A cafe manager recorded the number of takeaway coffees sold in one hour by 25 different baristas. The grouped frequency table below shows the results, with tally marks used to record the data.
Number of coffees (c)
Tally
Frequency
0-4
||||
4
5-9
|||| ||||
8
10-14
|||| |
?
15-19
||||
?
20-24
|||
3
(a)Use the tally marks to complete the frequency column for the classes 10-14 and 15-19.(2)
(b)Write down the modal class.(1)
(c)Work out how many baristas sold fewer than 10 coffees.(1)
(Total for Question 1 is 4 marks)
2
Here are four class intervals used in a grouped frequency table for the time, t minutes, spent on homework by a group of Year 10 students: 0 ≤ t < 10, 10 ≤ t < 20, 20 ≤ t < 30, 30 ≤ t < 40
(a)Write down the midpoint of the class 0 ≤ t < 10.(1)
(b)Write down the midpoint of the class 20 ≤ t < 30.(1)
(Total for Question 2 is 2 marks)
3
When drawing a frequency polygon for grouped data, which value from each class interval is plotted against the frequency?
(a)Which value from each class interval is plotted against the frequency?(1)
A) the lower class boundary
B) the upper class boundary
C) the midpoint of the class
D) the class width
(Total for Question 3 is 1 mark)
4
A student is drawing a frequency polygon.
(a)State which axis, horizontal or vertical, should be used to plot the frequency.(1)
(Total for Question 4 is 1 mark)
5
25 students were asked how many hours they spent revising last week. The grouped frequency table is shown, but one frequency is missing.
Hours (h)
Frequency
0 ≤ h < 2
5
2 ≤ h < 4
9
4 ≤ h < 6
x
6 ≤ h < 8
4
Total = 25
(a)Work out the value of x.(2)
(Total for Question 5 is 2 marks)
6
The table shows the mass, m kg, of 40 parcels delivered by a courier in one morning.
Mass (m kg)
Frequency
Midpoint
0 ≤ m < 5
6
?
5 ≤ m < 10
14
?
10 ≤ m < 15
12
?
15 ≤ m < 20
8
?
(a)Complete the midpoint column.(2)
(b)Write down the modal class.(1)
(Total for Question 6 is 3 marks)
7
Using the table and midpoints from Question 6, draw a frequency polygon on a grid with 'Mass (kg)' on the horizontal axis from 0 to 20 (gridlines every 1) and 'Frequency' on the vertical axis from 0 to 16 (gridlines every 1).
(a)Plot the points (midpoint, frequency) for each class and join them in order with straight lines.(2)
(b)Sara instead plots the points (0,6), (5,14), (10,12), (15,8), using the lower class boundaries. Explain why Sara's method is incorrect.(1)
(Total for Question 7 is 3 marks)
8
A frequency polygon for the finishing times, t minutes, of 50 runners in a fun run is formed by plotting the midpoint of each class against its frequency. The polygon passes through the points (10, 4), (20, 9), (30, 18), (40, 13), (50, 6), where each class has a width of 10 minutes.
(a)Write down the frequency of the class with midpoint 30 minutes.(1)
(b)Write down the modal class.(1)
(c)Show that the frequencies on the polygon total the 50 runners stated.(2)
(Total for Question 8 is 4 marks)
9
A grouped frequency table for the length, l cm, of 60 fish uses the class intervals 0 ≤ l < 15, 15 ≤ l < 30, 30 ≤ l < 45, 45 ≤ l < 60.
(a)Write down the class width used in this table.(1)
(b)A second table for the same data uses a class width of 5 cm, covering the same range of 0 to 60 cm. Work out how many class intervals the second table would need.(1)
(Total for Question 9 is 2 marks)
10
State two ways in which a frequency polygon is different from a bar chart when displaying grouped data.
(a)State two ways in which a frequency polygon is different from a bar chart.(2)
(Total for Question 10 is 2 marks)
11
For each statement about frequency polygons, write True or False. (i) The points on a frequency polygon are always joined using straight lines. (ii) A frequency polygon can only be used to display discrete data. (iii) Frequency polygons are useful for comparing two or more sets of grouped data on the same axes.
(i)The points on a frequency polygon are always joined using straight lines.(1)
(ii)A frequency polygon can only be used to display discrete data.(1)
(iii)Frequency polygons are useful for comparing two or more sets of grouped data on the same axes.(1)
(Total for Question 11 is 3 marks)
12
The frequency polygons for the waiting times, w minutes, of 50 customers at Shop A and 50 customers at Shop B are drawn on the same grid, using classes of width 4 minutes. Shop A passes through: (2, 5), (6, 15), (10, 20), (14, 8), (18, 2) Shop B passes through: (2, 10), (6, 18), (10, 12), (14, 6), (18, 4)
(a)Write down the modal class for Shop A.(1)
(b)Write down the modal class for Shop B.(1)
(c)Kofi says, 'Shop B's customers generally wait for a shorter time than Shop A's customers.' Use the frequency polygons to comment on whether Kofi is correct.(2)
(Total for Question 12 is 4 marks)
13
The frequency polygon for the points scored by 40 competitors in a quiz passes through the points (5, 6), (15, 10), (25, 14), (35, 7), (45, 3), where each class has a width of 10 points.
(a)Work out the number of competitors who scored 30 points or more.(2)
(Total for Question 13 is 2 marks)
14
The frequency polygon for the daily rainfall, r mm, recorded in a town over 30 days passes through the points (2, 3), (6, 11), (10, 9), (14, 5), (18, 2), where each class has a width of 4 mm.
(a)Write down the number of days with rainfall in the class 4 ≤ r < 8.(1)
(b)Work out the number of days with rainfall of at least 8 mm.(2)
(Total for Question 14 is 3 marks)
15
The table shows the age, y years, of 45 members at a gym.
Age (y years)
Frequency
Midpoint
10 ≤ y < 20
8
15
20 ≤ y < 30
16
25
30 ≤ y < 40
12
35
40 ≤ y < 50
9
45
(a)Draw a frequency polygon for this data.(2)
(b)Write down the modal class.(1)
(c)The gym manager claims, 'More than half of the members are under 30.' Determine, showing your working, whether the manager is correct.(2)
(Total for Question 15 is 5 marks)
16
Using the frequency table for gym members' ages from Question 15, find an estimate for the range of the ages.
(a)Find an estimate for the range of the ages.(2)
(Total for Question 16 is 2 marks)
17
A gardener measured the height, h cm, of 24 tomato plants. The heights are given below: 34, 41, 28, 52, 47, 39, 45, 31, 58, 42, 36, 49, 27, 55, 44, 38, 33, 46, 51, 29, 40, 37, 53, 43
(a)Using the class intervals 25 ≤ h < 35, 35 ≤ h < 45, 45 ≤ h < 55 and 55 ≤ h < 65, complete a grouped frequency table for this data.(2)
(b)Write down the midpoint of each class interval.(1)
(c)Draw a frequency polygon for this data on a grid with 'Height (cm)' on the horizontal axis from 20 to 70 and 'Frequency' on the vertical axis from 0 to 10.(2)
(d)Write down the modal class.(1)
(Total for Question 17 is 6 marks)
18
Two frequency polygons, for the test scores (out of 60) of 30 students in Class 10X and 30 students in Class 10Y, are drawn on the same grid, using classes of width 10 marks. Class 10X passes through: (5, 2), (15, 6), (25, 10), (35, 7), (45, 3), (55, 2) Class 10Y passes through: (5, 1), (15, 3), (25, 6), (35, 9), (45, 7), (55, 4)
(a)Write down the modal class for Class 10X.(1)
(b)Write down the modal class for Class 10Y.(1)
(c)Compare the two distributions of test scores, referring to the modal classes and to the number of students scoring 35 marks or more in each class.(1)
(Total for Question 18 is 3 marks)
19
The frequency polygon for the waiting time, t minutes, of 60 patients at a clinic passes through the points (5, 8), (15, a), (25, 22), (35, 11), (45, 4), where a is missing and each class has width 10 minutes.
(a)Work out the value of a.(2)
(b)Write down the modal class.(1)
(Total for Question 19 is 3 marks)
20
The tables show the queueing time, q minutes, for 50 customers at Supermarket P and 50 customers at Supermarket Q on a Saturday morning, using classes of width 3 minutes.
(a)On the same grid, draw and clearly label frequency polygons for Supermarket P and Supermarket Q.(3)
(b)A store manager claims that Supermarket Q generally has shorter queueing times than Supermarket P. Use the polygons to justify whether the manager's claim is correct.(2)
(Total for Question 20 is 5 marks)
Mark scheme · 2.13 Frequency Polygons
Question 1
(a) B1 10-14 frequency = 6
(a) B1 15-19 frequency = 4
(a) Answer: 6 and 4
(b) B1 5-9 cao
(b) Answer: 5-9
(c) B1 4 + 8 = 12 ft
(c) Answer: 12
Question 2
(a) B1 5 cao
(a) Answer: 5
(b) B1 25 cao
(b) Answer: 25
Question 3
(a) B1 C cao
(a) Answer: C) the midpoint of the class
Question 4
(a) B1 vertical (y-axis) cao
(a) Answer: Vertical axis (y-axis)
Question 5
(a) M1 25 - (5 + 9 + 4) oe
(a) A1 x = 7 cao
(a) Answer: 7
Question 6
(a) M1 correct method shown for at least one midpoint, e.g. (0+5)/2
(a) A1 all four midpoints correct: 2.5, 7.5, 12.5, 17.5
(a) Answer: 2.5, 7.5, 12.5, 17.5
(b) B1 5 ≤ m < 10 cao
(b) Answer: 5 ≤ m < 10
Question 7
(a) M1 all four points plotted correctly at (2.5,6), (7.5,14), (12.5,12), (17.5,8) ft from Q6(a)
(a) A1 points joined in the correct order with straight lines using a ruler
(a) Answer: Frequency polygon through (2.5,6), (7.5,14), (12.5,12), (17.5,8)
(b) B1 correct explanation, e.g. a frequency polygon must use the midpoint of each class, not the class boundary, because the boundary does not represent the class as a whole oe
(b) Answer: The points should be plotted at the midpoint of each class, not at the boundary.
Question 8
(a) B1 18 cao
(a) Answer: 18
(b) B1 25 ≤ t < 35 cao
(b) Answer: 25 ≤ t < 35
(c) M1 4 + 9 + 18 + 13 + 6
(c) A1 = 50 cso
(c) Answer: 50
Question 9
(a) B1 15 cao
(a) Answer: 15 cm
(b) B1 12 cao
(b) Answer: 12
Question 10
(a) C1 first valid difference, e.g. a frequency polygon joins points with straight lines instead of drawing bars oe
(a) C1 second valid difference, e.g. points are plotted at the midpoint of each class rather than a bar spanning the whole class width, or a frequency polygon makes it easier to compare two data sets on the same axes oe
(a) Answer: e.g. (1) uses straight lines joining points instead of bars; (2) allows two or more distributions to be compared clearly on the same axes.
Question 11
(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True
Question 12
(a) B1 8 ≤ w < 12 cao
(a) Answer: 8 ≤ w < 12
(b) B1 4 ≤ w < 8 cao
(b) Answer: 4 ≤ w < 8
(c) C1 correct judgement that Kofi is correct oe, supported by the modal class of Shop B (4-8) occurring earlier than Shop A (8-12)
(c) C1 correct numerical support, e.g. Shop B has 10+18=28 out of 50 customers waiting under 8 minutes compared with Shop A's 5+15=20 out of 50
(c) Answer: Kofi is correct; Shop B's waiting times are generally shorter.
Question 13
(a) M1 identify correct classes and add 7 + 3
(a) A1 10 cao
(a) Answer: 10
Question 14
(a) B1 11 cao
(a) Answer: 11
(b) M1 identify classes with midpoint 10, 14 and 18 and add 9+5+2
(b) A1 16 cao
(b) Answer: 16
Question 15
(a) M1 all four points plotted correctly at (15,8), (25,16), (35,12), (45,9)
(a) A1 points joined in order with straight lines
(a) Answer: Polygon through (15,8), (25,16), (35,12), (45,9)
(b) B1 20 ≤ y < 30 cao
(b) Answer: 20 ≤ y < 30
(c) M1 8 + 16 = 24
(c) A1 correct conclusion: 24 out of 45 is more than half (22.5), so the manager is correct cao
(c) Answer: Correct; 24 out of 45 members are under 30, which is more than half.
Question 16
(a) M1 50 - 10 oe (upper bound of last class minus lower bound of first class)
(a) A1 40 years cao
(a) Answer: 40 years
Question 17
(a) M1 at least three classes correctly tallied
(a) A1 all four frequencies correct: 6, 9, 7, 2
(a) Answer: 6, 9, 7, 2
(b) B1 30, 40, 50, 60 all correct
(b) Answer: 30, 40, 50, 60
(c) M1 points plotted correctly at (30,6), (40,9), (50,7), (60,2) ft from (a) and (b)
(c) A1 points joined in order with straight lines
(c) Answer: Polygon through (30,6), (40,9), (50,7), (60,2)
(d) B1 35 ≤ h < 45 ft
(d) Answer: 35 ≤ h < 45
Question 18
(a) B1 20 ≤ m < 30 cao
(a) Answer: 20 ≤ m < 30
(b) B1 30 ≤ m < 40 cao
(b) Answer: 30 ≤ m < 40
(c) C1 correct comparison with data reference, e.g. Class 10Y's modal class (30-40) is higher than Class 10X's (20-30), and more of 10Y scored 35+ (9+7+4=20) than 10X (7+3+2=12), so 10Y generally scored higher oe
(c) Answer: Class 10Y generally scored higher than Class 10X.
Question 19
(a) M1 60 - (8 + 22 + 11 + 4)
(a) A1 a = 15 cao
(a) Answer: 15
(b) B1 20 ≤ t < 30 ft
(b) Answer: 20 ≤ t < 30
Question 20
(a) M1 Supermarket P plotted correctly at (1.5,5), (4.5,14), (7.5,20), (10.5,8), (13.5,3) and joined with straight lines
(a) M1 Supermarket Q plotted correctly at (1.5,12), (4.5,18), (7.5,13), (10.5,5), (13.5,2) and joined with straight lines
(a) A1 both polygons fully correct and clearly labelled
(a) Answer: Two polygons as described above.
(b) C1 correct judgement: the manager is correct oe
(b) C1 correct data reference, e.g. Supermarket Q's modal class (3-6) is lower than Supermarket P's (6-9), and more of Q's customers queued under 6 minutes (12+18=30) than P's (5+14=19)
(b) Answer: The manager is correct; Supermarket Q's queueing times are generally shorter.