GCSE Maths Cumulative Frequency and Box Plots
Cumulative frequency and box plots summarise how data are distributed, rather than merely showing the most common value. Cumulative frequency records how many observations are at or below each class boundary, and a box plot represents the median, quartiles and extremes. Learning how to read these diagrams helps students compare spread as well as typical values.
Build a cumulative frequency table
Suppose grouped scores have frequencies 3 for 0–10, 5 for 10–20, 7 for 20–30, and 5 for 30–40, using non-overlapping class intervals. The cumulative frequencies at the upper class boundaries are 3, then 3 + 5 = 8, then 8 + 7 = 15, and finally 20. The final cumulative frequency must equal the total number of observations. If it does not, review the arithmetic before attempting a graph.
Plot the cumulative frequency curve
Put the measured score on the horizontal axis and cumulative frequency on the vertical axis. Plot the points for the upper class boundaries, such as (10,3), (20,8), (30,15) and (40,20). Depending on how the first class begins and the exam convention, include an appropriate starting point at cumulative frequency zero. Join the plotted points smoothly or as directed by the question. This graph is designed for reading approximate positions and totals, not individual category frequencies.
Estimate the median and quartiles
With 20 observations, the median is associated with cumulative frequency 10, the lower quartile with cumulative frequency 5, and the upper quartile with cumulative frequency 15 under the usual grouped-data graphical convention. Find those levels on the vertical axis, move across to the curve and down to read estimated score values. Because grouped data hide the exact observations, the resulting quartiles are estimates. Show your guide lines clearly on an exam diagram.
What a box plot shows
A box plot uses the minimum, lower quartile, median, upper quartile and maximum. The box extends from the lower to upper quartile, a line inside marks the median, and whiskers extend towards the extremes. The interquartile range is upper quartile minus lower quartile. It measures the spread of the middle half of the data, which can be more informative than the full range when an extreme value is present.
Compare two distributions properly
When comparing box plots, comment separately on centre and spread. One distribution might have a higher median but a smaller interquartile range. That supports a statement such as “the typical result is higher and the middle half is more consistent”, but it does not tell you that every value is higher. Read the same scale for both plots, distinguish the range from the interquartile range, and avoid unsupported claims about averages or outliers.
Common exam mistakes and checks
Do not add frequencies twice when constructing cumulative totals. Plot against the correct class boundary, and check that cumulative frequency never decreases. On a box plot, remember that the middle line is the median rather than the mean. When comparing groups, give numerical evidence where possible: a difference in medians and a difference in interquartile ranges. Always mention that quartiles from grouped cumulative-frequency curves are estimates.
Frequently asked questions
Can a cumulative frequency graph decrease?
No. The cumulative count remains the same or increases as the upper threshold rises.
Does a box plot show the mean?
No. Its five-number summary includes the median and quartiles, not the mean.
Related maths learning resources
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