Percentiles
A percentile is a value below which a given percentage of the data falls; for example, the 30th percentile is the value below which 30% of the data lies. Percentiles are used to see how one particular value compares with the rest of a data set, and are often read from a cumulative frequency graph. In GCSE Statistics, common percentiles include the median (50th percentile) and the quartiles (25th and 75th percentiles).
Before you start
Make sure you're comfortable with these topics first:
Method
- Order the data set from smallest to largest, or use a cumulative frequency table or graph if the data is already grouped.
- Find the total number of values, n, in the data set.
- To find the k-th percentile, calculate the position (k divided by 100) x n, reading this position along the cumulative frequency axis.
- If using a cumulative frequency graph, draw a horizontal line from that position on the vertical axis to the curve, then a vertical line down to the horizontal axis to read off the value.
- If working from a list, round the position to the nearest whole number and read off the corresponding data value.
- State the percentile clearly in context, for example 'the 90th percentile is 62 marks, so 90% of students scored 62 marks or fewer.'
Worked example
40 runners complete a race. A cumulative frequency graph of their finishing times shows that a cumulative frequency of 34 corresponds to a time of 52 minutes. Find the percentile represented by this point, and interpret it.
- The total number of runners is n = 40.
- The cumulative frequency at this point is 34.
- Calculate the percentage: (34 divided by 40) x 100 = 85.
- This means the point corresponds to the 85th percentile.
- Final answer: the 85th percentile is 52 minutes, so 85% of runners finished in 52 minutes or less (equivalently, 15% took longer than 52 minutes).
Practice questions
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Q1What percentage of data lies below the 40th percentile?Show answer
Answer: 40%.
Q2Which percentile is the same as the median?Show answer
Answer: The 50th percentile.
Q3In a data set of 200 values, at what position, counting from the smallest, would you expect to find the 25th percentile?Show answer
Answer: Around the 50th value (25% of 200 = 50).
Q4A cumulative frequency graph shows that a value of 18 corresponds to a cumulative frequency of 45 out of 60 total data points. Find the percentile.Show answer
Answer: 75th percentile (45/60 x 100 = 75).
Q5In a class of 30 students, a student's mark is at the 90th percentile. Roughly how many students scored less than or equal to that mark?Show answer
Answer: About 27 students (90% of 30 = 27).
Q6A data set of 50 exam scores has the 20th percentile at 42 marks and the 80th percentile at 78 marks. Explain what these two values tell you about the middle 60% of scores.Show answer
Answer: The middle 60% of scores lie between 42 and 78 marks (from the 20th to the 80th percentile).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A data set contains 80 values. Find the cumulative frequency position that corresponds to the 35th percentile.
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A cumulative frequency graph for the delivery times, in minutes, of 120 parcels shows that a cumulative frequency of 90 corresponds to a delivery time of 35 minutes. (a) Find the percentile that this represents. (b) Interpret your answer in context.
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A cumulative frequency graph for the heights of 200 sunflowers shows that the 10th percentile is 90 cm and the 90th percentile is 160 cm. (a) State how many sunflowers are shorter than 90 cm. (b) State how many sunflowers are taller than 160 cm. (c) Find the range of heights that contains the middle 80% of sunflowers.
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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 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 2 of GCSE Statistics Foundation Workbook 2, the whole course as one free printable PDF.
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