marks)

A gardener recorded the height, h (cm), of 48 sunflower plants grown from the same batch of seeds, using unequal class widths. Some entries in the table are missing.

Height, h (cm)	Frequency	Class width	Frequency density
0 ≤ h < 20	6	20	?
20 ≤ h < 40	22	20	?
40 ≤ h < 60	12	20	?
60 ≤ h < 100	8	40	?

(a)Complete the table by calculating the four missing frequency densities.(4)

(b)On the grid, draw a histogram to show this information.(3)

(Total for Question 11 is 7 marks)

The amount of money, m (GBP), spent by 90 customers in a bookshop one Saturday was recorded. The histogram shows the results.

(a)Write down the frequency density for the class 10 ≤ m < 15.(1)

(b)Work out the percentage of customers who spent less than £15.(3)

(Total for Question 12 is 4 marks)

A petrol station recorded the amount of fuel, f (litres), put into 40 cars filling up one morning. Some entries in the table are missing.

Fuel, f (litres)	Frequency	Class width	Frequency density
0 ≤ f < 20	?	20	?
20 ≤ f < 40	16	20	0.8
40 ≤ f < 50	12	10	1.2
50 ≤ f < 70	8	20	0.4

(a)Show that the total frequency for the three known classes (20 ≤ f < 40, 40 ≤ f < 50 and 50 ≤ f < 70) is 36.(1)

(b)Given that 40 cars were filled up in total, work out the frequency for the class 0 ≤ f < 20.(2)

(c)Hence work out the frequency density for the class 0 ≤ f < 20.(2)

(Total for Question 13 is 5 marks)

A conservation charity wants to find out how long, in minutes, visitors spend at a nature reserve's bird hide, in order to help decide the reserve's opening hours. They plan to record the time spent by a sample of 60 visitors, using unequal class widths because most visits are expected to be fairly short, with a few visitors staying much longer. The frequencies they collected are shown below.

Time, t (minutes)	Frequency	Class width	Frequency density
0 ≤ t < 10	12	10	?
10 ≤ t < 20	30	10	?
20 ≤ t < 40	14	20	?
40 ≤ t < 80	4	40	?

(a)(Plan) Write down a suitable question the charity could investigate using this data.(1)

(b)(Collect) Give one reason why the charity chose to record the times using unequal class widths, rather than four equal-width classes of 20 minutes each.(1)

(c)(Process) Complete the table above by calculating the frequency density for each class.(4)

(d)(Interpret) Write down the modal class, and use it to describe when visitors are typically most concentrated at the hide.(2)

(e)(Interpret) The reserve manager claims: 'More than half of all visitors spend less than 20 minutes at the bird hide.' Comment on whether this claim is supported, using figures from the table.(2)

(Total for Question 14 is 10 marks)

The times, in minutes, taken by 90 competitors to complete a fun run were recorded. The histogram shows the results.

(a)Estimate the number of competitors who took between 25 and 35 minutes to complete the fun run.(3)

(b)Explain why your answer to part (a) is an estimate rather than an exact value.(1)

(Total for Question 15 is 4 marks)

Mark scheme · S25 Histograms

Question 1

(i) B1 Equal cao
(i) Answer: Equal
(ii) B1 Unequal cao
(ii) Answer: Unequal

Question 2

(a) B1 8 cao
(a) Answer: 8
(b) B1 8 cao
(b) Answer: 8
(c) B1 6 ≤ t < 7 cao
(c) Answer: 6 ≤ t < 7
(d) M1 4 + 2, oe (adds the frequencies for 7≤t<8 and 8≤t<9)
(d) A1 6 cao
(d) Answer: 6

Question 3

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True

Question 4

(a) M1 18 / 3, oe
(a) A1 6 cao
(a) Answer: 6
(b) M1 52 / 8, oe
(b) A1 6.5 cao
(b) Answer: 6.5

Question 5

(a) M1 4 x 7, oe
(a) A1 28 cao
(a) Answer: 28
(b) M1 54 / 9, oe
(b) A1 6 cao
(b) Answer: 6

Question 6

(a) M1 24 / 5, oe
(a) A1 4.8 cao
(a) Answer: 4.8
(b) M1 15 / 10, oe
(b) A1 1.5 cao
(b) Answer: 1.5
(c) M1 3 / 15, oe
(c) A1 0.2 cao
(c) Answer: 0.2

Question 7

(a) B1 3.0 cao
(a) Answer: 3.0
(b) B1 5 ≤ d < 10 cao
(b) Answer: 5 ≤ d < 10
(c) M1 2.0 x 10, oe
(c) A1 20 cao
(c) Answer: 20
(d) M1 1.8 x 5 + 3.0 x 5, oe
(d) A1 24 cao
(d) Answer: 24

Question 8

B1 the modal class is the one with the greatest frequency density (tallest bar), not the greatest raw frequency, because the class widths are unequal
B1 5 ≤ d < 10 has a higher frequency density (3.0) than 10 ≤ d < 20 (2.0), so 5 ≤ d < 10 is actually the modal class, even though 10 ≤ d < 20 has more employees overall (20 compared with 15)
Answer: The colleague is wrong: 5 ≤ d < 10 has the higher frequency density (3.0 against 2.0), so it is the modal class, even though 10 ≤ d < 20 contains more employees.

Question 9

B1 since the class widths are unequal, bar heights must show frequency density (frequency / class width), not raw frequency, otherwise the classes cannot be compared fairly; the vertical axis should be labelled 'Frequency density'
B1 40 / 10 = 4 cao (correct frequency density for the class with frequency 40 and width 10)
Answer: The vertical axis should show frequency density, not frequency; the correct height for the last class is 40 / 10 = 4.

Question 10

M1 identifies the interval 3 to 5 kg has width 2, within the class 2≤w<5 (frequency density 8)
M1 8 x 2, oe
A1 16 cao
Answer: 16

Question 11

(a) B1 0.3 cao (6 / 20)
(a) B1 1.1 cao (22 / 20)
(a) B1 0.6 cao (12 / 20)
(a) B1 0.2 cao (8 / 40)
(a) Answer: 0.3, 1.1, 0.6, 0.2
(b) B1 a suitable linear scale used on the frequency density axis, starting at 0
(b) B1 all four bars drawn to the correct height (0.3, 1.1, 0.6, 0.2), ft from (a), tolerance ± half a small square
(b) B1 bars drawn with no gaps between them, chart fully labelled
(b) Answer: Histogram with bars of frequency density 0.3, 1.1, 0.6, 0.2 across the four classes, drawn touching with no gaps.

Question 12

(a) B1 5 cao
(a) Answer: 5
(b) M1 2 x 10 + 5 x 5 (= 20 + 25 = 45), oe
(b) M1 45 / 90, ft
(b) A1 50% cao
(b) Answer: 50%

Question 13

(a) B1 16 + 12 + 8 = 36, fully shown
(a) Answer: 36
(b) M1 40 - 36, ft from (a)
(b) A1 4 cao
(b) Answer: 4
(c) M1 4 / 20, ft from (b)
(c) A1 0.2 cao
(c) Answer: 0.2

Question 14

(a) B1 sensible statistical question, e.g. 'Do most visitors spend less than 20 minutes at the bird hide?'
(a) Answer: Do most visitors spend less than 20 minutes at the bird hide?
(b) B1 sensible reason, e.g. most visit times are expected to cluster in a fairly short range, so narrower classes there give more detail, while the few long visits can share a wider class without lots of very low frequencies
(b) Answer: Unequal widths let the narrow, closely-clustered range of typical visit times be shown in more detail.
(c) B1 1.2 cao
(c) B1 3.0 cao
(c) B1 0.7 cao
(c) B1 0.1 cao
(c) Answer: 1.2, 3.0, 0.7, 0.1
(d) B1 10 ≤ t < 20, ft from (c)
(d) B1 sensible interpretation, e.g. visit times are most concentrated between 10 and 20 minutes, more so than any other length of time relative to its class width
(d) Answer: 10 ≤ t < 20; this class has the greatest concentration of visit times.
(e) M1 12 + 30 = 42, and 42 / 60 (= 0.7 or 70%), oe
(e) A1 supported, since 70% is more than half, with correct figures quoted
(e) Answer: Supported: 42 out of 60 visitors (70%) spend under 20 minutes at the bird hide.