Cumulative frequency and box plots

Objective. Move between cumulative curves and comparative summaries.

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Learn: the key idea

Cumulative frequency and box plots summarise a distribution's spread by plotting running totals and then displaying key quartiles visually. A cumulative frequency curve is read at specific running totals to estimate the median (at the middle total) and quartiles (at a quarter and three-quarters of the total), giving the interquartile range as upper minus lower quartile. These five values (minimum, lower quartile, median, upper quartile, maximum) are then drawn as a box plot for easy visual comparison between two distributions. For a dataset of 40 values, the median is read at cumulative frequency 20. Comparing box plots focuses on both median position and interquartile range spread.

Worked example

A cumulative frequency curve for 60 students shows the lower quartile at 14 marks and the upper quartile at 38 marks. Find the interquartile range.

1 $Interquartile range = upper quartile - lower quartile$
2 $Upper quartile = 38$
3 $Lower quartile = 14$
4 $Interquartile range = 38 - 14 = 24$

Answer: 24 marks

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Common mistake. Students often read the median at half the frequency axis value rather than half the total frequency, giving a shifted estimate whenever frequencies are unevenly distributed. Always use half the total cumulative frequency.

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