A percentile splits the data so that a given percentage falls below it. To find the pth percentile from a cumulative frequency curve, work out p/100 × n where n is the total frequency, find that height on the vertical axis, go across to the curve and then straight down. Quartiles are just familiar percentiles: the lower quartile is the 25th, the median is the 50th and the upper quartile is the 75th.
If you can already find the median and the quartiles from a cumulative frequency curve, you can find any percentile, because it is the same procedure with a different fraction. The median is the 50th percentile. The lower quartile is the 25th. The upper quartile is the 75th. A question asking for the 30th percentile is asking you to do exactly what you already do, using 0.3 instead of 0.5.
The one thing to be careful about is direction. The pth percentile is the value below which p percent of the data falls. If a question asks about the top 10 percent, that is the 90th percentile, not the 10th.
Total frequency n
where the curve levels off
Position
p/100 × n on the vertical axis
Then
across to the curve, then straight down
Lower quartile
25th percentile, 0.25n
Median
50th percentile, 0.5n
Upper quartile
75th percentile, 0.75n
Note that for a cumulative frequency curve you use p/100 × n directly, without the +1 adjustment used for small ordered lists. The curve is treated as continuous data, so no adjustment is needed.
A cumulative frequency curve shows the times taken by 200 students to complete a task. Find the 30th percentile.
Solution
n = 200, read from where the curve flattens at the top.30/100 × 200 = 60 on the cumulative frequency axis.60 on the vertical axis and draw a horizontal line across to the curve.18 minutes. That is the 30th percentile.Draw the construction lines and leave them on the page. Examiners award marks for the method shown on the graph, not only for the number in the answer box. A correct answer with no lines drawn can score less than a slightly-off answer with clear working.
Using the same curve for 200 students, find the time exceeded by the slowest 10% of students.
Solution
90/100 × 200 = 180 on the cumulative frequency axis.27 minutes.The step students get wrong
Using the 10th percentile when the question says "the top 10 percent". Cumulative frequency always counts upwards from the bottom, so anything about the top of the distribution has to be converted first by subtracting from 100. Write down which percentile you actually need before you touch the graph.
Because the quartiles are percentiles, the interquartile range can also be described as the gap between the 25th and the 75th percentile. That is the middle half of the data, which is what makes it resistant to extreme values, unlike the range.
A percentile is a position, not a score. If a student is at the 80th percentile in a test, it does not mean they scored 80 marks. It means 80 percent of the cohort scored below them. Questions that ask you to "comment" are usually testing exactly this distinction, so phrase your answer in terms of the proportion of the group rather than the value itself.
What is the difference between a percentile and a percentage?
A percentage is a proportion of a total. A percentile is a position in an ordered data set. Scoring 80 percent means you got 80 marks out of 100; being at the 80th percentile means 80 percent of people scored below you. They are unrelated numbers.
Is the median the 50th percentile?
Yes. The median splits the data in half, so 50 percent falls below it. Everything you already do to find the median from a curve is the percentile method with p = 50.
Do I use n or n + 1 to find the position?
For a cumulative frequency curve, use n. The n + 1 adjustment belongs to small ordered lists of discrete values. The curve treats the data as continuous, so no adjustment is needed.
How do I handle a question about the top percentage?
Subtract from 100 first. The top 20 percent begins at the 80th percentile, and the top 5 percent begins at the 95th. Cumulative frequency counts from the bottom upwards, always.
How does this connect to the rest of E-Maths?
Percentiles come from the same curve you use for quartiles and box plots in cumulative frequency curves and box-and-whisker plots, and they are the tool behind any question comparing where one individual sits within two different data sets.
— Mr Gan Math Tuition
Mr. Gan works through reading the graph accurately so the marks come from the diagram, not from guesswork.
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