Cumulative frequency is a running total of the frequencies. Plot each cumulative frequency against the upper boundary of its class and join the points with a smooth curve. Read across from half the total to estimate the median, and from a quarter and three quarters to estimate the lower and upper quartiles. The interquartile range, IQR = upper quartile − lower quartile, measures the spread of the middle half of the data.
Cumulative frequency
- Cumulative frequency: add each frequency to the total so far
- Plot at the upper class boundary
- Median: read across at n ÷ 2
- Lower quartile at n ÷ 4; upper quartile at 3n ÷ 4
- IQR = UQ − LQ
Building a cumulative frequency table
Add up the frequencies as you go: each cumulative frequency is the total number of values up to the end of that class. For frequencies 6, 11, 9 and 4, the cumulative frequencies are 6, 17, 26 and 30. The last cumulative frequency always equals the total number of values.
Plotting the graph
Plot each cumulative frequency against the upper boundary of its class, because that is where all of those values have been counted. Join the points with a smooth curve, or with straight lines if the question says so. The graph should start at zero at the lower boundary of the first class.
Median, quartiles and the IQR
For n values, read across from n ÷ 2 to the curve and down to estimate the median. Read from n ÷ 4 for the lower quartile and 3n ÷ 4 for the upper quartile. For 100 values, read at 50, 25 and 75. The interquartile range is the upper quartile minus the lower quartile; it ignores extreme values, so it is a reliable measure of spread.
Reading other values
To estimate how many values are below a certain amount, read up from that value to the curve and across. To find how many are above it, subtract that reading from the total. Show your reading lines on the graph.
Box plots
A box plot shows five values: the minimum, lower quartile, median, upper quartile and maximum. The box runs from the lower to the upper quartile, with a line at the median. Box plots make it easy to compare the average and spread of two groups.
Worked examples
Example 1. Frequencies for four classes are 6, 11, 9 and 4. Write down the cumulative frequencies.
- Running totals: 6, 6 + 11, 17 + 9, 26 + 4.
Answer: 6, 17, 26, 30.
Example 2. A cumulative frequency graph shows 100 values. At which cumulative frequencies do you read the median and quartiles?
- Median at 100 ÷ 2.
- LQ at 100 ÷ 4 and UQ at 3 × 100 ÷ 4.
Answer: Median at 50, LQ at 25, UQ at 75.
Example 3. From a graph, the lower quartile is 21 minutes and the upper quartile is 34 minutes. Find the interquartile range.
- IQR = UQ − LQ = 34 − 21.
Answer: 13 minutes.
Example 4. A graph of 100 values shows that 62 values are 40 kg or less. How many values are more than 40 kg?
- Subtract from the total: 100 − 62.
Answer: 38 values.
Example 5. For a box plot, the data has minimum 12, LQ 21, median 27, UQ 34 and maximum 45. Find the range and the IQR.
- Range: 45 − 12.
- IQR: 34 − 21.
Answer: Range 33; IQR 13.
Common mistakes
- Plotting at the midpoint. Plot cumulative frequency at the upper class boundary.
- Reading the median at (n + 1) ÷ 2. For a cumulative frequency graph, use n ÷ 2.
- Giving the cumulative frequency as the answer. Read across to the curve, then down to the horizontal axis.
- Forgetting to subtract for “more than”. Read the number below the value, then subtract from the total.
- Mixing up the quartiles. The lower quartile is at n ÷ 4 and the upper at 3n ÷ 4.
How cumulative frequency is examined
Cumulative frequency graphs and box plots belong to the Higher tier of both Edexcel specifications, and the questions are usually worth 4 to 6 marks. Questions ask you to:
- complete a cumulative frequency table and plot the graph
- estimate the median, quartiles and IQR
- estimate how many values are above or below a given amount
- draw or compare box plots
The cumulative frequency worksheet has full answers, and the data and statistics engine gives more practice.
Use the median and IQR in comparing distributions. For unequal class widths, see histograms.
Frequently asked questions
What is cumulative frequency?
A running total of the frequencies, showing how many values are up to each class boundary.
How do you find the median from a cumulative frequency graph?
Read across from half the total frequency to the curve, then down to the axis.
What is the interquartile range?
The upper quartile minus the lower quartile; the spread of the middle half of the data.