Averages from Frequency Tables: Fluency and Exam Drill (GCSE Maths) - Worksheets, Questions and Revision

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GCSE · Statistics: averages and range

4.18D Averages from Frequency Tables: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 70 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Priya asked 20 pupils in her class how many siblings they have. The table shows her results.

Number of siblings (x): 0, 1, 2, 3, 4
Frequency (f): 6, 9, 3, 1, 1
Total number of pupils = 20

Write down the mode of the number of siblings.

(Total for Question 1 is 1 mark)

A hockey team's coach recorded the number of goals scored in each of 15 matches. The table shows the results.

Goals scored (x): 0, 1, 2, 3, 4
Frequency (f): 2, 5, 4, 3, 1
Total number of matches = 15

Write down the range of the number of goals scored.

(Total for Question 2 is 1 mark)

Callum recorded the number of text messages sent by 17 students during one evening. The table shows the results.

Messages sent (x): 0, 1, 2, 3, 4, 5
Frequency (f): 1, 3, 5, 4, 3, 1
Total number of students = 17

Find the median number of messages sent.

(Total for Question 3 is 2 marks)

Grace kept a record of the number of books read last month by 20 pupils in her form. The table shows the results.

Books read (x): 0, 1, 2, 3, 4
Frequency (f): 3, 7, 6, 3, 1
Total number of pupils = 20

Find the median number of books read.

(Total for Question 4 is 2 marks)

The table shows the number of pets owned by 10 pupils.

Number of pets (x): 0, 1, 2, 3
Frequency (f): 4, 3, 2, 1
Total number of pupils = 10

Work out the mean number of pets owned.

(Total for Question 5 is 2 marks)

24 pupils were asked how many siblings they have. One frequency, a, is missing from the table.

Number of siblings (x): 0, 1, 2, 3
Frequency (f): 5, a, 6, 2
Total number of pupils = 24

Find the value of a.

(Total for Question 6 is 2 marks)

A school recorded the time, t minutes, spent on homework by 30 pupils in one evening. The table shows the results.

Time, t (minutes): 0 < t ≤ 20, 20 < t ≤ 40, 40 < t ≤ 60, 60 < t ≤ 80, 80 < t ≤ 100
Frequency (f): 4, 9, 11, 4, 2
Total number of pupils = 30

Write down the modal class.

(Total for Question 7 is 1 mark)

Using the same table as Question 7 (time, t minutes, spent on homework by 30 pupils):

Time, t (minutes): 0 < t ≤ 20, 20 < t ≤ 40, 40 < t ≤ 60, 60 < t ≤ 80, 80 < t ≤ 100
Frequency (f): 4, 9, 11, 4, 2
Total number of pupils = 30

State the class that contains the median.

(Total for Question 8 is 2 marks)

Using the same table as Questions 7 and 8:

Time, t (minutes): 0 < t ≤ 20, 20 < t ≤ 40, 40 < t ≤ 60, 60 < t ≤ 80, 80 < t ≤ 100
Frequency (f): 4, 9, 11, 4, 2

Write down the midpoint of the class 20 < t ≤ 40.

(Total for Question 9 is 1 mark)

Using the same table as Questions 7 to 9 (time, t minutes, spent on homework by 30 pupils):

Time, t (minutes): 0 < t ≤ 20, 20 < t ≤ 40, 40 < t ≤ 60, 60 < t ≤ 80, 80 < t ≤ 100
Frequency (f): 4, 9, 11, 4, 2
Total number of pupils = 30

Calculate an estimate for the mean time spent on homework.

(Total for Question 10 is 3 marks)

Using the same table as Questions 7 to 10 (time, t minutes, spent on homework by 30 pupils):

Time, t (minutes): 0 < t ≤ 20, 20 < t ≤ 40, 40 < t ≤ 60, 60 < t ≤ 80, 80 < t ≤ 100
Frequency (f): 4, 9, 11, 4, 2

Find an estimate for the range of the times, using the class boundaries.

(Total for Question 11 is 2 marks)

Using the same table as Questions 7 to 11 (time, t minutes, spent on homework by 30 pupils):

Time, t (minutes): 0 < t ≤ 20, 20 < t ≤ 40, 40 < t ≤ 60, 60 < t ≤ 80, 80 < t ≤ 100
Frequency (f): 4, 9, 11, 4, 2

Write down the class width of the class 60 < t ≤ 80.

(Total for Question 12 is 1 mark)

The table shows the number of aces served by a tennis player in 10 games.

Aces served (x): 0, 1, 2, 3, 4
Frequency (f): 2, 4, 2, 1, 1
Total number of games = 10

Work out the mean number of aces per game.

(Total for Question 13 is 2 marks)

A delivery firm recorded the mass, m kg, of 25 parcels. The table shows the results.

Mass, m (kg): 0 < m ≤ 5, 5 < m ≤ 10, 10 < m ≤ 15, 15 < m ≤ 20, 20 < m ≤ 25
Frequency (f): 6, 10, 6, 2, 1
Total number of parcels = 25

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

(b)State the class that contains the median.(1)

(Total for Question 14 is 2 marks)

24 pupils were asked how many siblings they have, but the table has been damaged and one frequency, a, is unknown.

Number of siblings (x): 0, 1, 2, 3
Frequency (f): 2, 4, a, 1

Given that the mean number of siblings is 1.3, find the value of a. You must show your working.

(Total for Question 15 is 4 marks)

20 pupils in Class A sat a test out of 40. The table shows their scores, s.

Score, s: 0 < s ≤ 10, 10 < s ≤ 20, 20 < s ≤ 30, 30 < s ≤ 40
Frequency (f): 2, 5, 9, 4
Total number of pupils = 20

(a)Calculate an estimate for the mean score for Class A.(3)

(b)20 pupils in Class B sat the same test. Their estimated mean score was 27. Compare the performance of the two classes, referring to your answer to part (a).(1)

(Total for Question 16 is 4 marks)

A cafe recorded the time, m minutes, that 26 + b customers spent queuing at the counter. One frequency, b, is unknown.

Time, m (minutes): 0 < m ≤ 5, 5 < m ≤ 10, 10 < m ≤ 15, 15 < m ≤ 20
Frequency (f): 8, 12, b, 6

Given that the estimated mean queuing time is 10 minutes, find the value of b. You must show your working.

(Total for Question 17 is 4 marks)

30 pupils were asked how many hours of TV they watched last night. The table shows the results, but one frequency, c, is missing.

Hours watched (x): 0, 1, 2, 3, 4
Frequency (f): 6, 10, c, 4, 2
Total number of pupils = 30

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

(b)Write down the modal number of hours watched.(1)

(c)Find the median number of hours watched.(2)

(Total for Question 18 is 5 marks)

The table shows the annual salary, in thousands of pounds, of 10 employees at a small company.

Salary, s (GBP thousands): 20 ≤ s < 30, 30 ≤ s < 40, 40 ≤ s < 50, 50 ≤ s < 60, 200 ≤ s < 210
Frequency (f): 3, 4, 1, 1, 1
Total number of employees = 10

(a)Calculate an estimate for the mean salary.(3)

(b)State the class containing the median, and explain why the median might be a more suitable measure than the mean to describe a typical salary at this company.(2)

(Total for Question 19 is 5 marks)

25 learner drivers were asked how many driving lessons they took before passing their test. Two frequencies, p and q, are missing.

Number of lessons: 10-19, 20-29, 30-39, 40-49
Frequency (f): 4, p, q, 3
Total number of learner drivers = 25

Given that the modal class is 20-29, work out a possible pair of values for p and q, and give a reason for your answer.

(Total for Question 20 is 4 marks)

20 members of a gym class each completed a number of pull-ups in one minute. Two frequencies, p and q, are missing from the table.

Pull-ups completed (x): 2, 3, 4, 5, 6
Frequency (f): 2, p, q, 4, 1
Total number of members = 20

Given that the mean number of pull-ups is 3.9, find the values of p and q. You must show your working.

(Total for Question 21 is 6 marks)

A competition weighed 60 fish and recorded their length, l cm, in a grouped frequency table. Two of the frequencies are given in terms of n.

Length, l (cm): 0 < l ≤ 20, 20 < l ≤ 40, 40 < l ≤ 60, 60 < l ≤ 80
Frequency (f): n, 3n, 18, 2n
Total number of fish = 60

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

(b)Using your value of n, calculate an estimate for the mean length of the fish.(4)

(Total for Question 22 is 6 marks)

Mark scheme · 4.18D Averages from Frequency Tables: Fluency and Exam Drill

Question 1

B1 1 cao
Answer: 1

Question 2

B1 4 cao
Answer: 4

Question 3

M1 correct use of cumulative frequencies to find the 9th value (1, 4, 9, 13, 16, 17) oe
A1 2 cao
Answer: 2

Question 4

M1 correct use of cumulative frequencies to identify the 10th and 11th values (3, 10, 16, 19, 20) oe
A1 1.5 cao
Answer: 1.5

Question 5

M1 sum(fx) = 0(4) + 1(3) + 2(2) + 3(1) = 10 oe
A1 1 cao
Answer: 1

Question 6

M1 5 + 6 + 2 (= 13) oe, or 24 - 13
A1 11 cao
Answer: 11

Question 7

B1 40 < t ≤ 60 cao
Answer: 40 < t ≤ 60

Question 8

M1 correct use of cumulative frequencies to locate the 15th and 16th values (4, 13, 24, 28, 30) oe
A1 40 < t ≤ 60 cao
Answer: 40 < t ≤ 60

Question 9

B1 30 cao
Answer: 30

Question 10

M1 correct midpoints identified: 10, 30, 50, 70, 90
M1 sum(fx) = 10(4) + 30(9) + 50(11) + 70(4) + 90(2) = 1320 oe
A1 44 cao
Answer: 44 minutes

Question 11

M1 identifies the greatest possible value (100) and least possible value (0) oe
A1 100 cao
Answer: 100 minutes

Question 12

B1 20 cao
Answer: 20

Question 13

M1 sum(fx) = 0(2) + 1(4) + 2(2) + 3(1) + 4(1) = 15 oe
A1 1.5 cao
Answer: 1.5

Question 14

(a) B1 5 < m ≤ 10 cao
(a) Answer: 5 < m ≤ 10
(b) B1 5 < m ≤ 10 cao
(b) Answer: 5 < m ≤ 10

Question 15

M1 sum(fx) = 4 + 2a + 3 (= 7 + 2a) oe
M1 sum(f) = 7 + a oe
dM1 (7 + 2a) / (7 + a) = 1.3 leading to 7 + 2a = 9.1 + 1.3a, dependent on both previous M marks
A1 3 cao
Answer: a = 3

Question 16

(a) M1 correct midpoints identified: 5, 15, 25, 35
(a) M1 sum(fx) = 5(2) + 15(5) + 25(9) + 35(4) = 450 oe
(a) A1 22.5 cao
(a) Answer: 22.5
(b) B1 ft correct comparative conclusion: Class B has the higher estimated mean (27 > 22.5), so Class B performed better on average
(b) Answer: Class B performed better on average, since its estimated mean (27) is higher than Class A's estimated mean (22.5).

Question 17

M1 correct midpoints identified: 2.5, 7.5, 12.5, 17.5
M1 sum(fx) = 20 + 90 + 12.5b + 105 (= 215 + 12.5b) oe, and sum(f) = 26 + b oe
dM1 (215 + 12.5b) / (26 + b) = 10 leading to 215 + 12.5b = 260 + 10b, dependent on both previous M marks
A1 18 cao
Answer: b = 18

Question 18

(a) M1 6 + 10 + 4 + 2 (= 22) oe, or 30 - 22
(a) A1 8 cao
(a) Answer: 8
(b) B1 ft 1 cao, using their value of c
(b) Answer: 1
(c) M1 ft correct use of cumulative frequencies to identify the 15th and 16th values (6, 16, 24, 28, 30), using their value of c
(c) A1 ft 1 cao
(c) Answer: 1

Question 19

(a) M1 correct midpoints identified: 25, 35, 45, 55, 205
(a) M1 sum(fx) = 25(3) + 35(4) + 45(1) + 55(1) + 205(1) = 520 oe
(a) A1 52 (i.e. GBP 52,000) cao
(a) Answer: GBP 52,000 (estimated mean = 52 thousand pounds)
(b) B1 30 ≤ s < 40 cao
(b) B1 correct explanation referring to the one very high salary (200 ≤ s < 210) pulling the mean upward, while the median is not affected by this outlier
(b) Answer: Median class: 30 ≤ s < 40. The median is more suitable because the one very high salary (in the 200 ≤ s < 210 class) is an outlier that pulls the mean up to GBP 52,000, much higher than most employees earn; the median is not affected by this extreme value, so it gives a more typical figure.

Question 20

M1 4 + p + q + 3 = 25 leading to p + q = 18 oe
M1 recognises that for 20-29 to be the modal class, p must be greater than both q and 4
A1 any valid pair with p + q = 18 and p > q, e.g. p = 10 and q = 8
B1 correct reason given, e.g. p = 10 is the greatest frequency of the four classes (4, 10, 8, 3), so 20-29 is indeed the modal class
Answer: One possible answer: p = 10, q = 8 (any pair with p + q = 18 and p the greatest frequency is acceptable)

Question 21

M1 2 + p + q + 4 + 1 = 20 leading to p + q = 13 oe
M1 sum(fx) = 4 + 3p + 4q + 20 + 6 (= 30 + 3p + 4q) oe
M1 (30 + 3p + 4q) / 20 = 3.9 leading to 3p + 4q = 48 oe
dM1 correct substitution of p = 13 - q (or equivalent elimination) into 3p + 4q = 48, dependent on the two previous method marks
A1 q = 9 cao
A1 p = 4 cao
Answer: p = 4, q = 9

Question 22

(a) M1 n + 3n + 18 + 2n = 60 leading to 6n + 18 = 60 oe
(a) A1 7 cao
(a) Answer: n = 7
(b) M1 ft correct midpoints identified: 10, 30, 50, 70
(b) M1 ft correct frequencies from part (a): 7, 21, 18, 14
(b) M1 ft sum(fx) = 10(7) + 30(21) + 50(18) + 70(14) = 2580 oe
(b) A1 ft 43 cao
(b) Answer: 43 cm

Question 22

6 marks

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Averages from Frequency Tables: Fluency and Exam Drill (GCSE Maths) - Worksheets, Questions and Revision

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4.18D Averages from Frequency Tables: Fluency and Exam Drill

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