Averages from a Frequency Table - Worksheets, Questions and Revision

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GCSE · Statistics

S03 Averages from a Frequency Table

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A small bakery kept a record of the number of loaves of bread bought by each customer during one morning. The frequency table shows the results.

Number of loaves bought	1	2	3	4
Frequency	4	6	3	2

(a)Write down the modal number of loaves bought.(1)

(b)Work out the total number of customers recorded in the table.(2)

(c)Find the median number of loaves bought.(2)

(Total for Question 1 is 5 marks)

10 friends were asked how many pets they owned. The results are shown in the frequency table.

Number of pets owned	0	1	2	3
Frequency	3	4	2	1

(a)Write down the modal number of pets owned.(1)

(b)Work out how many friends were asked in total.(2)

(Total for Question 2 is 3 marks)

Decide whether each statement about averages from a frequency table is True or False, then answer part (iv).

(i)The mode of a frequency table is the value with the highest frequency.(1)

(ii)For a frequency table with n data values, the position of the median is the (n+1)th value.(1)

(iii)To find the mean from a frequency table, you should divide the sum of (frequency times value) by the number of different values (categories) in the table.(1)

(iv)Which calculation correctly finds the mean from a frequency table?(1)

A) Sum(x) divided by n
B) Sum(f times x) divided by Sum(f)
C) Sum(f) divided by Sum(x)
D) Sum(f times x) divided by the number of categories

(Total for Question 3 is 4 marks)

A car dealership recorded the number of cars sold each day over a 15-day period. The frequency table shows the results.

Cars sold	0	1	2	3
Frequency	2	5	6	2

(a)Write down the modal number of cars sold in a day.(1)

(b)Find the median number of cars sold in a day.(2)

(c)Calculate the mean number of cars sold per day, giving your answer correct to 2 decimal places.(3)

(Total for Question 4 is 6 marks)

22 pupils in a class were asked how many books they read last month. The frequency table shows the results.

Books read	0	1	2	3	4
Frequency	2	6	8	4	2

(a)Write down the modal number of books read.(1)

(b)Find the median number of books read.(2)

(c)Calculate the mean number of books read, giving your answer correct to 1 decimal place.(3)

(Total for Question 5 is 6 marks)

A hockey team recorded the number of goals it scored in each of 18 matches last season. The frequency table shows the results.

Goals scored	0	1	2	3	4
Frequency	3	5	6	3	1

(a)Write down the modal number of goals scored.(1)

(b)Find the median number of goals scored.(2)

(c)Calculate the mean number of goals scored per match, giving your answer correct to 2 decimal places.(3)

(Total for Question 6 is 6 marks)

A netball team's goals scored in each of its last 20 matches were recorded as a list:

1, 2, 1, 3, 2, 2, 4, 1, 2, 3, 2, 1, 2, 3, 4, 2, 1, 3, 2, 2

Goals scored	1	2	3	4
Frequency

(a)Complete the frequency table above by counting through the list of 20 results.(2)

(b)Write down the modal number of goals scored.(1)

(c)Find the median number of goals scored.(2)

(d)Calculate the mean number of goals scored per match.(3)

(Total for Question 7 is 8 marks)

The bar chart shows the number of siblings (brothers and sisters) reported by each of 20 pupils in a survey.

(a)Use the bar chart to write down the modal number of siblings.(1)

(b)Use the bar chart to find the median number of siblings.(2)

(c)Use the bar chart to calculate the mean number of siblings, giving your answer as a decimal.(3)

(Total for Question 8 is 6 marks)

A shoe shop manager records the shoe size of every pair sold in a week, to help decide which sizes to reorder. The frequency table shows the sizes sold.

Shoe size	4	5	6	7	8
Frequency	3	10	22	15	4

Which average (mode, median or mean) would be most useful to the manager when deciding which shoe size to order more of? Give a reason for your answer.

(Total for Question 9 is 2 marks)

The frequency table shows the number of text messages sent in one day by each of 50 students.

Number of texts sent	0	1	2	3	4	5	6	7	8	9
Frequency	2	3	5	6	8	9	7	5	3	2

(a)Write down the modal number of texts sent.(1)

(b)Find the median number of texts sent.(2)

(c)Calculate the mean number of texts sent, giving your answer correct to 1 decimal place.(3)

(Total for Question 10 is 6 marks)

A student wants to investigate whether pupils in her school complete enough homework, compared with a recommendation of an average of 90 minutes (3 half-hour sessions) of homework every night of the week, equivalent to 21 half-hour sessions per week. She plans to survey a sample of pupils about their homework.

(a)Write down one suitable question the student could ask on her data collection sheet, including a set of response boxes, to find out how many half-hour homework sessions each pupil completed last week.(1)

(b)She surveys 30 pupils. The frequency table shows the number of half-hour homework sessions each pupil completed last week.

Number of half-hour sessions	4	5	6	7	8
Frequency	3	8	10	6	3

Calculate the mean number of half-hour sessions completed, giving your answer correct to 1 decimal place.(3)

(c)The recommendation is equivalent to 21 half-hour sessions per week. Use the mean from part (b) to comment on whether pupils at this school are meeting the recommended amount of homework.(2)

(Total for Question 11 is 6 marks)

The frequency table shows the number of classes attended last month by members of a dance club. One frequency is unknown, shown as x. The mean number of classes attended is 4.

Classes attended	2	3	4	5	6
Frequency	3	6	7	x	2

(Total for Question 12 is 4 marks)

Two Year 11 classes, 11A and 11B, sat the same maths quiz, marked out of 5. The frequency tables show the number of pupils achieving each score.

11A

Score	1	2	3	4	5
Frequency	2	4	9	7	3

11B

Score	1	2	3	4	5
Frequency	1	3	6	10	5

(a)Calculate the mean score for 11A.(2)

(b)Calculate the mean score for 11B.(2)

(c)Compare the performance of the two classes in this quiz, using the means from parts (a) and (b).(1)

(d)Give a reason why it is fair to compare the means of 11A and 11B in this way.(1)

(Total for Question 13 is 6 marks)

A corner shop's till records show the number of items bought by each of 15 customers one morning.

Number of items bought	1	2	3	4	25
Frequency	4	5	3	2	1

(a)Calculate the mean number of items bought, giving your answer correct to 1 decimal place.(3)

(b)Find the median number of items bought.(2)

(c)One customer bought an unusually large number of items (25). Explain which average, the mean or the median, better represents a typical customer's purchase, and why.(2)

(Total for Question 14 is 7 marks)

60 employees at a company were asked how many days they worked from home last week. The frequency table shows the results, but two frequencies are unknown, shown as a and b. The mean number of days worked from home is 2.05.

Days worked from home	0	1	2	3	4
Frequency	8	a	15	b	6

(Total for Question 15 is 5 marks)

Mark scheme · S03 Averages from a Frequency Table

Question 1

(a) B1 2 cao
(a) Answer: 2
(b) M1 4+6+3+2, oe (adds all four frequencies)
(b) A1 15 cao
(b) Answer: 15
(c) M1 identifies the median position as the half(n+1)th = 8th value, using cumulative frequencies 4, 10, 13, 15
(c) A1 2 cao
(c) Answer: 2

Question 2

(a) B1 1 cao
(a) Answer: 1
(b) M1 3+4+2+1, oe
(b) A1 10 cao
(b) Answer: 10

Question 3

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 False cao
(iii) Answer: False
(iv) B1 B cao
(iv) Answer: B) Sum(f times x) divided by Sum(f)

Question 4

(a) B1 2 cao
(a) Answer: 2
(b) M1 identifies the median position as half(15+1) = 8th value, using cumulative frequencies 2, 7, 13, 15
(b) A1 2 cao
(b) Answer: 2
(c) M1 0x2+1x5+2x6+3x2 (=23), oe
(c) M1 23 divided by 15
(c) A1 1.53 awrt
(c) Answer: 1.53

Question 5

(a) B1 2 cao
(a) Answer: 2
(b) M1 identifies the median position as half(22+1) = 11.5th value (mean of 11th and 12th values), using cumulative frequencies 2, 8, 16, 20, 22
(b) A1 2 cao
(b) Answer: 2
(c) M1 0x2+1x6+2x8+3x4+4x2 (=42), oe
(c) M1 42 divided by 22
(c) A1 1.9 awrt
(c) Answer: 1.9

Question 6

(a) B1 2 cao
(a) Answer: 2
(b) M1 identifies the median position as half(18+1) = 9.5th value (mean of 9th and 10th values), using cumulative frequencies 3, 8, 14, 17, 18
(b) A1 2 cao
(b) Answer: 2
(c) M1 0x3+1x5+2x6+3x3+4x1 (=30), oe
(c) M1 30 divided by 18
(c) A1 1.67 awrt
(c) Answer: 1.67

Question 7

(a) M1 at least three of the four frequencies correctly tallied
(a) A1 all four frequencies correct: 5, 9, 4, 2
(a) Answer: 5, 9, 4, 2 (for 1, 2, 3, 4 goals respectively)
(b) B1 2 cao, ft from (a)
(b) Answer: 2
(c) M1 identifies the median position as half(20+1) = 10.5th value (mean of 10th and 11th values), using cumulative frequencies 5, 14, 18, 20
(c) A1 2 cao
(c) Answer: 2
(d) M1 1x5+2x9+3x4+4x2 (=43), oe
(d) M1 43 divided by 20
(d) A1 2.15 cao
(d) Answer: 2.15

Question 8

(a) B1 1 cao
(a) Answer: 1
(b) M1 identifies the median position as half(20+1) = 10.5th value, using cumulative frequencies 5, 12, 16, 19, 20
(b) A1 1 cao
(b) Answer: 1
(c) M1 0x5+1x7+2x4+3x3+4x1 (=28), oe
(c) M1 28 divided by 20
(c) A1 1.4 cao
(c) Answer: 1.4

Question 9

B1 identifies the mode as the most useful average for this purpose
B1 valid justification, e.g. the mode shows the shoe size that sold the most (size 6, with frequency 22), so it tells the manager which size to reorder more of; the mean or median size would not necessarily match a real, whole shoe size that needs restocking
Answer: Mode

Question 10

(a) B1 5 cao
(a) Answer: 5
(b) M1 identifies the median position as half(50+1) = 25.5th value, using cumulative frequencies 2, 5, 10, 16, 24, 33, 40, 45, 48, 50
(b) A1 5 cao
(b) Answer: 5
(c) M1 0x2+1x3+2x5+3x6+4x8+5x9+6x7+7x5+8x3+9x2 (=227), oe
(c) M1 227 divided by 50
(c) A1 4.5 awrt
(c) Answer: 4.5

Question 11

(a) B1 a suitable closed question with mutually exclusive, exhaustive response boxes for the number of half-hour sessions completed, e.g. 'How many half-hour homework sessions did you complete this week? Tick one box: 0-2, 3-4, 5-6, 7-8, 9 or more'
(a) Answer: Any suitable closed question with clear, non-overlapping response boxes covering all possible values.
(b) M1 4x3+5x8+6x10+7x6+8x3 (=178), oe
(b) M1 178 divided by 30
(b) A1 5.9 awrt
(b) Answer: 5.9
(c) M1 compares the calculated mean (5.9) with the recommended figure (21)
(c) A1 correct conclusion in context, e.g. the mean of 5.9 half-hour sessions per week is much lower than the recommended 21, so on average pupils at this school are completing far less homework than recommended
(c) Answer: The mean (5.9) is well below the recommended 21 sessions, so pupils are not meeting the recommendation.

Question 12

M1 total frequency = 18+x, oe (sums the known frequencies: 3+6+7+2=18)
M1 Sum(f times x) = 64+5x, oe (2x3+3x6+4x7+6x2=64, plus 5x from the unknown term)
M1 sets up 64+5x = 4(18+x), oe, and solves
A1 x = 8 cao
Answer: x = 8

Question 13

(a) M1 1x2+2x4+3x9+4x7+5x3 (=80), oe
(a) A1 3.2 cao
(a) Answer: 3.2
(b) M1 1x1+2x3+3x6+4x10+5x5 (=90), oe
(b) A1 3.6 cao
(b) Answer: 3.6
(c) B1 correct comparative statement referencing both means in context, e.g. 11B had a higher mean score (3.6) than 11A (3.2), so on average pupils in 11B performed better in this quiz
(c) Answer: 11B performed better on average (mean 3.6 compared with 3.2 for 11A).
(d) B1 valid reason, e.g. both classes have the same number of pupils (25) and sat the same quiz marked out of the same total, so comparing their means is a fair like-for-like comparison
(d) Answer: Both classes have the same number of pupils (25) and sat the identical quiz, so the comparison is fair.

Question 14

(a) M1 1x4+2x5+3x3+4x2+25x1 (=56), oe
(a) M1 56 divided by 15
(a) A1 3.7 awrt
(a) Answer: 3.7
(b) M1 identifies the median position as half(15+1) = 8th value, using cumulative frequencies 4, 9, 12, 14, 15
(b) A1 2 cao
(b) Answer: 2
(c) B1 identifies the median as the better representative average
(c) B1 valid justification, e.g. the single high value of 25 is an outlier that pulls the mean up to 3.7, higher than 14 of the 15 customers actually bought, whereas the median (2) is not affected by this extreme value and better reflects a typical customer
(c) Answer: Median, because it is not affected by the outlier of 25 items.