Histograms and Frequency Density (GCSE Statistics) - Worksheets, Questions and Revision

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H16 Histograms and Frequency Density

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

In a grouped frequency table with unequal class widths, the frequency density for a class is found using the formula:

frequency density = frequency ÷ class width

(a)A class has frequency 24 and class width 6. Work out the frequency density for this class.(2)

(b)A different class has frequency 45 and class width 9. Work out the frequency density for this class.(2)

(Total for Question 1 is 4 marks)

A cinema manager recorded how long, q minutes, each of 70 customers queued at the kiosk before a late-night screening. She grouped the times using unequal class widths, because most customers queued for a fairly short time, with far fewer queuing for a long time.

Time, q (minutes)	Frequency	Class width	Frequency density
0 ≤ q < 10	12	10	1.2
10 ≤ q < 20	18	10	1.8
20 ≤ q < 40	20	20	?
40 ≤ q < 70	15	30	?
70 ≤ q < 90	5	20	?

(a)Complete the table by calculating the three missing frequency densities.(3)

(b)Write down the class with the highest frequency density.(1)

(c)Suggest one reason why the manager used unequal class widths for the queue times, rather than five equal-width classes of 18 minutes each.(1)

(Total for Question 2 is 5 marks)

A weather station recorded the daily rainfall, r mm, on each of 50 days one spring. The histogram shows the results.

(a)Write down the frequency density for the class 6 < r ≤ 10.(1)

(b)Write down the class with the highest frequency density.(1)

(c)Work out the number of days with rainfall between 2mm and 6mm.(2)

(Total for Question 3 is 4 marks)

For each statement about histograms, write down whether it is True or False.

(i)In a histogram, the area of each bar represents the frequency for that class.(1)

(ii)If every class in a histogram has the same width, the tallest bar always shows the class with the most data values.(1)

(iii)In a histogram with unequal class widths, the tallest bar always shows the class with the most data values.(1)

(iv)The vertical axis of a histogram should always be labelled 'Frequency'.(1)

(Total for Question 4 is 4 marks)

A cafe manager recorded how long, w minutes, each of a sample of customers waited to be served one lunchtime. The histogram shows the results.

(a)Work out the number of customers who waited between 2 and 5 minutes.(2)

(b)Work out the number of customers who waited between 8 and 14 minutes.(2)

(c)Hence work out how many times as many customers waited between 2 and 5 minutes as waited between 8 and 14 minutes.(2)

(Total for Question 5 is 6 marks)

The times, t seconds, taken by 60 runners to complete a sprint drill were recorded, using unequal class widths because sprint times cluster closely together while a few slower runners take much longer. Some entries in the table are missing.

Time, t (seconds)	Frequency	Class width	Frequency density
10 ≤ t < 12	9	2	4.5
12 ≤ t < 14	?	2	6
14 ≤ t < 17	21	3	?
17 ≤ t < 22	?	5	1.2
22 ≤ t < 30	?	8	?

(a)Complete the table by finding the frequency for the class 12 ≤ t < 14, and the frequency density for the class 14 ≤ t < 17.(2)

(b)Complete the table by finding the frequency for the class 17 ≤ t < 22.(1)

(c)Given that the total frequency is 60, work out the frequency for the class 22 ≤ t < 30, and hence its frequency density.(3)

(d)Explain why it would not be appropriate to compare the classes in this table using their frequencies alone.(1)

(Total for Question 6 is 7 marks)

For each part, use the formula frequency density = frequency ÷ class width.

(a)A class has frequency 36 and frequency density 4.5. Work out the class width.(2)

(b)A different class has class width 12 and frequency density 3.5. Work out the frequency for this class.(2)

(Total for Question 7 is 4 marks)

A quality inspector measured the diameter, x mm, of 80 ball bearings taken from a production line. The histogram shows the results.

(a)Show that the number of ball bearings with diameter greater than 10mm is 40.(2)

(b)Hence find the probability that a ball bearing chosen at random from the 80 tested has diameter greater than 10mm. Give your answer as a decimal.(2)

(c)State one assumption needed for your answer to part (b) to be valid.(1)

(Total for Question 8 is 5 marks)

A school's PE department wants to investigate how long, t minutes, Year 10 students spend on physical activity after school on a typical day. They plan to survey a sample of 60 students and record the times using a grouped frequency table with unequal class widths. The data they collected is shown below.

Time, t (minutes)	Frequency	Class width	Frequency density
0 ≤ t < 15	9	15	?
15 ≤ t < 30	36	15	?
30 ≤ t < 60	12	30	?
60 ≤ t < 90	3	30	?

(a)(Plan) Give one reason why the department might choose unequal class widths for this survey, rather than dividing the times into equal-width classes.(1)

(b)(Process) Complete the table above by calculating the frequency density for each class.(4)

(c)(Interpret) State the modal class in terms of frequency density, and use it to describe when Year 10 students are typically most active.(2)

(d)(Interpret) A teacher claims: 'More than half of the students surveyed are active for under 30 minutes.' Use the table to comment on whether this claim is supported.(2)

(Total for Question 9 is 9 marks)

A vet recorded the mass, m kg, of 80 dogs brought in for a routine check-up, using the grouped frequency table below.

Mass, m (kg)	Frequency	Class width	Frequency density
0 < m ≤ 10	8	10	0.8
10 < m ≤ 20	20	10	?
20 < m ≤ 30	24	10	?
30 < m ≤ 45	18	15	?
45 < m ≤ 65	10	20	?

(a)Complete the table by calculating the four missing frequency densities.(4)

(b)On the grid, draw a histogram to show this information.(3)

(Total for Question 10 is 7 marks)

A market trader sells phone cases and records his sales on a histogram. He says: 'Looking at my sales histogram, the tallest bar is for cases priced £10 to £15, so I sold more cases in that price range than in any other.'

(a)Work out the number of phone cases sold priced £10 to £15.(2)

(b)Work out the number of phone cases sold priced £15 to £30.(2)

(c)Evaluate the trader's claim, using your answers to parts (a) and (b).(2)

(Total for Question 11 is 6 marks)

A forestry researcher recorded the age, y years, of 150 trees in a plantation. Four of the classes are shown on the histogram below; the bar for the class 40 < y ≤ 60 has been left out.

(a)Work out the total frequency represented by the four classes shown on the histogram.(3)

(b)Given that there are 150 trees in total, work out the frequency for the missing class 40 < y ≤ 60.(2)

(c)Hence work out the frequency density that should be used to draw the missing bar.(2)

(Total for Question 12 is 7 marks)

A courier company recorded the delivery time, d minutes, for a sample of 100 parcels. The histogram shows the results.

(a)Write down the class with the highest frequency density.(1)

(b)Use the histogram to complete a frequency table for the delivery times.(5)

(c)Using midpoints, estimate the mean delivery time.(3)

(d)Explain why your answer to part (c) is an estimate rather than an exact value.(1)

(Total for Question 13 is 10 marks)

A recycling centre recorded the weight, w kg, of 100 bags of glass collected in one week. The histogram shows the results.

(a)Show that the frequency for the class 10 < w ≤ 20 is 40.(2)

(b)Show that the median weight lies in the class 10 < w ≤ 20.(2)

(c)Use linear interpolation to estimate the median weight.(3)

(Total for Question 14 is 7 marks)

A gym recorded the length of members' gym sessions, in minutes, separately for 114 weekday visits and 78 weekend visits. The histograms show the results.

(a)Work out the number of weekday sessions lasting between 45 and 60 minutes.(2)

(b)Work out the number of weekend sessions lasting between 45 and 75 minutes.(2)

(c)Make two comparisons between the weekday and weekend histograms, referring to frequency density values from each chart.(2)

(Total for Question 15 is 6 marks)

An estate agent recorded the floor area, A m², of 100 properties sold in a town during one year.

Floor area, A (m²)	Frequency	Class width	Frequency density
40 < A ≤ 60	20	20	?
60 < A ≤ 80	30	20	?
80 < A ≤ 100	25	20	?
100 < A ≤ 140	20	40	?
140 < A ≤ 180	5	40	?

(a)Complete the table by calculating the frequency density for each class.(5)

(b)Estimate the number of properties with floor area greater than 90 m².(4)

(c)Hence estimate the probability that a randomly selected property from this sample has floor area greater than 90 m².(2)

(Total for Question 16 is 11 marks)

Mark scheme · H16 Histograms and Frequency Density

Question 1

(a) M1 24 ÷ 6, oe
(a) A1 4 cao
(a) Answer: 4
(b) M1 45 ÷ 9, oe
(b) A1 5 cao
(b) Answer: 5

Question 2

(a) B1 1.0 cao (20 ÷ 20)
(a) B1 0.5 cao (15 ÷ 30)
(a) B1 0.25 cao (5 ÷ 20)
(a) Answer: 1.0, 0.5, 0.25
(b) B1 10 ≤ q < 20 cao, ft from (a)
(b) Answer: 10 ≤ q < 20
(c) B1 sensible reason, e.g. most queue times are short and cluster together, so narrower classes there give more detail, while the few long queue times can be grouped into wider classes without losing useful information
(c) Answer: Narrower classes give more detail where most of the data lies; the few long queue times can share a wider class.

Question 3

(a) B1 2 cao
(a) Answer: 2
(b) B1 2 < r ≤ 6 cao
(b) Answer: 2 < r ≤ 6
(c) M1 7.5 × 4, oe
(c) A1 30 cao
(c) Answer: 30

Question 4

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

Question 5

(a) M1 8 × 3, oe
(a) A1 24 cao
(a) Answer: 24
(b) M1 0.5 × 6, oe
(b) A1 3 cao
(b) Answer: 3
(c) M1 24 ÷ 3, ft from (a) and (b)
(c) A1 8 cao
(c) Answer: 8

Question 6

(a) B1 12 cao (6 × 2)
(a) B1 7 cao (21 ÷ 3)
(a) Answer: 12, 7
(b) B1 6 cao (1.2 × 5)
(b) Answer: 6
(c) M1 60 - (9 + 12 + 21 + 6), oe
(c) A1 12 cao
(c) B1 1.5 cao, ft (12 ÷ 8)
(c) Answer: 12, 1.5
(d) B1 the class widths are unequal, so a class can have a higher frequency simply because it covers a wider range of times, not because times are more concentrated there; frequency density accounts for this
(d) Answer: The class widths are unequal, so frequency alone does not show how concentrated the data is.

Question 7

(a) M1 36 ÷ 4.5, oe
(a) A1 8 cao
(a) Answer: 8
(b) M1 3.5 × 12, oe
(b) A1 42 cao
(b) Answer: 42

Question 8

(a) M1 4.8 × 5 + 1.6 × 10, oe
(a) A1 24 + 16 = 40, fully shown
(a) Answer: 40
(b) M1 40 ÷ 80, ft from (a)
(b) A1 0.5 cao
(b) Answer: 0.5
(c) B1 each ball bearing is equally likely to be chosen (selection is at random), oe
(c) Answer: Each ball bearing is equally likely to be selected.

Question 9

(a) B1 sensible reason, e.g. most students' activity times are expected to cluster in a fairly narrow range, so narrower classes there give more detail, while few students are active for a very long time so those can share a wider class
(a) Answer: Unequal widths let the narrow, closely-clustered range of typical times be shown in more detail.
(b) B1 0.6 cao
(b) B1 2.4 cao
(b) B1 0.4 cao
(b) B1 0.1 cao
(b) Answer: 0.6, 2.4, 0.4, 0.1
(c) B1 15 ≤ t < 30, ft from (b)
(c) B1 sensible interpretation, e.g. activity times are most concentrated between 15 and 30 minutes, more so than any other length of time relative to its class width
(c) Answer: 15 ≤ t < 30; this class has the greatest concentration of activity times.
(d) M1 9 + 36 = 45, and 45 ÷ 60 (= 0.75 or 75%), oe
(d) A1 supported, since 75% is more than half, with correct figures quoted
(d) Answer: Supported: 45 out of 60 students (75%) are active for under 30 minutes.

Question 10

(a) B1 2.0 cao
(a) B1 2.4 cao
(a) B1 1.2 cao
(a) B1 0.5 cao
(a) Answer: 2.0, 2.4, 1.2, 0.5
(b) B1 a suitable linear scale used on the frequency density axis, starting at 0
(b) B1 all five bars drawn to the correct height (0.8, 2.0, 2.4, 1.2, 0.5), ft from (a), tolerance ± half a small square
(b) B1 bars drawn with no gaps between them, chart fully labelled
(b) Answer: Histogram with bars of frequency density 0.8, 2.0, 2.4, 1.2, 0.5 across the five classes, drawn touching with no gaps.

Question 11

(a) M1 8 × 5, oe
(a) A1 40 cao
(a) Answer: 40
(b) M1 3 × 15, oe
(b) A1 45 cao
(b) Answer: 45
(c) B1 the claim is not correct/is false
(c) B1 correct reasoning: although £10-£15 has the tallest bar (highest frequency density), £15-£30 is a wider price band and actually accounts for more cases sold, 45 compared with 40, ft from (a) and (b)
(c) Answer: The claim is false: 45 cases were sold priced 15-30 pounds, more than the 40 sold priced 10-15 pounds, even though 10-15 pounds has the taller bar.

Question 12

(a) M1 1.0 × 20, 3.0 × 20, 0.8 × 30 and 0.1 × 60, oe (at least 2 correct)
(a) M1 20 + 60 + 24 + 6, oe (sum of all four)
(a) A1 110 cao
(a) Answer: 110
(b) M1 150 - 110, ft from (a)
(b) A1 40 cao
(b) Answer: 40
(c) M1 40 ÷ 20, ft from (b)
(c) A1 2.0 cao
(c) Answer: 2.0

Question 13

(a) B1 20 < d ≤ 30 cao
(a) Answer: 20 < d ≤ 30
(b) B1 10 cao (1.0 × 10)
(b) B1 25 cao (2.5 × 10)
(b) B1 30 cao (3.0 × 10)
(b) B1 20 cao (1.0 × 20)
(b) B1 15 cao (0.75 × 20)
(b) Answer: 10, 25, 30, 20, 15
(c) M1 midpoints 5, 15, 25, 40, 60 identified
(c) M1 sum of (frequency × midpoint) = 2875, oe
(c) A1 28.75 (minutes) cao
(c) Answer: 28.75 minutes
(d) B1 the exact delivery time of each parcel within a class is not known, only the class it falls in, so the midpoint is used as an approximation for every value in that class
(d) Answer: The exact times within each class are unknown, so the midpoint is used as an approximation.

Question 14

(a) M1 4.0 × 10, oe
(a) A1 40, fully shown
(a) Answer: 40
(b) M1 cumulative frequency before this class: (3.2 × 5) + (4.8 × 5) = 16 + 24 = 40, which is less than 50 (half of 100)
(b) A1 cumulative frequency through this class: 40 + 40 = 80, which is at least 50, so the 50th value (the median) falls in 10 < w ≤ 20
(b) Answer: Median lies in 10 < w ≤ 20, since the cumulative frequency passes 50 within this class.
(c) M1 (50 - 40) ÷ 40, oe
(c) M1 their fraction × 10 (= 2.5)
(c) A1 10 + 2.5 = 12.5 (kg) cao
(c) Answer: 12.5 kg

Question 15

(a) M1 1.6 × 15, oe
(a) A1 24 cao
(a) Answer: 24
(b) M1 1.4 × 30, oe
(b) A1 42 cao
(b) Answer: 42
(c) B1 the weekday histogram's highest frequency density (3.2) is in the 30-45 minute class, showing most weekday sessions are relatively short
(c) B1 the weekend histogram's highest frequency density (1.4) is in the 45-75 minute class, showing weekend sessions typically last longer than weekday sessions
(c) Answer: Weekday sessions are concentrated in the shorter 30-45 minute class (FD 3.2); weekend sessions are concentrated in the longer 45-75 minute class (FD 1.4), so weekend sessions tend to last longer.

Question 16

(a) B1 1.0 cao
(a) B1 1.5 cao
(a) B1 1.25 cao
(a) B1 0.5 cao
(a) B1 0.125 cao
(a) Answer: 1.0, 1.5, 1.25, 0.5, 0.125
(b) M1 90 recognised as lying halfway through the 80 < A ≤ 100 class
(b) M1 (100 - 90) ÷ 20 (= 0.5) × 25 (= 12.5), oe, assuming values are spread evenly through the class
(b) M1 12.5 + 20 + 5, oe (adding the two classes fully above 100)
(b) A1 37.5 (accept 37 or 38) cao
(b) Answer: 37.5 properties (accept 37 or 38)
(c) M1 37.5 ÷ 100, ft from (b)
(c) A1 0.375 cao
(c) Answer: 0.375

Question 16

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Histograms and Frequency Density (GCSE Statistics) - Worksheets, Questions and Revision

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