A box plot for the masses, in grams, of some apples has a minimum of 82, a lower quartile of 96, a median of 104, an upper quartile of 119 and a maximum of 140.
Work out the interquartile range.
Box Plots - Edexcel GCSE Higher Maths
Typically 4-6 marks when it appears, usually split between drawing and comparing · Spec 1MA1
A box plot reduces a whole data set to five numbers: the smallest value, the lower quartile, the median, the upper quartile and the largest value. The box runs from the lower quartile to the upper quartile with the median marked inside it, and a whisker reaches out from each end of the box to an extreme value. Each of the four sections holds a quarter of the data, so a wide section means the values there are spread thinly, not that there are more of them. On Edexcel 1MA1 box plots sit in spec ref S4, in the part printed in bold for Higher tier only, along with quartiles and the interquartile range.
Two shapes of question come up again and again. In the first you are handed five values in a table, or a set of clues such as a lowest mark of 10, a lower quartile of 33, an interquartile range of 35 and a range of 65, and you draw the plot on a printed scale. In the second a box plot is already drawn for one group, a second group is described, and the part that carries the marks asks you to compare the two distributions. The pairing with a cumulative frequency graph is common as well: take the median and both quartiles off the curve, then use the smallest and largest values quoted in the question to finish the plot. Any of the three papers can carry it, and a calculator changes almost nothing, since the arithmetic is subtraction.
The comparison is where marks disappear. A mark scheme wants one statement about the median and one about the spread, each carrying a figure and each phrased about what was measured: the median wait at clinic B is 7 minutes longer, and its interquartile range is 2 minutes smaller, so the waits there vary less. Two statements about spread score one mark of the two. So does a bare pair of readings with no comparative word in the sentence, and so does anything about a single individual, such as the fastest runner or the tallest plant, because one person is not a distribution. Fix the quartiles first, then write the two sentences to that pattern.
Worked example
Here are the weights, in grams, of 11 tomatoes picked from plant A, written in order.
62, 71, 74, 78, 80, 83, 85, 89, 92, 96, 118
(a) Work out the median, the lower quartile and the upper quartile of these weights, and hence the interquartile range.
(b) A box plot for 11 tomatoes picked from plant B has a minimum of 58 g, a lower quartile of 76 g, a median of 88 g, an upper quartile of 100 g and a maximum of 124 g. Compare the distribution of the weights from plant A with the distribution of the weights from plant B.
[5 marks]- There are 11 weights and the list is already ordered, so the median is the value in position . Counting along, that is 83 g.B1: median 83 g, taken by position from a sorted list rather than from the middle of the range
- The quartiles sit at positions and . The 3rd weight is 74 g and the 9th is 92 g.M1: both quartile positions worked out before any subtraction happens
- Interquartile range g.A1: 18 g, a single number, upper quartile minus lower quartile
- Average: plant B has a median of 88 g against 83 g for plant A, so the plant B tomatoes were 5 g heavier on average.B1: medians compared, with the difference quoted and the tomatoes named
- Spread: plant B's interquartile range is g against 18 g for plant A, so the middle half of the plant A weights is more tightly grouped.B1: a comparison of spread; a second remark about averages would leave this mark unearned
Practice questions
These are original questions in Edexcel 1MA1 Higher style. There are no diagrams here, so each box plot is written out as its five numbers in the order minimum, lower quartile, median, upper quartile, maximum, and you work from those exactly as you would from a plot drawn on a printed scale. Where a question applies the 1.5 times interquartile range test for an outlier, the rule is stated in the stem, because 1MA1 asks you to take outliers into account without fixing a definition of one.
No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.
Question 1 Warm-up [2 marks] Question 2 Warm-up [2 marks] For a set of test marks the lowest mark is 10 and the range is 65. Work out the highest mark.Question 3 Exam pace [2 marks] A cumulative frequency curve for the distances, in km, travelled to work by 120 employees passes through the points (0, 0), (10, 20), (20, 60), (30, 100), (40, 114) and (50, 120). Estimate the median distance, reading between the plotted points in a straight line.Question 4 Exam pace [3 marks] A value is called an outlier if it is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile. A set of data has a lower quartile of 24 and an upper quartile of 40. Work out the upper boundary, that is the upper quartile plus 1.5 times the interquartile range.Question 5 Stretch [3 marks] The box plot for the marks of 80 girls has a lower quartile of 32, a median of 41 and an upper quartile of 55. The box plot for the marks of 120 boys has a lower quartile of 28, a median of 44 and an upper quartile of 52. Estimate the total number of these 200 students who scored more than the upper quartile for their own group.Question 6 Stretch [4 marks] Here are the masses, in kg, of 13 parcels, written in order. 2.1, 2.4, 2.6, 2.9, 3.0, 3.3, 3.5, 3.6, 3.9, 4.2, 4.4, 4.8, 5.6 Work out the interquartile range.
Common mistakes examiners see
Reading the whisker ends as the quartiles, so a plot with a minimum of 12, a lower quartile of 20, an upper quartile of 35 and a maximum of 48 produces an interquartile range of .
The interquartile range is the width of the box: . The whisker ends are the smallest and largest values, and subtracting those gives the range, which is a different question with a different answer.
Treating a range or an interquartile range as a value to plot. Given a lower quartile of 33 and an interquartile range of 35, writing the upper quartile as 35; given a lowest mark of 10 and a range of 65, writing the highest mark as 65.
Add. Upper quartile , highest mark . A range is the size of a gap, so it has to be added to the value at the bottom of that gap before anything can be plotted.
Comparing individuals instead of distributions: 'the quickest boy beat the quickest girl', or 'the tallest Year 11 girl is 8 cm taller than the tallest Year 7 girl'.
Compare the medians, then compare the interquartile ranges. A value at the end of a whisker describes one person, and it earns nothing on a compare-the-distributions question however accurately it is read.
Offering two comparisons of the same kind, such as 'club A's range is bigger' followed by 'club A's times are more spread out', which is one point made twice and scores 1 of 2.
Check that your two sentences name different measures: one median, one range or interquartile range. If the word spread appears in both sentences, rewrite the first one to be about the median.
Reading a long whisker as 'most of the values are up here', or assuming the median line must sit halfway across the box.
Every section of a box plot holds a quarter of the values. A long upper whisker means the top 25% is stretched across a wide interval, and a median line close to the lower quartile means the second quarter is squeezed into a narrow one.
Frequently asked questions
- Are box plots on the Foundation paper?
- No. Within spec ref S4 the words covering box plots, quartiles and the interquartile range are the Higher-only part, so a Foundation candidate is asked for a median and a range but never for a quartile. Box plots join cumulative frequency graphs and histograms with unequal class widths on the list of statistics that only Higher students meet.
- What is the difference between the range and the interquartile range?
- The range is the full width of the box plot, from one whisker end to the other, so a single freak value moves it. The interquartile range is the width of the box, upper quartile minus lower quartile, and it covers only the middle half of the data, which is why it is the fairer measure of spread when one value is extreme. Both are a single number, never a pair of values.
- How do you get both marks on a compare the distributions question?
- Write two sentences. The first compares the medians and quotes the difference, for example that the median mark for class B is 6 higher. The second compares the spread and quotes it too, for example that the interquartile range for class B is 4 smaller, so its marks vary less. Four numbers written down with no comparison between them score nothing.
- Do I need the 1.5 times IQR rule for outliers on Edexcel GCSE Maths?
- S4 asks you to take outliers into account, but the 1MA1 subject content does not define a test for one, so you are not expected to produce the boundaries from memory the way a GCSE Statistics or A level student would. A question that wants the rule applied states the rule. What 1MA1 does expect is that you spot an extreme value and can say what it does to the range or the mean.