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Either paper 4 marks
The box plot shows information about the times taken by a group of students to solve a puzzle.
(a)State the median time.
(b)Calculate the range of the times.
(c)Calculate the interquartile range of the times.
Show answer
(a) \(25\ \text{minutes}\)
(b) \(24\ \text{minutes}\)
(c) \(11\ \text{minutes}\)The median is the line drawn inside the box.
Reading straight down from it to the scale gives \(25\) minutes.
The range is the largest value take away the smallest, and those are the two ends of the whiskers: \(10\) and \(34\).
\(34 - 10 = 24\) minutes
The two ends of the box are the quartiles. Reading them off the scale, the lower quartile is \(18\) and the upper quartile is \(29\).
Interquartile range \(= 29 - 18 = 11\) minutes
The interquartile range uses only the middle half of the data, so one unusually large value cannot stretch it the way it stretches the range.
- (c) M1upper quartile \(29\) and lower quartile \(18\) read from the ends of the box
- (c) A1interquartile range \(11\) minutes
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Either paper 5 marks
Here are the numbers of goals scored by eleven hockey players last season.
\(16,\ 19,\ 10,\ 30,\ 6,\ 33,\ 43,\ 38,\ 14,\ 27,\ 23\)
(a)Find the median.
(b)Find the interquartile range.
(c)A box plot is to be drawn for this data.
Write down the five values it needs.
Show answer
(a) \(23\ \text{goals}\)
(b) \(19\ \text{goals}\)
(c) \(6\), \(14\), \(23\), \(33\), \(43\)In order the eleven values are
\(6,\ 10,\ 14,\ 16,\ 19,\ 23,\ 27,\ 30,\ 33,\ 38,\ 43\)
With eleven values the median is the 6th one, because there are five below it and five above it.
Median \(= 23\) goals
The median splits the ordered list into a lower half of five values and an upper half of five values.
The lower quartile is the middle of the lower half, the 3rd value: \(14\).
The upper quartile is the middle of the upper half, the 9th value: \(33\).
Interquartile range \(= 33 - 14 = 19\) goals
A box plot needs the smallest value, the lower quartile, the median, the upper quartile and the largest value, in that order:
\(6\), \(14\), \(23\), \(33\), \(43\)
The whiskers reach out to \(6\) and \(43\), the box runs from \(14\) to \(33\), and the median is drawn inside it at \(23\).
- (b) M1lower quartile \(14\) and upper quartile \(33\)
- (b) A1interquartile range \(19\)
- (c) B1median \(23\) and quartiles \(14\) and \(33\)
- (c) B1smallest \(6\) and largest \(43\) found from the ordered data
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Either paper 2 marks
The box plot shows information about the marks scored by a class in a science test.
One of the students had a mark of \(14\) marks.
Is that mark in the bottom 25% of the marks? Give a reason for your answer.
Show answer
NoThe bottom 25% of the data is everything below the lower quartile, which is \(13\) marks.
\(14\) marks is above \(13\) marks, so the answer is No.
- M1the lower quartile, \(13\) marks
- A1No, comparing \(14\) marks with \(13\) marks
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Either paper 4 marks
In a survey, 50 patients at each of four surgeries were timed from arriving to seeing a doctor.
The box plots show the results.
(a)At which surgery were the waiting times most consistent, judged by the interquartile range?
Give a reason for your answer.
(b)On average, which surgery had the lowest waiting times?
Give a reason for your answer.
Show answer
(a) Elm Road. The interquartile range for Elm Road is \(4\) minutes, which is smaller than the other three (the next smallest is \(7\) minutes).
(b) Greenlane. The median for Greenlane is \(32\) minutes, which is lower than the other three (the next lowest is \(35\) minutes).An average is read from the median (the line inside each box); consistency is read from the interquartile range (the width of each box). Read both for all four plots.
Medians (minutes): Elm Road \(35\), Fairview \(39\), Greenlane \(32\), Hillcrest \(38\).
Interquartile ranges (minutes): Elm Road \(36 - 32 = 4\), Fairview \(44 - 37 = 7\), Greenlane \(37 - 28 = 9\), Hillcrest \(43 - 25 = 18\).
This part is about the interquartile range, and Elm Road has the smallest interquartile range: \(4\) minutes, against \(7\) minutes for the next smallest.
An average is read from the median and consistency from the interquartile range, the width of the box. This part is about the median.
Greenlane has the lowest median: \(32\) minutes, against \(35\) minutes for the next lowest.
Only the median answers this part. The plot with the lowest minimum (which is also the box that starts furthest left) and the plot with the shortest overall range are not Greenlane, and an extreme value says nothing about the middle half of the data.
- (a) B1Elm Road
- (a) B1reason quoting the interquartile range of \(4\) minutes against the next smallest of \(7\) minutes
- (b) B1Greenlane
- (b) B1reason quoting the median of \(32\) minutes against the next lowest of \(35\) minutes
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Either paper 4 marks
The box plots show information about the waiting times at two health centres one morning.
(a)Find the interquartile range of the waiting times for Oakfield.
(b)Compare the waiting times for Oakfield with the waiting times for Riverside.
Make one comparison about the average and one about the spread.
Show answer
(a) \(8\ \text{minutes}\)
(b) The median for Riverside is higher, \(21\) minutes against \(16\) minutes, so on average the waiting times for Riverside were greater. The interquartile range for Oakfield is smaller, \(8\) minutes against \(11\) minutes, so the waiting times for Oakfield were more consistent.The ends of the Oakfield box are the quartiles: \(13\) and \(21\).
Interquartile range \(= 21 - 13 = 8\) minutes
One comparison about the average and one about the spread, each backed up by a number off the diagram.
Average: the median for Riverside is \(21\) minutes and the median for Oakfield is \(16\) minutes, so the waiting times for Riverside were greater on average.
Spread: the interquartile range for Oakfield is \(8\) minutes and for Riverside it is \(11\) minutes, so the waiting times for Oakfield were more consistent.
Comparing the two ranges would also be a comparison of spread, but the interquartile range is the safer one: it ignores the two whisker ends, where a single unusual value sits.
- (a) M1quartiles \(13\) and \(21\) read for Oakfield
- (a) A1interquartile range \(8\) minutes
- (b) B1median for Riverside higher, \(21\) minutes against \(16\) minutes
- (b) B1interquartile range for Oakfield smaller, \(8\) minutes against \(11\) minutes, so more consistent
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Either paper 3 marks
The table shows information about the marks scored by a class in a test. No diagram has been drawn.
Smallest mark \(7\) marks Lower quartile \(14\) marks Median \(19\) marks Range \(24\) marks Interquartile range \(10\) marks (a)Work out the largest mark.
(b)Work out the upper quartile.
(c)A box plot is to be drawn from this information.
Write down, in order, the five values it needs.
Show answer
(a) \(31\ \text{marks}\)
(b) \(24\ \text{marks}\)
(c) \(7\), \(14\), \(19\), \(24\), \(31\)The range is the largest value take away the smallest, so the largest is the smallest plus the range.
\(7 + 24 = 31\) marks
The interquartile range is the upper quartile take away the lower quartile, so the upper quartile is the lower quartile plus the interquartile range.
\(14 + 10 = 24\) marks
Three of the five are given and two came from parts (a) and (b):
\(7\), \(14\), \(19\), \(24\), \(31\)
The range and the interquartile range are *not* two of the five — they are the distances between them.
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Either paper 2 marks
The box plot shows information about the marks scored by a class in a science test.
Marcus looks at the box plot and says:
"More students have a mark between the lower quartile and the median than between the median and the upper quartile, because that part of the box is wider."
Is he correct? Explain your answer.
Show answer
NoHe is not correct.
However the box plot is drawn, each of its four sections — the two whiskers and the two halves of the box — always holds exactly one quarter of the students: that is what the lower quartile, median and upper quartile mean. A wider section only shows that those students are more spread out, not that there are more of them.
- M1However the box plot is drawn, each of its four sections — the two whiskers and the two halves of the box — always holds exactly one quarter of the students: that is what the lower quartile, median and upper quartile mean. A wider section only shows that those students are more spread out, not that there are more of them.
- A1No — he is not correct
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Either paper 3 marks
The box plot shows information about the heights of the tomato plants grown in a greenhouse trial.
Work out what percentage of the range is covered by the interquartile range.
Show answer
\(20\ \%\)The range is the largest value take away the smallest:
\(42 - 2 = 40\) cm
The interquartile range is the upper quartile take away the lower quartile:
\(16 - 8 = 8\) cm
As a percentage of the range:
\(\dfrac{ 8 }{ 40 } \times 100 = 20\%\)
- M1range \(42 - 2 = 40\)
- M1interquartile range \(16 - 8 = 8\) divided by the range \(40\), times 100
- A1\(20\)%
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Either paper 5 marksThe box plot shows information about the scores achieved by students on a quiz in 2022. The table shows information about the scores achieved by students on a quiz in 2021.
marks Lower quartile 19 Median 25 Upper quartile 31 (a)In 2022, what was the lowest score?
(b)Use the data to comment on this statement.
"On average, scores were higher in 2022 than in 2021."
(c)Use the data to comment on this statement.
"The scores were equally consistent in 2022 and 2021."
Show answer
(a) \(15\ \text{marks}\)
(b) Agree. The median for 2022 is \(28\) marks and the median for 2021 is \(25\) marks, so on average the scores were higher in 2022 than in 2021.
(c) Agree. The interquartile range for 2022 is \(33 - 21 = 12\) marks and for 2021 it is \(31 - 19 = 12\) marks.The left end of the whisker is the smallest value: \(15\) marks
The average is compared using the medians.
2022 (box plot): the line in the box is at \(28\) marks.
2021 (table): the median is \(25\) marks.
The median for 2022 is greater than the median for 2021, so on average the scores were higher in 2022 than in 2021. The statement says they were higher, so the answer is Agree.
Consistency is compared using the interquartile ranges, because the table gives no range.
2022 (box plot): \(33 - 21 = 12\) marks.
2021 (table): \(31 - 19 = 12\) marks.
Comparing them, the interquartile ranges are equal, so the scores were equally consistent in both years. The statement is true, so the answer is Agree.
- (b) M1median \(28\) marks in 2022 and \(25\) marks in 2021
- (b) A1Agree: the scores were higher in 2022 than in 2021
- (c) M1interquartile range \(12\) marks in 2022 and \(12\) marks in 2021
- (c) A1Agree, comparing \(12\) marks with \(12\) marks
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Either paper 3 marksHere is some information about the times, in minutes, that some students spent on homework one evening.
Draw a box plot to show this information.Shortest time 8 minutes Longest time 30 minutes Upper quartile 24 minutes Median 18 minutes Interquartile range 14 minutes You cannot draw on this page. Draw this on paper, then show the answer and check your drawing against the model.
Show answer
A box plot with minimum 8, lower quartile 10, median 18, upper quartile 24 and maximum 30 minutesThe table gives four of the five values a box plot needs. The fifth, the lower quartile, comes from the interquartile range:
\(\text{lower quartile} = \text{upper quartile} - \text{interquartile range} = 24 - 14 = 10\) minutes
Then draw the box plot against the scale:
- whiskers out to the smallest value, \(8\), and the largest, \(30\) - the box from the lower quartile, \(10\), to the upper quartile, \(24\) - a line across the box at the median, \(18\)
The scale is ruled every 2 minutes, so each value is on a line.
- B1three of the five values plotted correctly
- B1four of the five values plotted correctly, for example with the lower quartile at \(10\)
- B1a fully correct box plot: whiskers at \(8\) and \(30\), box from \(10\) to \(24\), median at \(18\)