Percentiles and Interpercentile Range
A percentile is the value below which a given percentage of a data set falls when the values are arranged in order, so the 30th percentile has 30% of the data below it. The interpercentile range, often the 10th to 90th, is the difference between two chosen percentiles and measures spread while ignoring extreme values. It is estimated from a cumulative frequency graph on GCSE Statistics Higher tier.
Before you start
Make sure you're comfortable with these topics first:
Method
- Draw or read a cumulative frequency diagram, with cumulative frequency on the vertical axis and the variable on the horizontal axis.
- To find the pth percentile, calculate its position as (p/100) x n, where n is the total frequency.
- Find that position on the cumulative frequency axis (or table), then interpolate within the class it falls in to estimate the value.
- To find an interpercentile range, find both percentile values in this way, then subtract the lower one from the upper one.
- Give the answer to a sensible precision for the scale of the data, with units, and check it is smaller than the full range of the data.
Worked example
The table shows the finishing times, in minutes, of 100 runners in a fun run: 0 < t <= 10, frequency 5; 10 < t <= 20, frequency 20; 20 < t <= 30, frequency 25; 30 < t <= 40, frequency 30; 40 < t <= 50, frequency 20. Estimate the 10th percentile, the 90th percentile, and the 10-90 interpercentile range.
- n = 100, so the 10th percentile is at position 0.10 x 100 = 10th value, and the 90th percentile is at position 0.90 x 100 = 90th value.
- The cumulative frequencies are 5, 25, 50, 80, 100 at the class boundaries 10, 20, 30, 40, 50.
- The 10th value falls in the class 10 < t <= 20 (cumulative frequency goes from 5 to 25 there): 10th percentile = 10 + (10 - 5)/20 x 10 = 12.5 minutes.
- The 90th value falls in the class 40 < t <= 50 (cumulative frequency goes from 80 to 100 there): 90th percentile = 40 + (90 - 80)/20 x 10 = 45 minutes.
- Interpercentile range (10th to 90th) = 45 - 12.5 = 32.5 minutes.
- Final answer: 10th percentile = 12.5 minutes, 90th percentile = 45 minutes, 10-90 interpercentile range = 32.5 minutes.
Practice questions
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Q1In a data set of 40 values arranged in order, which position corresponds to the 25th percentile?Show answer
Answer: 10th value (25/100 x 40 = 10)
Q2A cumulative frequency graph shows 60 values. Which position corresponds to the 80th percentile?Show answer
Answer: 48th value (0.80 x 60 = 48)
Q3A data set has an interquartile range of 18. Would the 10-90 interpercentile range usually be smaller or larger than this? Explain.Show answer
Answer: Larger - the 10-90 interpercentile range covers 80% of the data (10th to 90th percentile) while the IQR covers only the middle 50% (25th to 75th), so it usually spans a wider gap
Q4Heights of 50 plants (cm): 0 < h <= 10, freq 5, cumulative 5; 10 < h <= 20, freq 20, cumulative 25; 20 < h <= 30, freq 15, cumulative 40; 30 < h <= 40, freq 10, cumulative 50. Estimate the 20th percentile.Show answer
Answer: 12.5 cm (position 0.2 x 50 = 10th value; 10 + (10-5)/20 x 10 = 12.5)
Q5Weekly household spending for 40 households has cumulative frequencies: up to 50 pounds, 10; up to 100 pounds, 25; up to 150 pounds, 35; up to 200 pounds, 40. Estimate the 75th percentile.Show answer
Answer: 125 pounds (position 0.75 x 40 = 30th value; 100 + (30-25)/10 x 50 = 125)
Q6Class A has a 10th percentile of 42 marks and a 90th percentile of 88 marks. Class B has a 10th percentile of 55 marks and a 90th percentile of 79 marks. Compare the spread of the middle 80% of marks in each class.Show answer
Answer: Class A's interpercentile range is 88 - 42 = 46 marks; Class B's is 79 - 55 = 24 marks, so Class B's middle 80% of marks are much less spread out (more consistent) than Class A's
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
Delivery times, in minutes, for 60 parcels have cumulative frequencies: up to 10, 5; up to 20, 25; up to 30, 45; up to 40, 55; up to 50, 60. Estimate the 30th percentile.
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The ages, in years, of 120 gym members have cumulative frequencies: 16 <= a < 20, 15; 20 <= a < 30, 60; 30 <= a < 40, 100; 40 <= a < 60, 120. Estimate the interpercentile range between the 20th and 80th percentiles.
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The 10th percentile of the prices of second-hand cars at a dealership is 3200 pounds and the 90th percentile is 11800 pounds. Calculate the 10-90 interpercentile range and briefly interpret what it shows about the spread of prices.
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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 percentiles and interpercentile range worksheet pack - 18 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 4 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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