GCSE Maths: Box plots and outliers (Higher)
Part of the GCSE Maths statistics chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Box plots and outliers
A box plot (or box-and-whisker diagram) shows the main features of a distribution using five numbers, called the five-number summary: the smallest value, the lower quartile (Q1), the median (Q2), the upper quartile (Q3) and the largest value. The box goes from Q1 to Q3 with a line at the median. The whiskers are lines from the box to the smallest and the largest values. The box holds the middle half of the data. Each whisker and each half of the box holds one quarter of the data.
Reading a box plot. The median is the line in the box. The range is the largest minus the smallest. The IQR is the length of the box (Q3 − Q1). About 25% of the data lies in each section: from the smallest value to Q1, from Q1 to the median, from the median to Q3, and from Q3 to the largest value.
Comparing box plots. Compare the medians (a typical value), and compare the IQRs (the lengths of the boxes) or the ranges (the overall spread). Always write your comparison in context.
Outliers. An outlier is a value that is far from the rest of the data. A common rule says a value is an outlier if it is more than 1.5 × IQR above Q3 or below Q1. In a box plot an outlier is shown as a cross, and the whisker then goes to the most extreme value that is not an outlier. Outliers might be genuine unusual values or mistakes, so we should check them rather than just deleting them.
Key ideas
- Five-number summary: smallest, Q1, median, Q3, largest.
- The box is the middle 50% of the data. Each of the four sections holds 25%.
- Range = largest − smallest. IQR = length of the box = Q3 − Q1.
- Compare box plots using medians and IQRs, in context.
- Outlier rule: below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. Show outliers as crosses.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. A box plot is drawn from five numbers: the smallest value, the lower quartile, the median, the upper quartile and the largest value.
- Write the data in order. We need the ordered data to find the quartiles.23, 34, 38, 41, 45, 48, 52, 55, 60, 66, 78
- Find the median, lower quartile and upper quartile. Q1 is the median of the lower half and Q3 is the median of the upper half.Median = 48, Q1 = 38, Q3 = 60
- Write down the smallest and largest values. These are the ends of the whiskers.Smallest = 23, largest = 78
- Draw a scale and then the box plot. Draw a box from Q1 to Q3 with a line at the median, then whiskers to the smallest and largest values.
Check. The box holds the middle half of the data, and the line is inside the box. ✓
Watch out. The five numbers must be in order: smallest < Q1 < median < Q3 < largest, and the scale must be even.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Read the median and find the IQR of each box plot, then compare them.
- Read the medians. The median is the line inside each box.Class A: 48
Class B: 49 - Find the IQRs. IQR = Q3 − Q1, the length of the box.Class A: 60 − 38 = 22
Class B: 57 − 42 = 15 - Find the ranges. Largest minus smallest.Class A: 78 − 23 = 55
Class B: 64 − 30 = 34 - Write the comparison. Use the medians for the average and the IQRs for the spread, in context.The classes have similar median marks (A: 48, B: 49) but class B is more consistent: its IQR is 15 compared with 22.
Check. Class B has the shorter box and the shorter whiskers. ✓
Watch out. Compare an average and a spread, and use the words of the question.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. An outlier is a value that is far from the rest. One rule says a value is an outlier if it is more than 1.5 × IQR below Q1 or above Q3.
- Work out the IQR. IQR = Q3 − Q1.21 − 14.5 = 6.5
- Work out 1.5 × IQR. This is the allowed distance from the quartiles.1.5 × 6.5 = 9.75
- Work out the lower limit. Take it away from Q1.14.5 − 9.75 = 4.75
- Work out the upper limit. Add it to Q3.21 + 9.75 = 30.75
- Compare each value with the limits. A value below the lower limit or above the upper limit is an outlier.Outliers: 60
Check. Every other value lies between 4.75 and 30.75. ✓
Watch out. The limits are measured from Q1 and Q3, not from the median.
(Q1 = 14.5, median = 17.5, Q3 = 21, smallest = 12.)
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. The outlier is plotted as a cross. The upper whisker ends at the largest value that is not an outlier.
- Find the largest value that is not an outlier. The value 60 is an outlier. The next largest value is 22.Whisker goes up to 22.
- Write the five numbers for the box plot. Use the smallest value, Q1, median, Q3 and the largest non-outlier.12, 14.5, 17.5, 21, 22
- Draw the box plot and mark 60 with a cross. A cross shows that the value is an outlier.
Check. The cross at 60 is far from the end of the whisker at 22. ✓
Watch out. Do not extend the whisker to the outlier: mark it separately with a cross.
Watch out
A box plot does not show the mean, and a longer box does not mean there is more data. A box plot also does not show how many values there are. Compare the box plots’ medians and IQRs, and mention both.Exercise 24N Box plots and outliers
Non-calculatorKey words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. A box plot is drawn from five numbers: the smallest value, the lower quartile, the median, the upper quartile and the largest value.
- Write the data in order. We need the ordered data to find the quartiles.8, 11, 14, 15, 17, 19, 22, 25, 30
- Find the median, lower quartile and upper quartile. Q1 is the median of the lower half and Q3 is the median of the upper half.Median = 17, Q1 = 12.5, Q3 = 23.5
- Write down the smallest and largest values. These are the ends of the whiskers.Smallest = 8, largest = 30
- Draw a scale and then the box plot. Draw a box from Q1 to Q3 with a line at the median, then whiskers to the smallest and largest values.
Check. The box holds the middle half of the data, and the line is inside the box. ✓
Watch out. The five numbers must be in order: smallest < Q1 < median < Q3 < largest, and the scale must be even.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. A box plot is drawn from five numbers: the smallest value, the lower quartile, the median, the upper quartile and the largest value.
- Write the data in order. We need the ordered data to find the quartiles.38, 41, 43, 45, 47, 50, 52, 60
- Find the median, lower quartile and upper quartile. Q1 is the median of the lower half and Q3 is the median of the upper half.Median = 46, Q1 = 42, Q3 = 51
- Write down the smallest and largest values. These are the ends of the whiskers.Smallest = 38, largest = 60
- Draw a scale and then the box plot. Draw a box from Q1 to Q3 with a line at the median, then whiskers to the smallest and largest values.
Check. The box holds the middle half of the data, and the line is inside the box. ✓
Watch out. The five numbers must be in order: smallest < Q1 < median < Q3 < largest, and the scale must be even.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. The median is the line inside the box.
- Find group A’s box. The box runs from 20 to 38.
- Read the line inside it. Median = 26
Check. 26 lies between 20 and 38. ✓
Watch out. The median is the line, not the end of the box.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. IQR = Q3 − Q1: the length of the box.
- Group A. The box runs from 20 to 38.38 − 20 = 18
- Group B. The box runs from 22 to 34.34 − 22 = 12
Check. Group A’s box is longer. ✓
Watch out. Use the ends of the box, not the whiskers.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Range = largest − smallest: the ends of the whiskers.
- Read the ends of B’s whiskers. Smallest 15, largest 48
- Subtract. 48 − 15 = 33
Check. 15 + 33 = 48. ✓
Watch out. The range uses the whisker ends, not the box.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Compare medians and IQRs.
- Medians. 30 > 26.Group B took longer on average.
- Spread. B’s box (12) is shorter than A’s (18), and its range (33) is smaller than A’s (43).Group B is more consistent.
- Write it in context. On average group B took longer (median 30 minutes compared with 26 minutes), but the times of group B were more consistent.
Check. Each sentence uses numbers from the diagram. ✓
Watch out. Write about the times in the context of the puzzle.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Q3 has three quarters of the data below it.
- Think about the quartile. Three quarters of the data is at or below Q3.One quarter is above.
- Write the fraction. 14
Check. The upper whisker holds the top 25%. ✓
Watch out. Each section of a box plot holds a quarter of the data.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. This section is one quarter of the data.
- Decide what fraction of the data this section holds. The quartiles and median cut the data into four equal sections, and this is one of them.One quarter
- Find a quarter of 80. Divide by 4.80 ÷ 4 = 20
Check. Four sections of 20 make 80. ✓
Watch out. The sections are not equal in length, but they each hold the same number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Think about what the five numbers tell us.
- What is shown? The smallest, Q1, median, Q3 and largest.Five numbers.
- What is not shown? The same five numbers could come from 8 plants or from 800 plants.The sample size is not shown.
Check. Nothing on the diagram counts the values. ✓
Watch out. Box plots show spread and position, not the amount of data.
Data: 31, 33, 35, 37, 38, 40, 41, 42, 44, 46, 78. For this data Q1 = 35 and Q3 = 44.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. An outlier is a value that is far from the rest. One rule says a value is an outlier if it is more than 1.5 × IQR below Q1 or above Q3.
- Work out the IQR. IQR = Q3 − Q1.44 − 35 = 9
- Work out 1.5 × IQR. This is the allowed distance from the quartiles.1.5 × 9 = 13.5
- Work out the lower limit. Take it away from Q1.35 − 13.5 = 21.5
- Work out the upper limit. Add it to Q3.44 + 13.5 = 57.5
- Compare each value with the limits. A value below the lower limit or above the upper limit is an outlier.Outliers: 78
Check. Every other value lies between 21.5 and 57.5. ✓
Watch out. The limits are measured from Q1 and Q3, not from the median.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. An outlier is a value that is far from the rest. One rule says a value is an outlier if it is more than 1.5 × IQR below Q1 or above Q3.
- Work out the IQR. IQR = Q3 − Q1.56 − 47 = 9
- Work out 1.5 × IQR. This is the allowed distance from the quartiles.1.5 × 9 = 13.5
- Work out the lower limit. Take it away from Q1.47 − 13.5 = 33.5
- Work out the upper limit. Add it to Q3.56 + 13.5 = 69.5
- Compare each value with the limits. A value below the lower limit or above the upper limit is an outlier.No value is outside the limits, so there are no outliers.
Check. Every other value lies between 33.5 and 69.5. ✓
Watch out. The limits are measured from Q1 and Q3, not from the median.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. The IQR is the length of the box, and the median is the line inside it.
- Compare the IQRs. The IQR is the length of the box.P’s box is 6 long and Q’s box is 15 long.
- Compare the medians. They are equal.The median lines are at the same place on the scale.
- Write the comparison. Q’s middle 50% is more spread out than P’s, but the two data sets have the same typical value.
Check. A longer box means a larger IQR. ✓
Watch out. The medians are equal, so only the spread differs.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. A quarter of the data is above the upper quartile, so 52 must be Q3.
- Find 25% of 60. A quarter of 60.60 ÷ 4 = 15 values
- Say what 52 is. The value with 25% of the data above it is the upper quartile.52 is the upper quartile (the right-hand end of the box).
Check. 3 quarters of 60 is 45 values at or below 52. ✓
Watch out. 25% above means Q3.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Use range = largest − smallest and IQR = Q3 − Q1, then draw the five numbers.
- Find the largest value. Largest = smallest + range.10 + 40 = 50
- Find the upper quartile. Q3 = Q1 + IQR.20 + 16 = 36
- Write the five-number summary. In order: smallest, Q1, median, Q3, largest.10, 20, 28, 36, 50
- Draw the box plot. Box from 20 to 36 with a line at 28, whiskers to 10 and 50.
Check. 50 − 10 = 40 and 36 − 20 = 16. ✓
Watch out. Put the five numbers in order and check the scale.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Think about what the length of the box tells us.
- What does the length of the box show? It shows the interquartile range: how spread out the middle half of the data is.A longer box means more spread.
- Does it show the number of values? No. Each section holds a quarter of the data, whatever the size of the sample.The number of people cannot be seen.
- Conclude. Aisha is wrong.
Check. Two groups of different sizes can have identical box plots. ✓
Watch out. Box plots show spread and position, not the number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. The median has half the data below it, and the quartiles cut the data into quarters.
- Runners faster than 24 minutes. Half of the 40 runners have a time below the median.40 ÷ 2 = 20
- Runners between 20 and 31 minutes. The middle half of the data lies between the quartiles.40 ÷ 2 = 20
Check. Each quarter of the data has 10 runners; the middle two quarters have 20. ✓
Watch out. Between the quartiles is half of the data.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read the median and quartiles from the graph, then draw a box plot using the five numbers.
- Read the median and quartiles. Median at 40, lower quartile at 20 and upper quartile at 60 on the cumulative frequency axis.Q1 ≈ 23.5, median ≈ 31.5, Q3 ≈ 39.5
- Write the smallest and largest times. These are given in the question.Smallest = 2, largest = 58
- Draw the box plot. Box from Q1 to Q3 with a line at the median, whiskers to 2 and 58.
Check. The five numbers are in order: 2 < Q1 < median < Q3 < 58. ✓
Watch out. Read the quartiles at n ÷ 4, n ÷ 2 and 3n ÷ 4.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Each section of a box plot holds a quarter of the data, so the length of a section shows how spread out that quarter is.
- Look at the section between Q1 and the median. It is short but holds a quarter of the data.These values are bunched close together.
- Look at the section between the median and Q3. It is long but also holds a quarter of the data.These values are spread out.
- Conclude. The data is bunched at the lower end and spread out towards the higher end.
Check. A short section means values close together; a long section means values far apart. ✓
Watch out. Every section holds the same number of values, however long it is.