Cumulative frequency graphs

A cumulative frequency graph shows a running total of the frequencies in grouped data. You plot each cumulative frequency against the upper end of its class and join the points, which gives an S-shaped curve. From the graph you can estimate the median, the quartiles, the interquartile range and how many values lie above or below any point. It is also called a cumulative frequency diagram, curve or ogive.

What is cumulative frequency?

Cumulative means "adding up as you go". The cumulative frequency of a class is its own frequency plus all the frequencies before it. Here are the times 80 students took to finish a puzzle:

The last cumulative frequency always equals the total, here 80. If it doesn't, an addition has gone wrong. The cumulative frequency 45 means 45 students took 30 minutes or less.

How to draw a cumulative frequency graph

1. Add a cumulative frequency column to the table.
2. Draw the axes: the data (here time) along the bottom, cumulative frequency up the side, from 0 to the total.
3. Plot each cumulative frequency at the upper class boundary, the end of its class: (10, 6), (20, 20), (30, 45) and so on.
4. Plot a starting point at the lower end of the first class with cumulative frequency 0, here (0, 0).
5. Join the points with a smooth curve. Some textbooks join them with straight lines instead, which is called a cumulative frequency polygon.

Why the upper boundary? Cumulative frequency 20 means 20 students took up to 20 minutes. You only know all of them have finished by the end of the class, so that is where the point goes.

Here x is the time in minutes and y is the cumulative frequency. This version joins the points with straight lines so the readings below can be checked exactly. On a smooth curve your readings will be very close, and answers read from any graph are estimates.

Finding the median from a cumulative frequency graph

The median is the middle value. With n values, find n ÷ 2 on the cumulative frequency axis, go across to the graph, then down to the data axis. Here n = 80, so go across from 40.

40 lies between the points (20, 20) and (30, 45). Going from 20 to 45 takes 10 minutes, so each student is 10 ÷ 25 = 0.4 minutes, and 40 is 20 students past the point at 20 minutes: 20 + 20 × 0.4 = 28 minutes. On paper, you would just read it off your graph.

Quartiles and the interquartile range

• Lower quartile (LQ): go across from n ÷ 4 = 20. That meets the graph at 20 minutes.
• Upper quartile (UQ): go across from 3n ÷ 4 = 60. That meets the graph at 37.5 minutes.
• Interquartile range (IQR): UQ − LQ = 37.5 − 20 = 17.5 minutes.

The IQR measures how spread out the central 50% of values are. Unlike the range, it isn't thrown off by one very fast or very slow student. In England, AQA's GCSE Maths specification includes cumulative frequency graphs, and quartiles and the interquartile range, as ways to compare distributions.

Reading other values from the graph

How many are above a value?

How many students took more than 45 minutes? Go up from 45 to the graph and across: the cumulative frequency is about 70.5, so about 70 or 71 students took 45 minutes or less. The rest took longer: 80 − 70.5 ≈ 10 students. Remember to subtract from the total: the graph gives "less than" directly.

What percentage are below a value?

What percentage finished within 15 minutes? Up from 15 gives a cumulative frequency of 13, and 13 ÷ 80 × 100 = 16.25%.

Percentiles

The 90th percentile is the value 90% of the way through the data. 90% of 80 is 72, so go across from 72: it meets the graph at about 46 minutes. So 90% of students finished in about 46 minutes or less.

From a cumulative frequency graph to a box plot

A box plot needs five numbers: the lowest value, LQ, median, UQ and highest value. The graph gives the middle three. For grouped data you don't know the exact lowest and highest values, so use the ends of the first and last classes, here 0 and 60, unless the question tells you otherwise. The box plot calculator draws one from these values.

Comparing two distributions

Exam questions often give two cumulative frequency graphs, such as two classes doing the same puzzle, and ask you to compare them. Make two comparisons, each with numbers and in context:

• Average: "Class A's median of 28 minutes is lower than Class B's 33 minutes, so Class A were faster on average."
• Spread: "Class A's IQR of 17.5 minutes is larger than Class B's 12 minutes, so Class A's times were less consistent."

The shape of the graph

A cumulative frequency graph always goes up or stays flat, never down, because you keep adding. It is steepest where the data is most bunched together: here between 20 and 30 minutes, the class with the highest frequency. Where it is flat, very few values fall in that range.

Common mistakes

• Plotting at the midpoints. That is for frequency polygons. Cumulative frequency goes at the upper class boundary.
• Plotting frequency instead of cumulative frequency. The graph should end at the total.
• Forgetting the starting point at the lower end of the first class, with cumulative frequency 0.
• Reading the median from the time axis. Start from n ÷ 2 on the cumulative frequency axis.
• Giving "less than" when the question says "more than". Subtract from the total.

To practise statistics questions in a short timed paper, try today's GCSE Maths Higher paper. Statistics won't be in every paper, as each one mixes topics.

Read next

Maths · StatisticsFrequency polygons: how to draw and compare themHow to draw a frequency polygon from grouped data, compare two on one set of axes, and tell them apart from histograms and other frequency diagrams.