Cumulative Frequency Graphs

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A cumulative frequency graph (also called an ogive) is used when data is grouped into class intervals. You plot running totals of frequency against upper class boundaries, join the points with a smooth curve, then read off estimates for the median, quartiles and interquartile range.

This GCSE guide covers the full method: what cumulative frequency means, how to draw the graph step by step, how to interpret it, teacher’s guide, common mistakes to avoid, and five exam-style questions with worked solutions. Download the free PDF worksheet (same five questions, ready to print) or follow the video tutorial to teach or revise the topic.

What is cumulative frequency?

Cumulative frequency is the running total of the frequencies in a grouped table. Each row tells you how many values fall in that class interval; the cumulative frequency column tells you how many values are less than or equal to the upper boundary of that interval.

For example, if 5 apples have diameter 100–120 mm and 25 have diameter 120–140 mm, then 30 apples have diameter less than or equal to 140 mm. That running total is what you plot on a cumulative frequency graph.

Grouped data is useful for large data sets, but you no longer know every individual value. A cumulative frequency graph brings back what you need for GCSE questions: you can estimate the median, quartiles, interquartile range, percentages and comparisons between distributions.

Frequency vs cumulative frequency

• Frequency: how many values lie in one class interval (e.g. 25 apples between 120 mm and 140 mm).
• Cumulative frequency: the total number of values up to the end of that interval (e.g. 30 apples with diameter ≤ 140 mm).

On the graph, frequency would be shown with a bar chart or histogram. Cumulative frequency is shown with a smooth S-shaped curve (an ogive) because the totals only ever increase.

Teacher’s Guide

Students can learn the method for drawing cumulative frequency graphs without ever understanding why they exist. Instead, we should begin with why we draw them. Cumulative frequency graphs answer questions such as: What percentage of the data lies above or below a certain value? or Where does someone rank? Are they in the top 10% or the bottom 25%?

Once students see that cumulative frequency is used to compare distributions, find percentiles, and make decisions based on thresholds, the graph has a genuine purpose. It becomes something worth drawing, rather than simply another graph they have to produce.

I like to give students a handout of the example I am modelling at the front of the class. As they work alongside me, they can focus on the mathematics rather than rushing to copy from the board. They can clearly see the graph, know exactly where to plot each point, and leave the explanation with a correct, annotated example to refer to when working independently later in the lesson.

Video: plot and interpret a cumulative frequency graph

Watch the tutorial below to see the full GCSE method modelled from start to finish: completing the cumulative frequency column, plotting upper class boundaries, joining the points with a smooth curve, and reading estimates back from the graph. The video works through three examples a cycling times question, an age distribution, and a ratio problem about pass marks.

Want to practise alongside the video? Members can download the PDF for their students.

What the video covers

• Example 1: Cycling finishing times (60 students): complete the cumulative frequency table, plot the ogive using upper time boundaries, then estimate the median finishing time (about 1.9 hours, or 1 hour 54 minutes).
• Example 2: Ages of 100 people: draw the cumulative frequency graph from a given table, estimate the median age (about 50 years), then estimate how many people are older than 55 (about 35).
• Example 3: Minimum pass mark (120 students): read from a cumulative frequency graph when the ratio of students who failed to students who passed is 3 : 5 giving 45 failures and a minimum pass mark of about 35.

The written step-by-step guide below follows the same order as the video. Use the video for live modelling in class, or the text sections for revision and independent study.

How to draw a cumulative frequency graph (step by step)

Follow these five steps whenever you are asked to draw a cumulative frequency graph from a grouped frequency table at GCSE.

Step 1: Complete the cumulative frequency column

Add a cumulative frequency column to the table. The first cumulative frequency equals the first frequency. Each following entry adds the next frequency onto the running total.

For Marcus’s apples (80 apples in total):

Step 2: Plot upper class boundaries on the horizontal axis

Plot cumulative frequency against the upper class boundary of each interval, not the class midpoint. For the first interval, also plot a starting point at the lower boundary with cumulative frequency 0.

For Marcus’s apples, plot: (100, 0), (120, 5), (140, 30), (160, 60), (180, 75), (200, 80).

Step 3: Draw and label the axes

Label the horizontal axis with the variable and units (e.g. diameter, mm). Label the vertical axis cumulative frequency. Choose a sensible scale so the full curve fits on the grid and points can be read accurately.

Step 4: Plot the points accurately

Mark each point with a small cross or dot. Check that the final cumulative frequency equals the total number of values in the data set (80 apples here).

Step 5: Join with a smooth curve

Connect the points with a smooth curve, not straight line segments between each point. The finished graph should look S-shaped. Extend the curve to the axes if the question grid requires it.

How to read values from a cumulative frequency graph

Once the cumulative frequency graph is drawn, GCSE questions usually ask you to estimate the median, quartiles or interquartile range. All answers are estimates because the data is grouped.

Finding the median

For n values, the median is at cumulative frequency n ÷ 2. Draw a horizontal line from that value on the vertical axis to the curve, then down to the horizontal axis to read the median.

With 80 apples, draw a line from cumulative frequency 40, then read the median diameter from the horizontal axis.

Finding the quartiles and interquartile range

• Lower quartile (LQ) at cumulative frequency n ÷ 4
• Upper quartile (UQ) at cumulative frequency 3n ÷ 4
• Interquartile range (IQR) = UQ − LQ

For 80 apples: LQ is at CF 20 (≈ 134 mm) and UQ is at CF 60 (≈ 160 mm), so IQR ≈ 160 − 134 = 26 mm.

Reverse questions: percentages and frequencies

Some questions give a value on the horizontal axis and ask for a percentage or number of values. Draw a vertical line up to the curve, then across to the cumulative frequency axis.

To estimate the percentage of apples with diameter less than 150 mm: read a cumulative frequency of about 46.5 at 150 mm, then (46.5 ÷ 80) × 100 ≈ 58%.

To find how many shop assistants earn more than £530: read the cumulative frequency at £530 (about 72.5), then subtract from the total: 80 − 72.5 ≈ 8 assistants.

Cumulative frequency and box plots

A cumulative frequency graph gives the median and quartiles needed for a box plot. Read LQ and UQ from the graph, use the given minimum and maximum values for the whiskers, and draw the box from LQ to UQ with a vertical line at the median.

In the swimming club question, the median time is about 68 seconds, LQ ≈ 53 s, UQ ≈ 76 s, with lowest 28 s and highest 96 s giving a box plot with values 28, 53, 68, 76, 96.

Common mistakes with cumulative frequency graphs

• Plotting class midpoints instead of upper boundaries: examiners expect points at the end of each interval (and a starting point at the lower boundary with CF 0).
• Joining points with straight spikes: the curve should be smooth. Straight segments between each point are not a correct ogive.
• Confusing frequency with cumulative frequency: the vertical axis shows running totals, not the frequency in a single class.
• Wrong quartile positions: for n values use n/2, n/4 and 3n/4 on the cumulative frequency axis, not the horizontal axis.
• Forgetting answers are estimates: grouped data means median, quartiles and percentages are approximate. Use language such as “about” or “approximately” unless the mark scheme states otherwise.
• Mixing up “less than” and “more than”: cumulative frequency counts values up to a boundary. For “more than £530”, subtract the reading at 530 from the total.
• Evaluating claims without showing working: when a question asks whether a statement is correct, show the percentage or count you read from the graph before giving your conclusion.

GCSE cumulative frequency checklist

Use this list to check you can tackle any cumulative frequency graph question on a GCSE paper.

• I can explain what cumulative frequency means and how it differs from frequency.
• I can complete a cumulative frequency column from a grouped table.
• I can plot points at upper class boundaries and join them with a smooth curve.
• I can estimate the median, lower quartile and upper quartile from a graph.
• I can calculate and interpret the interquartile range.
• I can estimate a percentage or number of values above or below a given value.
• I can use a cumulative frequency graph to draw a box plot when minimum and maximum are given.
• I can evaluate whether a written claim about the data is supported by the graph.
• I can combine cumulative frequency with probability (e.g. selecting two students above a threshold).

Free cumulative frequency worksheet (PDF)

Exam Style Questions

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