What you'll learn
This topic covers cumulative frequency graphs and box plots — two ways of summarising and comparing grouped data. In this guide you will learn how to build a cumulative frequency table and graph, how to read off the median and quartiles, how to draw a box plot, and how to compare distributions. These are key higher-tier statistics skills.
Key terms and definitions
Cumulative frequency — a running total of frequencies up to each class boundary.
Median — the middle value (the 50th percentile).
Quartiles — values dividing the data into four equal parts (Q1, Q2, Q3).
Interquartile range (IQR) — Q3 − Q1, a measure of spread.
Box plot — a diagram showing minimum, quartiles and maximum.
Core concepts
Building cumulative frequency
Cumulative frequency is a running total of the frequencies. Add each class frequency to the total so far, and plot the cumulative frequency against the upper class boundary of each group. The points are joined with a smooth curve.
Reading the median and quartiles
On the curve, the median is read at half the total frequency, the lower quartile (Q1) at a quarter, and the upper quartile (Q3) at three-quarters. Read across to the curve and down to the value. The IQR = Q3 − Q1 measures the spread of the middle half.
Drawing box plots
A box plot shows five numbers: the minimum, lower quartile, median, upper quartile and maximum. The box spans Q1 to Q3 with a line at the median, and "whiskers" reach out to the minimum and maximum. It gives an instant picture of centre and spread.
Comparing distributions
To compare two data sets, compare a measure of average (usually the median) and a measure of spread (usually the IQR or range). A smaller IQR means more consistent data. Always write comparisons in context.
Estimating with the curve
You can use the cumulative frequency curve to estimate how many values are above or below a given figure (e.g. how many scored more than 60), by reading the cumulative frequency at that value and subtracting from the total.
Worked examples
Example 1: Median position
60 pieces of data are plotted. At what cumulative frequency do you read the median?
Half of 60 = 30.
Example 2: IQR
Q1 = 18 and Q3 = 31. Find the interquartile range.
31 − 18 = 13.
Example 3: Reading above a value
From 80 values, 55 are at most 40. How many are more than 40?
80 − 55 = 25.
Common mistakes and how to avoid them
Plotting at the wrong point. Use the upper class boundary, not the midpoint.
Forgetting to make it cumulative. Use a running total, not the raw frequency.
Reading the median at the wrong height. It is at half the total.
Confusing range and IQR. IQR uses the quartiles, not the extremes.
Comparing without context. Refer to what the data represents.
Exam technique for Cumulative Frequency and Box Plots
Make a running total and plot against upper boundaries.
Read median and quartiles at ½, ¼ and ¾ of the total.
Find the IQR as Q3 − Q1.
Draw box plots with the five summary values.
Compare an average and a spread, in context.
Quick revision summary
Cumulative frequency is a running total of frequencies, plotted against the upper class boundary and joined with a smooth curve. From the curve, read the median at half the total, the lower quartile at a quarter, and the upper quartile at three-quarters, giving the interquartile range IQR = Q3 − Q1 as a measure of spread. A box plot displays five numbers — minimum, Q1, median, Q3, maximum — with a box from Q1 to Q3 and whiskers to the extremes. To compare distributions, compare an average (median) and a spread (IQR), always in context, with a smaller IQR meaning more consistency. You can also estimate how many values lie above or below a figure by reading the curve and subtracting. The common errors are plotting at the midpoint, not making the total cumulative, reading the median at the wrong height, and confusing range with IQR. Build a running total, read at ½/¼/¾, find the IQR, and compare in context.