Box Plots from Cumulative Frequency and Outliers
A box plot built from cumulative frequency data is a diagram showing the five-number summary, minimum, lower quartile, median, upper quartile and maximum, estimated from a cumulative frequency graph rather than raw data. An outlier is a value more than 1.5 times the interquartile range beyond a quartile, marked as a separate point rather than at the end of a whisker. This is assessed on GCSE Statistics Higher tier.
Before you start
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Method
- From a cumulative frequency graph or table, estimate the median (position n/2), lower quartile (position n/4) and upper quartile (position 3n/4), interpolating within grouped data if needed.
- Record the five-number summary: minimum, lower quartile, median, upper quartile, maximum.
- Calculate the interquartile range: IQR = Q3 - Q1.
- Calculate the outlier boundaries: lower boundary = Q1 - 1.5 x IQR, upper boundary = Q3 + 1.5 x IQR.
- Test the minimum, maximum and any other suspicious values against these boundaries: any value beyond a boundary is an outlier.
- Draw the box plot with the box from Q1 to Q3 (median marked inside), whiskers extending to the most extreme values that are NOT outliers, and outliers marked individually beyond the whiskers.
Worked example
The cumulative frequency table below shows delivery times, in minutes, for 80 parcels: 0 < t <= 20, frequency 10; 20 < t <= 40, frequency 20; 40 < t <= 60, frequency 40; 60 < t <= 100, frequency 10. The minimum delivery time was 2 minutes and the maximum was 95 minutes. Estimate the quartiles, then determine whether the minimum and maximum should be plotted as outliers.
- With n = 80, the lower quartile is at position n/4 = 20, the median at position n/2 = 40, and the upper quartile at position 3n/4 = 60.
- The cumulative frequencies are 10, 30, 70, 80 at the class boundaries 20, 40, 60, 100. The lower quartile falls in the class 20 < t <= 40 (cumulative frequency 10 to 30): Q1 = 20 + (20 - 10)/20 x 20 = 30 minutes.
- The median falls in the class 40 < t <= 60 (cumulative frequency 30 to 70): median = 40 + (40 - 30)/40 x 20 = 45 minutes.
- The upper quartile also falls in the class 40 < t <= 60: Q3 = 40 + (60 - 30)/40 x 20 = 55 minutes.
- IQR = 55 - 30 = 25. Lower boundary = 30 - 1.5 x 25 = -7.5. Upper boundary = 55 + 1.5 x 25 = 92.5. The minimum (2) is above -7.5, so it is not an outlier; the maximum (95) is above 92.5, so it is an outlier.
- Final answer: Q1 = 30, median = 45, Q3 = 55 minutes; the maximum value of 95 minutes is an outlier and should be plotted as a separate point, with the whisker drawn only to the highest non-outlier value.
Practice questions
Try each question, then tap to reveal the answer.
Q1A data set has Q1 = 12 and Q3 = 20. Calculate the IQR and the upper outlier boundary.Show answer
Answer: IQR = 8; upper boundary = 20 + 1.5 x 8 = 32
Q2A data set has Q1 = 20 and Q3 = 32. Is a value of 55 an outlier? Use the 1.5 x IQR rule.Show answer
Answer: Yes; IQR = 12, upper boundary = 32 + 1.5 x 12 = 50, and 55 is greater than 50
Q3A data set has Q1 = 15 and Q3 = 45. Calculate both outlier boundaries.Show answer
Answer: Lower boundary = 15 - 1.5 x 30 = -30; upper boundary = 45 + 1.5 x 30 = 90
Q4A cumulative frequency table gives Q1 = 18, median = 25, Q3 = 34 for the ages of 60 people at a conference. Calculate the IQR and state the upper outlier boundary.Show answer
Answer: IQR = 34 - 18 = 16; upper boundary = 34 + 1.5 x 16 = 58
Q5Using the data in the previous question, a delegate aged 65 attends. Is this delegate's age an outlier? Explain.Show answer
Answer: Yes; 65 is greater than the upper boundary of 58
Q6A cumulative frequency graph for 100 test scores gives Q1 = 40, median = 58, Q3 = 70, minimum = 5, maximum = 98. Determine which of the minimum and maximum should be plotted as individual outliers.Show answer
Answer: Neither; IQR = 30, boundaries are -5 and 115, and both 5 and 98 lie within this range, so the whiskers extend directly to 5 and 98
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A data set has Q1 = 24 and Q3 = 40. Calculate the IQR and both outlier boundaries.
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The masses, in kg, of 50 parcels have quartiles Q1 = 3.2 kg and Q3 = 7.6 kg. The heaviest parcel has a mass of 15 kg. (a) Calculate the IQR and the upper outlier boundary. (b) Determine whether the heaviest parcel is an outlier.
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A cumulative frequency table for the times, in minutes, of 120 runners gives Q1 = 45, median = 55, Q3 = 65. The slowest runner finished in 140 minutes. (a) Calculate the IQR. (b) Calculate the upper outlier boundary. (c) Determine whether the slowest runner should be shown as an outlier, and explain how their result should be plotted.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the box plots from cumulative frequency and outliers worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 15 of GCSE Statistics Higher Workbook 1, the whole course as one free printable PDF.
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