How do you use cumulative frequency curves and box plots to find and compare the median, quartiles and spread?
Construct and read cumulative frequency curves, find the median and quartiles, calculate the interquartile range, and draw and compare box plots (Higher tier).
A CCEA GCSE Mathematics Higher answer on cumulative frequency and box plots, covering constructing and reading cumulative frequency curves, finding the median and quartiles, calculating the interquartile range, and drawing and comparing box plots.
Reviewed by: AI editorial process; not yet individually human-reviewed
Have a quick question? Jump to the Q&A page
Jump to a section
What this dot point is asking
Cumulative frequency curves and box plots are Higher-tier CCEA tools for finding and comparing the median, quartiles and spread of larger data sets. You must construct and read cumulative frequency curves, find the median and the lower and upper quartiles, calculate the interquartile range, and draw and compare box plots. The comparison of two box plots is a classic reasoning question, and the interquartile range is the key measure of spread that resists outliers.
Cumulative frequency
Cumulative frequency is the running total of frequencies: at each class, add the frequency of that class to the total so far. The final cumulative frequency equals the total number of data values, .
Reading the median and quartiles
On a cumulative frequency curve, the positions of the median and quartiles are fixed fractions of the total.
The curve can also answer "how many were less than (or more than) a given value", by reading up from the value to the curve and across to the cumulative axis.
The interquartile range
The interquartile range (IQR) is the upper quartile minus the lower quartile, . It measures the spread of the middle 50 percent of the data and, unlike the full range, is not distorted by a single extreme value. A smaller IQR means more consistent data.
Box plots
A box plot (box-and-whisker diagram) is a compact summary of five values: the minimum, lower quartile, median, upper quartile and maximum. The box spans the quartiles with the median marked inside, and the whiskers reach to the minimum and maximum.
Comparing data with box plots
Two box plots on the same scale are compared by their medians (which set is higher on average) and by their IQRs (which set is more consistent). As always, state the comparison in context, for example "Group B had a higher median, so scored more on average, but a larger IQR, so was less consistent."
Why this matters
Cumulative frequency and box plots handle the larger, grouped data sets that the simple averages cannot summarise neatly, and they make the spread visible at a glance. They are a dependable Higher-tier topic that builds directly on averages and spread, and the comparison questions reward the AO2 and AO3 interpretation CCEA values. The IQR's resistance to outliers is the reason it is preferred for skewed data.
Exam-style practice questions
Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
CCEA 20203 marksA cumulative frequency curve is drawn for 80 students. State the cumulative frequency values you would read across from to find the median and the lower quartile. (Higher, calculator.)Show worked answer →
Cumulative frequency positions are based on the total, .
The median is at , so read across from a cumulative frequency of 40.
The lower quartile is at , so read across from 20.
Marks are for the median position 40 and the lower-quartile position 20. Using as for a small ordered list is not how a cumulative frequency curve is read.
CCEA 20223 marksA data set has lower quartile , median and upper quartile . Find the interquartile range and explain what it measures. (Higher, non-calculator.)Show worked answer →
The interquartile range is the upper quartile minus the lower quartile.
.
It measures the spread of the middle 50 percent of the data, which is why it is not affected by extreme values at either end.
Marks are for the subtraction giving 16 and for the explanation about the middle half of the data. Confusing the IQR with the full range is the usual error.
Related dot points
- Find the mean, median, mode and range of a data set, estimate the mean from grouped data, find the modal class, and use averages and range to compare two distributions.
A CCEA GCSE Mathematics answer on averages and spread, covering the mean, median, mode and range, estimating the mean from grouped data, finding the modal class, and comparing two distributions using an average and a measure of spread.
- Understand the data-handling cycle, distinguish types of data and sampling, use frequency and two-way tables, and draw and interpret bar charts, pie charts, pictograms, frequency polygons and histograms.
A CCEA GCSE Mathematics answer on collecting and representing data, covering the data-handling cycle, types of data and sampling, frequency and two-way tables, and bar charts pie charts pictograms frequency polygons and histograms.
- Plot scatter graphs, describe the type and strength of correlation, draw a line of best fit, use it to estimate values, and understand interpolation, extrapolation and the difference between correlation and causation.
A CCEA GCSE Mathematics answer on scatter graphs and correlation, covering plotting paired data, describing the type and strength of correlation, drawing and using a line of best fit, interpolation and extrapolation, and the difference between correlation and causation.
- Use the probability scale from 0 to 1, find probabilities of single events, use that probabilities sum to 1, apply the mutually exclusive addition rule, and list outcomes with sample space diagrams.
A CCEA GCSE Mathematics answer on probability basics, covering the probability scale, single-event probability, the fact that probabilities sum to 1, mutually exclusive events and the addition rule, and listing outcomes with sample space diagrams.
- Use relative frequency to estimate probability from experimental data, understand how estimates improve with more trials, identify bias, and calculate expected frequencies.
A CCEA GCSE Mathematics answer on relative frequency, covering estimating probability from experimental data, how estimates improve with more trials, recognising a biased experiment, and calculating expected frequencies.
Sources & how we know this
- CCEA GCSE Mathematics specification (2210) — CCEA (2017)