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GCSE · Statistics

S05 Estimated Mean from Grouped Data

EDEXCEL 1ST0 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A call centre recorded the length, in minutes, of 40 phone calls. The frequency table shows the results, with an empty column for the midpoint of each class interval.

Class interval (minutes)	Frequency	Midpoint
0 < t ≤ 10	6
10 < t ≤ 20	14
20 < t ≤ 30	12
30 < t ≤ 40	8
Total	40

(a)Complete the table by finding the midpoint of each class interval.(4)

(b)Write down the class interval that contains the mode.(1)

(Total for Question 1 is 5 marks)

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

(i)The midpoint of a class interval is found by adding the lower and upper boundaries of the interval and dividing by 2.(1)

(ii)An estimated mean calculated from a grouped frequency table will always be exactly equal to the true mean of the original raw data.(1)

(Total for Question 2 is 2 marks)

The table shows the mass, in kg, of 30 parcels sent by a delivery company.

Class interval (kg)	Frequency	Midpoint	fx
0 < m ≤ 5	4
5 < m ≤ 10	10
10 < m ≤ 15	9
15 < m ≤ 20	5
20 < m ≤ 25	2
Total	30

(a)Complete the table to show the midpoint and the value of fx for each class interval.(3)

(b)Work out an estimate for the mean mass of the parcels.(2)

(Total for Question 3 is 5 marks)

The table shows the time, in minutes, that 25 customers spent in a coffee shop.

Class interval (minutes)	Frequency	Midpoint	fx
0 < t ≤ 5	3
5 < t ≤ 10	8
10 < t ≤ 15	7
15 < t ≤ 20	5
20 < t ≤ 25	2
Total	25

(a)Complete the table to show the midpoint and the value of fx for each class interval.(3)

(b)Work out an estimate for the mean time that a customer spent in the coffee shop.(2)

(Total for Question 4 is 5 marks)

The table shows the amount of electricity used, in kWh, by 35 households in one week.

Class interval (kWh)	Frequency	Midpoint	fx
0 < e ≤ 20	6
20 < e ≤ 40	14
40 < e ≤ 60	10
60 < e ≤ 80	5
Total	35

(a)Complete the table to show the midpoint and the value of fx for each class interval.(3)

(b)Work out an estimate for the mean amount of electricity used per household.(2)

(Total for Question 5 is 5 marks)

In Question 5(b) you calculated an estimated mean of 38 kWh for the electricity used by the 35 households. Explain why this value is described as an estimate rather than the exact mean of the 35 households' electricity use.

(Total for Question 6 is 2 marks)

The table shows the annual mileage, in miles, of 50 cars. The class intervals are not all the same width.

Class interval (miles)	Frequency	Midpoint	fx
0 < m ≤ 5,000	6
5,000 < m ≤ 10,000	18
10,000 < m ≤ 20,000	16
20,000 < m ≤ 40,000	10
Total	50

(a)Write down the midpoint of the class interval 10,000 < m ≤ 20,000.(1)

(b)Complete the table and use it to calculate an estimate for the mean annual mileage.(4)

(Total for Question 7 is 5 marks)

The table shows the heights, in metres, of 60 sunflowers grown for a science project.

Class interval (m)	Frequency	Midpoint	fx
1.0 < h ≤ 1.5	8
1.5 < h ≤ 2.0	22
2.0 < h ≤ 2.5	24
2.5 < h ≤ 3.0	6
Total	60

(a)Complete the table to show the midpoint and the value of fx for each class interval.(3)

(b)Work out an estimate for the mean height of the sunflowers. Give your answer to 3 significant figures.(2)

(Total for Question 8 is 5 marks)

The bar chart shows the diameter, in cm, of 45 plant pots produced by a garden centre.

(a)Use the bar chart to complete the frequency table below.

Class interval (cm)	Frequency
0 < d ≤ 10
10 < d ≤ 20
20 < d ≤ 30
30 < d ≤ 40
Total	45

Write the missing frequencies in the table.(2)

(b)Work out an estimate for the mean diameter of the plant pots.(3)

(Total for Question 9 is 5 marks)

The table below repeats the completed frequency table for the plant pot diameters from Question 9.

Class interval (cm)	Frequency
0 < d ≤ 10	18
10 < d ≤ 20	5
20 < d ≤ 30	15
30 < d ≤ 40	7
Total	45

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

(b)Work out the class interval that contains the median diameter.(2)

(c)The exact mode and exact median of the original 45 diameters cannot be found from the grouped table. Explain why.(2)

(Total for Question 10 is 5 marks)

The table shows the time, in minutes, that 60 members of a gym spent on a treadmill during one visit. The frequency for the class 30 < t ≤ 40 is missing and is shown as x.

Class interval (minutes)	Frequency
0 < t ≤ 10	8
10 < t ≤ 20	14
20 < t ≤ 30	12
30 < t ≤ 40	x
40 < t ≤ 50	6
Total	60

(a)Work out the value of x.(2)

(b)Using x = 20, work out an estimate for the mean time spent on the treadmill.(3)

(Total for Question 11 is 5 marks)

The tables show the amount spent, in £, by 40 customers at each of two coffee shops, Shop A and Shop B, on the same Saturday.

Shop A

Class interval (£)	Frequency
0 < x ≤ 2	6
2 < x ≤ 4	16
4 < x ≤ 6	14
6 < x ≤ 8	4
Total	40

Shop B

Class interval (£)	Frequency
0 < x ≤ 2	10
2 < x ≤ 4	10
4 < x ≤ 6	12
6 < x ≤ 8	8
Total	40

(a)Work out an estimate for the mean amount spent at Shop A.(3)

(b)Work out an estimate for the mean amount spent at Shop B.(3)

(c)Using your answers to parts (a) and (b), which coffee shop had the higher mean spend per customer? Give a reason using the figures you have calculated.(2)

(Total for Question 12 is 8 marks)

A researcher wants to find out how long customers wait in a queue at a post office. She plans to record the waiting time, in minutes, of a sample of customers on a busy Saturday morning.

(a)Give one reason why the researcher should record the waiting times of a sample of customers, rather than every customer who visits the post office in a year.(1)

(b)She records the waiting times of 50 customers and groups the results into the table below.

Class interval (minutes)	Frequency	Midpoint	fx
0 < w ≤ 5	8
5 < w ≤ 10	20
10 < w ≤ 15	14
15 < w ≤ 20	8
Total	50

Complete the table and use it to calculate an estimate for the mean waiting time.(4)

(c)The post office claims that, on average, customers wait 8 minutes or less. Using your answer to part (b), does the data support this claim? Give a reason.(2)

(Total for Question 13 is 7 marks)

A researcher recorded the time, in minutes, that shoppers spent in a supermarket. The table shows some of the results, where y is the number of shoppers in the class 10 < t ≤ 20. The estimated mean time spent was 16 minutes.

Class interval (minutes)	Frequency	Midpoint
0 < t ≤ 10	8	5
10 < t ≤ 20	y	15
20 < t ≤ 30	12	25

Show that y = 20.

(Total for Question 14 is 3 marks)

Two students each grouped the same 20 sprint times, in seconds, using different class widths, to estimate the mean.

Student A's table

Class interval (s)	Frequency
12 < t ≤ 13	2
13 < t ≤ 14	5
14 < t ≤ 15	7
15 < t ≤ 16	4
16 < t ≤ 17	2
Total	20

Student B's table

Class interval (s)	Frequency
12 < t ≤ 14	7
14 < t ≤ 16	11
16 < t ≤ 18	2
Total	20

The true mean of the original 20 sprint times, calculated before grouping, was 14.6 seconds.

(a)Use Student A's table to work out an estimate for the mean sprint time.(3)

(b)Use Student B's table to work out an estimate for the mean sprint time.(3)

(c)The true mean of the 20 sprint times was 14.6 seconds. Which student's estimate is closer to the true mean? Comment on whether using narrower class intervals always gives a more accurate estimate of the mean.(3)

(Total for Question 15 is 9 marks)

Mark scheme · S05 Estimated Mean from Grouped Data

Question 1

(a) B1 midpoint 5 for 0 < t ≤ 10, cao
(a) B1 midpoint 15 for 10 < t ≤ 20, cao
(a) B1 midpoint 25 for 20 < t ≤ 30, cao
(a) B1 midpoint 35 for 30 < t ≤ 40, cao
(a) Answer: 5, 15, 25, 35
(b) B1 10 < t ≤ 20 cao
(b) Answer: 10 < t ≤ 20

Question 2

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

Question 3

(a) B1 all midpoints correct: 2.5, 7.5, 12.5, 17.5, 22.5
(a) B1 all fx values correct: 10, 75, 112.5, 87.5, 45
(a) B1 Σfx = 330
(a) Answer: Midpoints 2.5, 7.5, 12.5, 17.5, 22.5; fx values 10, 75, 112.5, 87.5, 45; Σfx = 330
(b) M1 330 ÷ 30, oe (Σfx ÷ Σf)
(b) A1 11 cao
(b) Answer: 11 kg

Question 4

(a) B1 all midpoints correct: 2.5, 7.5, 12.5, 17.5, 22.5
(a) B1 all fx values correct: 7.5, 60, 87.5, 87.5, 45
(a) B1 Σfx = 287.5
(a) Answer: Midpoints 2.5, 7.5, 12.5, 17.5, 22.5; fx values 7.5, 60, 87.5, 87.5, 45; Σfx = 287.5
(b) M1 287.5 ÷ 25, oe (Σfx ÷ Σf)
(b) A1 11.5 cao
(b) Answer: 11.5 minutes

Question 5

(a) B1 all midpoints correct: 10, 30, 50, 70
(a) B1 all fx values correct: 60, 420, 500, 350
(a) B1 Σfx = 1330
(a) Answer: Midpoints 10, 30, 50, 70; fx values 60, 420, 500, 350; Σfx = 1330
(b) M1 1330 ÷ 35, oe (Σfx ÷ Σf)
(b) A1 38 cao
(b) Answer: 38 kWh

Question 6

B2 full explanation, e.g. the exact amount of electricity used by each household within a class interval is not known, so the midpoint is used to represent every household in that class; since not every household will have used exactly the midpoint value, the calculated mean is only an estimate
B1 partial explanation, e.g. states that the individual data values inside each class interval are unknown/not used, without linking this to why the mean is only an estimate

Question 7

(a) B1 15,000 cao
(a) Answer: 15,000 miles
(b) M1 all midpoints correct: 2,500, 7,500, 15,000, 30,000
(b) M1 all fx values correct: 15,000, 135,000, 240,000, 300,000
(b) M1 690,000 ÷ 50, oe
(b) A1 13,800 cao
(b) Answer: 13,800 miles

Question 8

(a) B1 all midpoints correct: 1.25, 1.75, 2.25, 2.75
(a) B1 all fx values correct: 10, 38.5, 54, 16.5
(a) B1 Σfx = 119
(a) Answer: Midpoints 1.25, 1.75, 2.25, 2.75; fx values 10, 38.5, 54, 16.5; Σfx = 119
(b) M1 119 ÷ 60, oe
(b) A1 1.98 cao (awrt 1.98)
(b) Answer: 1.98 m (3 s.f.)

Question 9

(a) B1 at least two frequencies correct, read from the bar chart
(a) B1 all four frequencies correct: 18, 5, 15, 7
(a) Answer: 18, 5, 15, 7
(b) M1 midpoints 5, 15, 25, 35 used
(b) M1 Σfx = 785, oe (18×5 + 5×15 + 15×25 + 7×35)
(b) A1 17.4 cao (awrt 17.4)
(b) Answer: 17.4 cm (1 d.p.)

Question 10

(a) B1 0 < d ≤ 10 cao
(a) Answer: 0 < d ≤ 10
(b) M1 cumulative frequencies 18, 23, 38, 45 found and the middle position (23rd of 45 values) identified
(b) A1 10 < d ≤ 20 cao
(b) Answer: 10 < d ≤ 20
(c) B1 states that the table only records how many diameters fall in each class interval, not the individual diameters themselves
(c) B1 concludes that only the class containing the mode/median can be identified from the table, not the exact value

Question 11

(a) M1 8+14+12+6, oe (adds the four known frequencies)
(a) A1 x = 20 cao
(a) Answer: x = 20
(b) M1 midpoints 5, 15, 25, 35, 45 used
(b) M1 Σfx = 1,520, oe
(b) A1 25.3 cao (awrt 25.3)
(b) Answer: 25.3 minutes (1 d.p.)

Question 12

(a) M1 midpoints 1, 3, 5, 7 used
(a) M1 Σfx = 152, oe
(a) A1 £3.80 cao
(a) Answer: £3.80
(b) M1 midpoints 1, 3, 5, 7 used
(b) M1 Σfx = 156, oe
(b) A1 £3.90 cao
(b) Answer: £3.90
(c) B1 Shop B cao
(c) B1 correct comparison referencing both means, e.g. Shop B's estimated mean spend (£3.90) is £0.10 higher than Shop A's (£3.80)
(c) Answer: Shop B

Question 13

(a) B1 any suitable reason, e.g. it would take too long/be too costly/be impractical to record every customer who visits in a year
(b) B1 all midpoints correct: 2.5, 7.5, 12.5, 17.5
(b) B1 all fx values correct: 20, 150, 175, 140
(b) M1 485 ÷ 50, oe
(b) A1 9.7 cao
(b) Answer: 9.7 minutes
(c) B1 No cao
(c) B1 correct reason, e.g. the estimated mean waiting time (9.7 minutes) is greater than the claimed 8 minutes
(c) Answer: No

Question 14

M1 Σfx = 8×5 + 15y + 12×25 = 340 + 15y and Σf = 20 + y set up
M1 correct equation formed: (340+15y) ÷ (20+y) = 16, leading to 340+15y = 320+16y
A1 y = 20 shown correctly, cao
Answer: y = 20

Question 15

(a) M1 midpoints 12.5, 13.5, 14.5, 15.5, 16.5 used
(a) M1 Σfx = 289, oe
(a) A1 14.45 cao
(a) Answer: 14.45 seconds
(b) M1 midpoints 13, 15, 17 used
(b) M1 Σfx = 290, oe
(b) A1 14.5 cao
(b) Answer: 14.5 seconds
(c) B1 Student B's estimate (14.5 s) is closer to the true mean (14.6 s) than Student A's (14.45 s), cao
(c) B1 states that narrower class intervals generally tend to give a more accurate estimate, because the midpoint represents a smaller range of possible values
(c) B1 recognises that this is not guaranteed for every data set, e.g. here the wider intervals happened to give the closer estimate, so narrower classes do not always produce a better estimate
(c) Answer: Student B

Estimated Mean from Grouped Data - Worksheets, Questions and Revision

S05 Estimated Mean from Grouped Data

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