Revision Library

revisionlibrary.co.uk

GCSE · Statistics

S16 Grouped Frequency Tables

EDEXCEL 1ST0 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

20 pupils were timed doing their homework one evening. The time, in minutes, taken by each pupil is shown below.

4, 12, 27, 9, 15, 22, 31, 18, 6, 24, 11, 29, 3, 16, 20, 33, 8, 14, 26, 19

The class intervals for a grouped frequency table have been started below.

Time (minutes)	Frequency
0 < t ≤ 10
10 < t ≤ 20
20 < t ≤ 30
30 < t ≤ 40
Total

(a)Complete the grouped frequency table for the 20 homework times.(2)

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

(Total for Question 1 is 3 marks)

A museum recorded the age, in years, of 32 visitors one afternoon. The grouped frequency table shows the results.

Age (years)	Frequency
0 < a ≤ 20	9
20 < a ≤ 40	13
40 < a ≤ 60	7
60 < a ≤ 80	3
Total	32

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

(b)Write down the frequency of the class 40 < a ≤ 60.(1)

(c)Work out the total number of visitors in the classes 0 < a ≤ 20 and 20 < a ≤ 40 combined.(2)

(d)Work out the percentage of the 32 visitors who were over 40 years old.(2)

(Total for Question 2 is 6 marks)

For each statement about grouped frequency tables, write down whether it is True or False.

(i)The frequency column of a grouped frequency table always adds up to the total number of data values collected.(1)

(ii)Once data has been grouped into class intervals, you can always identify the exact largest and smallest values in the original data set.(1)

(Total for Question 3 is 2 marks)

25 students in a reading challenge recorded the number of pages they read last weekend. The grouped frequency table shows the results.

Pages read	Frequency
0 < p ≤ 20	4
20 < p ≤ 40	9
40 < p ≤ 60	8
60 < p ≤ 80	4
Total	25

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

(b)Work out how many students read more than 40 pages.(2)

(Total for Question 4 is 3 marks)

A delivery company recorded the mass, in kg, of 30 parcels. The grouped frequency table shows the results.

Mass (kg)	Frequency
0 < m ≤ 5	6
5 < m ≤ 10	11
10 < m ≤ 15	9
15 < m ≤ 20	4
Total	30

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

(b)One parcel has a mass of 12 kg. Write down the class interval that contains this parcel.(1)

(c)Work out how many parcels weigh 10 kg or less.(2)

(Total for Question 5 is 4 marks)

20 pupils were asked how many text messages they sent yesterday. The results are shown below.

3, 14, 27, 8, 35, 19, 42, 11, 25, 6, 31, 16, 39, 22, 9, 44, 17, 28, 5, 33

(a)Using a class width of 10, starting with 0 < x ≤ 10, complete a grouped frequency table for this data.

Texts sent	Frequency
0 < x ≤ 10
10 < x ≤ 20
20 < x ≤ 30
30 < x ≤ 40
40 < x ≤ 50
Total

(2)

(b)A classmate suggests using a class width of 5 instead of 10. Give one advantage and one disadvantage of using this narrower class width.(2)

(Total for Question 6 is 4 marks)

The grouped frequency table shows the rainfall, in mm, recorded over 40 days.

Rainfall (mm)	Frequency
0 < r ≤ 5	9
5 < r ≤ 10	14
10 < r ≤ 15	10
15 < r ≤ 20	7
Total	40

(a)Complete a cumulative frequency column for this table.

Rainfall (mm)	Frequency	Cumulative frequency
0 < r ≤ 5	9
0 < r ≤ 10	14
0 < r ≤ 15	10
0 < r ≤ 20	7

(2)

(b)Write down the number of days with rainfall of 10 mm or less.(1)

(c)Work out the number of days with rainfall of more than 10 mm.(2)

(Total for Question 7 is 5 marks)

A garden centre recorded the amount, in £, spent by 40 customers one Saturday. The grouped frequency table shows the results.

Amount spent (£)	Frequency
0 < m ≤ 10	7
10 < m ≤ 20	15
20 < m ≤ 30	12
30 < m ≤ 40	6
Total	40

(a)One customer spent £24. Write down the class interval that contains this customer.(1)

(b)Work out the number of customers who spent £20 or less.(2)

(c)Work out the number of customers who spent more than £20.(1)

(Total for Question 8 is 4 marks)

The grouped frequency table shows the age, in years, of 60 runners in a marathon.

Age (years)	Frequency
20 < a ≤ 30	8
30 < a ≤ 40	22
40 < a ≤ 50	19
50 < a ≤ 60	11
Total	60

An estimate of the median can be found using linear interpolation:

median ≈ L + ((n ÷ 2 − F) ÷ f) × w

where L is the lower boundary of the class containing the median, F is the cumulative frequency before that class, f is the frequency of that class, and w is its class width.

(a)Complete a cumulative frequency column for this table.

Age (years)	Frequency	Cumulative frequency
20 < a ≤ 30	8
20 < a ≤ 40	22
20 < a ≤ 50	19
20 < a ≤ 60	11

(2)

(b)n ÷ 2 = 60 ÷ 2 = 30. Use your cumulative frequencies to write down the class interval that contains the median age.(1)

(c)Use linear interpolation to estimate the median age of the runners.(2)

(Total for Question 9 is 5 marks)

A garden centre recorded the diameter, in cm, of 50 sunflowers grown from the same batch of seeds. The grouped frequency table shows the results.

Diameter (cm)	Frequency
10 < d ≤ 15	6
15 < d ≤ 20	18
20 < d ≤ 25	20
25 < d ≤ 30	6
Total	50

A frequency polygon can be drawn by plotting the midpoint of each class against its frequency, and joining the points with straight lines.

(a)Write down the coordinates of the point that would be plotted for the class 15 < d ≤ 20.(2)

(b)All four class widths in this table are equal, which is why the midpoints can be plotted directly. If the class widths were not all equal, state what quantity would need to be plotted on the vertical axis of a chart instead of frequency, and explain why.(2)

(c)On the grid provided, plot the midpoint of each class against its frequency and join the points in order with straight lines to draw a frequency polygon.(2)

(Total for Question 10 is 6 marks)

The grouped frequency table shows the test scores, as a percentage, of 45 students. The class widths are not all equal.

Score (%)	Frequency
0 < s ≤ 40	5
40 < s ≤ 60	14
60 < s ≤ 75	16
75 < s ≤ 100	10
Total	45

(a)One student scored 58%. Write down the class interval that contains this student.(1)

(b)Work out the number of students who scored 60% or less.(2)

(c)Work out the number of students who scored more than 75%.(1)

(d)Explain why it would be misleading to draw a bar chart with bar heights equal to the frequencies in this table, and state what quantity should be used for the bar heights instead.(2)

(Total for Question 11 is 6 marks)

A PE teacher wants to investigate how many minutes Year 10 students spend exercising outside school each week.

(a)Plan: write down a suitable question the teacher could ask each student to collect this data.(1)

(b)Collect and process: the teacher recorded the following times, in minutes, from 20 students.

15, 45, 90, 30, 120, 60, 10, 75, 105, 40, 55, 95, 20, 80, 130, 65, 35, 100, 50, 85

Using a class width of 30, starting with 0 < t ≤ 30, complete a grouped frequency table for this data.

Time (minutes)	Frequency
0 < t ≤ 30
30 < t ≤ 60
60 < t ≤ 90
90 < t ≤ 120
120 < t ≤ 150
Total

(2)

(c)Interpret: write down the modal class from your completed table.(1)

(d)Interpret: use the modal class to comment on what this suggests about Year 10 students' exercise habits outside school.(1)

(Total for Question 12 is 5 marks)

In Question 1, 20 pupils' homework times were grouped into a frequency table. Explain why the exact mean of the 20 homework times cannot be calculated from that grouped frequency table, even though an estimate of the mean could be found.

(Total for Question 13 is 2 marks)

Two market stalls, Stall A and Stall B, each recorded the amount, in £, spent by 40 customers one Saturday.

Stall A

Amount spent (£)	Frequency
0 < m ≤ 10	6
10 < m ≤ 20	10
20 < m ≤ 30	16
30 < m ≤ 40	8
Total	40

Stall B

Amount spent (£)	Frequency
0 < m ≤ 10	14
10 < m ≤ 20	15
20 < m ≤ 30	8
30 < m ≤ 40	3
Total	40

(a)Write down the modal class for Stall A and the modal class for Stall B.(2)

(b)By comparing the modal classes, make one comparison between the amount spent at Stall A and at Stall B.(2)

(Total for Question 14 is 4 marks)

The grouped frequency table shows the distance travelled to work, in km, by 50 employees at a company. The frequency for the class 15 < d ≤ 25 is unknown and is shown as x.

Distance (km)	Frequency
0 < d ≤ 5	12
5 < d ≤ 15	20
15 < d ≤ 25	x
25 < d ≤ 40	6
Total	50

(a)Find the value of x.(2)

(b)Use linear interpolation to estimate the median distance travelled to work.(3)

(Total for Question 15 is 5 marks)

Mark scheme · S16 Grouped Frequency Tables

Question 1

(a) B1 at least three of the four frequencies tallied correctly
(a) B1 all four frequencies correct: 5, 8, 5, 2 (total 20)
(a) Answer: 5, 8, 5, 2
(b) B1 10 < t ≤ 20 cao
(b) Answer: 10 < t ≤ 20

Question 2

(a) B1 20 < a ≤ 40 cao
(a) Answer: 20 < a ≤ 40
(b) B1 7 cao
(b) Answer: 7
(c) M1 9 + 13, oe
(c) A1 22 cao
(c) Answer: 22
(d) M1 (7 + 3) / 32 x 100, oe
(d) A1 31.25% cao (accept 31.3%)
(d) Answer: 31.25%

Question 3

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

Question 4

(a) B1 20 < p ≤ 40 cao
(a) Answer: 20 < p ≤ 40
(b) M1 8 + 4, oe
(b) A1 12 cao
(b) Answer: 12

Question 5

(a) B1 5 < m ≤ 10 cao
(a) Answer: 5 < m ≤ 10
(b) B1 10 < m ≤ 15 cao
(b) Answer: 10 < m ≤ 15
(c) M1 6 + 11, oe
(c) A1 17 cao
(c) Answer: 17

Question 6

(a) B1 at least four of the five frequencies tallied correctly
(a) B1 all five frequencies correct: 5, 5, 4, 4, 2 (total 20)
(a) Answer: 5, 5, 4, 4, 2
(b) B1 valid advantage, e.g. narrower classes show more detail/pattern in how the data is spread within the original range 0-50
(b) B1 valid disadvantage, e.g. more classes are needed so the table becomes longer, and some classes may end up with very low or zero frequencies
(b) Answer: Advantage: shows more detail. Disadvantage: more classes, some with low frequencies.

Question 7

(a) B1 at least two of the four cumulative frequencies correct
(a) B1 all four cumulative frequencies correct: 9, 23, 33, 40
(a) Answer: 9, 23, 33, 40
(b) B1 23 cao
(b) Answer: 23
(c) M1 40 - 23, oe
(c) A1 17 cao
(c) Answer: 17

Question 8

(a) B1 20 < m ≤ 30 cao
(a) Answer: 20 < m ≤ 30
(b) M1 7 + 15, oe
(b) A1 22 cao
(b) Answer: 22
(c) B1 18 cao, ft 40 minus their (b)
(c) Answer: 18

Question 9

(a) B1 at least two of the four cumulative frequencies correct
(a) B1 all four cumulative frequencies correct: 8, 30, 49, 60
(a) Answer: 8, 30, 49, 60
(b) B1 30 < a ≤ 40 cao
(b) Answer: 30 < a ≤ 40
(c) M1 30 + ((30 - 8) / 22) x 10, oe (correct substitution: L=30, F=8, f=22, w=10)
(c) A1 40 cao
(c) Answer: 40

Question 10

(a) B1 midpoint 17.5 cao
(a) B1 (17.5, 18) cao
(a) Answer: (17.5, 18)
(b) B1 identifies frequency density (frequency divided by class width)
(b) B1 valid reason, e.g. so that the area of each bar (in a histogram) represents the frequency fairly when class widths differ
(b) Answer: Frequency density, so that bar area (not height) represents frequency fairly.
(c) B1 all four points plotted correctly: (12.5, 6), (17.5, 18), (22.5, 20), (27.5, 6)
(c) B1 points joined in order with straight lines
(c) Answer: Points (12.5, 6), (17.5, 18), (22.5, 20), (27.5, 6) joined with straight lines.

Question 11

(a) B1 40 < s ≤ 60 cao
(a) Answer: 40 < s ≤ 60
(b) M1 5 + 14, oe
(b) A1 19 cao
(b) Answer: 19
(c) B1 10 cao
(c) Answer: 10
(d) B1 identifies that a bar for a wider class (e.g. 0 < s ≤ 40 or 75 < s ≤ 100) would look taller/shorter than a fair comparison simply because it covers a wider range of scores, not because scores are more or less common there
(d) B1 states that frequency density (frequency divided by class width) should be used for the bar heights instead
(d) Answer: Frequency density (frequency divided by class width) should be used instead of frequency.

Question 12

(a) B1 a suitable, answerable question with a numerical response, e.g. 'How many minutes in total did you spend exercising outside school last week?' oe
(a) Answer: e.g. 'How many minutes did you spend exercising outside school last week?'
(b) B1 at least three of the five frequencies tallied correctly
(b) B1 all five frequencies correct: 4, 6, 5, 4, 1 (total 20)
(b) Answer: 4, 6, 5, 4, 1
(c) B1 30 < t ≤ 60 cao, ft from (b)
(c) Answer: 30 < t ≤ 60
(d) B1 sensible comment in context, e.g. most Year 10 students exercise for between half an hour and an hour a week outside school, though a few exercise for much longer, oe
(d) Answer: e.g. Most students exercise between 30 and 60 minutes a week outside school.

Question 13

B1 recognises that once data is grouped, the exact individual values are no longer known, only the class each value falls into
B1 so the exact total of the 20 values cannot be found, meaning the exact mean cannot be calculated, only estimated using each class's midpoint
Answer: The individual raw values are lost once grouped, so the exact sum (and hence exact mean) cannot be found; only an estimate can be made using midpoints.

Question 14

(a) B1 Stall A: 20 < m ≤ 30 cao
(a) B1 Stall B: 10 < m ≤ 20 cao
(a) Answer: Stall A: 20 < m ≤ 30; Stall B: 10 < m ≤ 20
(b) B1 valid comparison referencing both modal classes, e.g. Stall A's modal class (£20-£30) is higher than Stall B's modal class (£10-£20)
(b) B1 comparison made in the context of spending, e.g. suggesting customers at Stall A typically spent more than customers at Stall B
(b) Answer: Stall A's modal class is higher than Stall B's, suggesting customers typically spent more at Stall A than at Stall B.

Question 15

(a) M1 12 + 20 + 6, oe (=38)
(a) A1 x = 12 cao
(a) Answer: x = 12
(b) M1 identifies the median class as 5 < d ≤ 15 using cumulative frequencies 12, 32, 44, 50 and n ÷ 2 = 50 ÷ 2 = 25
(b) M1 correct substitution: 5 + ((25 - 12) / 20) x 10, oe
(b) A1 11.5 cao
(b) Answer: 11.5 km

Grouped Frequency Tables - Worksheets, Questions and Revision

S16 Grouped Frequency Tables

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15