GCSE Cumulative Frequency And Box Plots

1.	Mary recorded the heights, in centimetres, of the girls in her class.
	She put the heights in order.
132	144	150	152	160	162	162	167	
167	170	172	177	181	182	182		
(a)	Find
(i)	the lower quartile,
(ii)	the upper quartile.
(2)
(b)	On the grid, draw a box plot for this data.

(3)
(Total 5 marks)
2.	The table shows information about the heights of 40 bushes.
Height (h cm)	Frequency	
170 £ h < 175		5	
175 £ h < 180		18	
180 £ h < 185		12	
185 £ h < 190		4	
190 £ h < 195		1	

(a)	Complete the cumulative frequency table.
Height
(h cm)	CumulativeFrequency	
170 £ h < 175		
170 £ h < 180		
170 £ h < 185		
170 £ h < 190		
170 £ h < 195		
(1) 
(b)	On the grid, draw a cumulative frequency graph for your table.

(2)
(c)	Use the graph to find an estimate for the median height of the bushes.
(1)
(Total 4 marks)
3.	40 boys each completed a puzzle.
The cumulative frequency graph below gives information about the times it took them to complete the puzzle.
(a)	Use the graph to find an estimate for the median time
(1)

For the boys
	the minimum time to complete the puzzle was 9 seconds
and	the maximum time to complete the puzzle was 57 seconds.
(b)	Use this information and the cumulative frequency graph to draw a box plot showing information about the boy’s times.

(3)
	The box plot below shows information about the times taken by 40 girls to complete the same puzzle.

(c)	Make two comparisons between the boys’ times and the girls’ times.
(2)
(Total 6 marks)

4.	The table gives information about the ages of 160 employees of an IT company.
Age (A) in years	Frequency	
15 < A £ 25	44	
25 < A £ 35	56	
35 < A £ 45	34	
45 < A £ 55	19	
55 < A £ 65	7	
(a)	Write down the modal class interval.
(1)
(b)	Complete the cumulative frequency table.
Age (A) in years	Cumulative Frequency	
15 < A £ 25		
15 < A £ 35		
15 < A £ 45		
15 < A £ 55		
15 < A £ 65		
(1)
(c)	On the grid below, draw a cumulative frequency graph for your table.
(2)
(d)	Use your graph to find an estimate for
(i)	the median age of the employees,
(i)	the interquartile range of the ages of the employees.
(3)
	Another IT company has 80 employees.
	The age of the youngest employee is 24 years.
The age of the oldest employee is 54 years.
	The median age is 38 years.
The lower quartile age is 30 years.
The upper quartile age is 44 years.
(e)	On the grid below, draw a box plot to show information about the ages of the employees.
(2)

	Diagram for part (e).

(Total 9 marks)

5.	The box plot gives information about the distribution of the heights of all the trees in a wood.
 
(a)	Write down the median height of the trees.
...
(1)
(b)	Work out the interquartile range of the heights of the trees.
(1)
	There are 300 trees in the wood.
(c)	Work out the number of trees in the wood with a height of 17 m or more.
(2)
(Total 4 marks)
6.	The incomplete box plot and table show some information about some marks.

Mark	
Lowest mark	5	
Lower quartile		
Median	30	
Upper quartile	35	
Highest mark	55	

(a)	Use the information in the table to complete the box plot.
(2)
(b)	Use the information in the box plot to complete the table.
(1)
(Total 3 marks) 
7.	The box plot gives information about the distribution of the weights of bags on a plane.
		
(a)	Jean says the heaviest bag weighs 23 kg.
	She is wrong.
Explain why.
(1)
(b)	Write down the median weight.
............................. kg
(1)
(c)	Work out the interquartile range of the weights.
............................. kg
(1)
	There are 240 bags on the plane.
(d)	Work out the number of bags with a weight of 10 kg or less.
(2)
(Total 5 marks)
8.	Lucy did a survey about the amounts of money spent by 120 men during their summer holidays.
	The cumulative frequency table gives some information about the amounts of money spent by the 120 men.

Amount (£A) spent	Cumulative frequency	
0 ≤ A < 100	13	
0 ≤ A < 150	25	
0 ≤ A < 200	42	
0 ≤ A < 250	64	
0 ≤ A < 300	93	
0 ≤ A < 350	110	
0 ≤ A < 400	120	

(a)	On the grid, draw a cumulative frequency diagram.

(2) 
(b)	Use your cumulative frequency diagram to estimate the median.
(2)
	A survey of the amounts of money spent by 200 women during their summer holidays gave a median of £205
(c)	Compare the amounts of money spent by the women with the amounts of money spent by the men.
(1)
(Total 5 marks)

9.	The table shows information about the amount spent by 100 customers in a supermarket.

Amount spent (£n)	Frequency	
0 < n ≤ 20	18	
20 < n ≤ 40	22	
40 < n ≤ 60	35	
60 < n ≤ 80	15	
80 < n ≤ 100	8	
100 < n ≤ 120	2	
 
(a)	Complete the cumulative frequency table for this information.

Amount spent (£n)	Cumulative
frequency	
0 < n ≤ 20	18	
0 < n ≤ 40		
0 < n ≤ 60		
0 < n ≤ 80		
0 < n ≤ 100		
0 < n ≤ 120		
(1)

(b)	On the grid, draw a cumulative frequency graph for your table.

(2) 
(c)	Use your graph to find an estimate for the median amount spent.
(1)
(Total 4 marks)
10.	An operator took 100 calls at a call centre.
The table gives information about the time (t seconds) it took the operator to answer each call.
Time (t seconds)	Frequency	
0 < t ≤ 10	16	
10 < t ≤ 20	34	
20 < t ≤ 30	32	
30 < t ≤ 40	14	
40 < t ≤ 50	4	

(a)	Complete the cumulative frequency table.
Time (t seconds)	Cumulative Frequency	
0 < t ≤ 10	16	
0 < t ≤ 20		
0 < t ≤ 30		
0 < t ≤ 40		
0 < t ≤ 50		
(1)

(b)	On the grid, draw a cumulative frequency graph for your table.
(2)
(c)	Use your graph to find an estimate for the number of calls the operator took more than
18 seconds to answer.
(2)
(Total 5 marks)

11.	Mrs Raja set work for the students in her class.
She recorded the time taken, in minutes, for each student to do the work.
She used her results to work out the information in the table.
Minutes	
Shortest time	4	
Lower quartile	14	
Median	26	
Upper quartile	30	
Longest time	57	
 
	On the grid, draw a box plot to show the information in the table.

(Total 2 marks) 
12.	The table shows information about the ages of the 240 people at a club.
Age (t years)	Frequency	
15 £ t < 20	95	
20 £ t < 25	90	
25 £ t < 30	35	
30 £ t < 35	15	
35 £ t < 40	5	

(a)	Complete the cumulative frequency table.
Age (t years)	Cumulative
frequency	
15 £ t < 20		
15 £ t < 25		
15 £ t < 30		
15 £ t < 35		
15 £ t < 40		
(1) 
(b)	On the grid, draw the cumulative frequency graph for your table.

(2)
(c)	Use your graph to find an estimate for the median age of the people.
(1)
(Total 4 marks)
13.	The table gives information about the ages of 160 employees of an IT company.
Age (A) in years	Frequency	
15 < A £ 25	44	
25 < A £ 35	56	
35 < A £ 45	34	
45 < A £ 55	19	
55 < A £ 65	7	

(a)	Complete the cumulative frequency table.
Age (A) in years	Cumulative Frequency	
15 < A £ 25		
15 < A £ 35		
15 < A £ 45		
15 < A £ 55		
15 < A £ 65		
(1) 
(b)	On the grid below, draw a cumulative frequency graph for your table.
(2)
(c)	Use your graph to find an estimate for
(i)	the median age of the employees,
(ii)	the interquartile range of the ages of the employees.
(3)
	Another IT company has 80 employees.
	The age of the youngest employee is 24 years.
The age of the oldest employee is 54 years.
	The median age is 38 years.
The lower quartile age is 30 years.
The upper quartile age is 44 years.
(d)	On the grid, draw a box plot to show information about the ages of the employees.
(2)

	Diagram for part (d).

(Total 8 marks)

14.	Here are four cumulative frequency diagrams.

	Here are four box plots.

	For each box plot, write down the letter of the appropriate cumulative
frequency diagram.
(Total 2 marks)
15.	A company tested 100 batteries.
	The table shows information about the time in hours that the batteries lasted.
Time(t hours)	Frequency	
50 ≤ t < 55	12	
55 ≤ t < 60	21	
60 ≤ t < 65	36	
65 ≤ t < 70	23	
70 ≤ t < 75	8	

(a)	Complete the cumulative frequency table.
(1)
Time (t hours)	Cumulative frequency	
50 ≤ t < 55	12	
50 ≤ t < 60		
50 ≤ t <65		
50 < t < 70		
50 ≤ t < 75		
(b)	On the grid, draw a cumulative frequency graph for your completed table.
(2)
(c)	Use your completed graph to find an estimate for the median time.

(1)
(Total 4 marks)
16.	30 students took part in a National Science quiz.
The quiz was in two parts.
The cumulative frequency graph on the grid below gives information about the marks scored in Part One.
The lowest mark was 5 and the highest mark was 47.

(a)	In the space provided on the grid, draw a box plot using the cumulative frequency graph for the results of Part One.
(3)
	The diagram also shows a box plot for the results of Part Two.
Use the box plots to compare the two distributions.
(b)	Give two differences between them.
(2)
(Total 5 marks)

17.	The table shows information about the number of hours that 120 children used a computer last week.
Number of hours
(h)	Frequency	
0 < h ≤ 2	10	
2 < h ≤ 4	15	
4 < h ≤ 6	30	
6 < h ≤ 8	35	
8 < h ≤ 10	25	
10 < h ≤ 12	5	

(a)	Work out an estimate for the mean number of hours that the children used a computer.
Give your answer correct to two decimal places.
(4)
(b)	Complete the cumulative frequency table.
Number of hours
(h)	Cumulative
frequency	
0 < h ≤ 2	10	
0 < h ≤ 4		
0 < h ≤ 6		
0 < h ≤ 8		
0 < h ≤ 10		
0 < h ≤ 12		
(1)
(c)	On the grid, draw a cumulative frequency graph for your table.

(2)
(d)	Use your graph to find an estimate for the number of children who used a computer for less than 7 hours last week.
(2)
(Total 9 marks)