Cumulative Frequency
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Revision notes for Edexcel GCSE Maths Cumulative Frequency. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Cumulative Frequency

What you'll learn

  • How to change a frequency table into a cumulative frequency table.
  • How to plot a cumulative frequency graph using upper boundaries.
  • How to estimate the median and quartiles from the graph.
  • How cumulative frequency links to box plots.

1. From Frequency to Cumulative Frequency

A frequency is how many times something happens. For example, if 8 people take between 5 and 10 seconds, the frequency for that group is 8.

When data is in groups, each group is called a class interval. For example, 5<t≤105 < t \le 105<t≤10 means the time is more than 5 seconds and up to 10 seconds.

Definition

Cumulative frequency

Cumulative frequency means the running total of the frequencies. You keep adding the frequencies as you move down the table.

Example

Making a cumulative frequency table

  1. Imagine these grouped times for 42 people:

    Time takenFrequency
    more than 0 up to 5 seconds7
    more than 5 up to 10 seconds11
    more than 10 up to 15 seconds13
    more than 15 up to 20 seconds8
    more than 20 up to 25 seconds3

  2. Start with the first cumulative frequency: 7.

  3. Add each new frequency to the total so far:

A running-total flow diagram shows how each frequency is added to make the cumulative frequencies.

$$
\begin{aligned}
7 + 11 &= 18\\
18 + 13 &= 31\\
31 + 8 &= 39\\
39 + 3 &= 42
\end{aligned}
$$

4. The cumulative frequencies are:

| Time up to | Cumulative frequency |
|---|---:|
| 5 seconds | 7 |
| 10 seconds | 18 |
| 15 seconds | 31 |
| 20 seconds | 39 |
| 25 seconds | 42 |

5. Check the final cumulative frequency is 42, the same as the total number of people.

Tip

Quick check

The last cumulative frequency should equal the total frequency. If it does not, one of your additions is wrong.

2. Plotting a Cumulative Frequency Graph

A cumulative frequency graph shows the running total on the vertical axis. An axis is one of the numbered lines on a graph.

Definition

Upper boundary

The upper boundary, or upper bound, is the largest value included in a class interval. For 5<t≤105 < t \le 105<t≤10, the upper boundary is 10.

Key Idea

What to plot

For cumulative frequency graphs, plot the upper boundary against the cumulative frequency. Also include the starting point at the lower boundary with cumulative frequency 0.

Example

Plotting the graph

  1. Use the cumulative frequencies from the previous example.

  2. Put time in seconds on the horizontal axis and cumulative frequency on the vertical axis.

  3. Plot the starting point (0, 0) because up to 0 seconds, nobody has been counted.

  4. Plot the cumulative frequency points: (5, 7), (10, 18), (15, 31), (20, 39), (25, 42).

Cumulative frequency points are plotted using the upper boundary on the horizontal axis and the running total on the vertical axis.

  1. Join the points with a smooth increasing curve.

The completed cumulative frequency graph is a smooth increasing curve through the plotted points.

Common Mistake

Plotting the midpoint

Do not plot the middle of each class interval. For cumulative frequency, you plot the upper boundary, such as 5, 10, 15 and so on.

For cumulative frequency graphs, use the upper boundary of each class interval rather than the midpoint.

3. Estimating the Median and Quartiles

The graph helps you estimate key positions in the data.

Definition

Median and quartiles

  • The median is the middle value.
  • The lower quartile, written Q1Q_1Q1​, is one quarter of the way through the data.
  • The upper quartile, written Q3Q_3Q3​, is three quarters of the way through the data.
  • The interquartile range, or IQR, is Q3−Q1Q_3 - Q_1Q3​−Q1​. It measures the spread of the middle 50% of the data.

To find these from a cumulative frequency graph, use the total frequency, usually called NNN.

  • Lower quartile position: N4\frac{N}{4}4N​
  • Median position: N2\frac{N}{2}2N​
  • Upper quartile position: 3N4\frac{3N}{4}43N​
Example

Estimating quartiles from a graph

  1. A cumulative frequency graph represents 60 batteries and their lifetimes. So N=60N=60N=60.

  2. Find the positions for Q1Q_1Q1​, the median and Q3Q_3Q3​:

    Q1:604=15median:602=30Q3:3×604=45\begin{aligned} Q_1 &: \frac{60}{4} = 15\\ \text{median} &: \frac{60}{2} = 30\\ Q_3 &: \frac{3 \times 60}{4} = 45 \end{aligned}Q1​medianQ3​​:460​=15:260​=30:43×60​=45​

  3. On the vertical cumulative frequency axis, start at 15, 30 and 45. For each one, read across to the curve, then down to the horizontal axis.

Quartiles are estimated by reading across from cumulative frequency positions to the curve, then down to the data axis.

  1. Suppose the graph gives these estimates: Q1=22Q_1=22Q1​=22, median 31, and Q3=41Q_3=41Q3​=41 minutes.

  2. Use the quartile estimates to find the IQR in minutes:

    IQR=41−22=19\text{IQR} = 41 - 22 = 19IQR=41−22=19
Common Mistake

These are estimates

Because the data is grouped and the curve is drawn by hand, medians and quartiles read from a cumulative frequency graph are usually estimates.

4. From Cumulative Frequency to Box Plots

A box plot is a diagram that summarises a set of data using five important values.

Definition

Box plot

A box plot shows the five-number summary: lowest value, lower quartile, median, upper quartile and highest value.

The range is the highest value minus the lowest value. It shows the overall spread.

Example

Drawing a box plot from summary values

  1. Suppose a set of times has lowest value 4 seconds, Q1=10Q_1=10Q1​=10 seconds, median 14 seconds, Q3=21Q_3=21Q3​=21 seconds and highest value 30 seconds.

  2. Draw a horizontal number line with a sensible scale, such as 0 to 30 seconds.

  3. Draw a box from 10 to 21.

  4. Put a vertical line inside the box at 14 for the median.

  5. Draw a whisker from 4 to 10 and another whisker from 21 to 30.

A box plot places the minimum, quartiles, median and maximum on a single number line.

Common Mistake

Forcing the median into the centre

The median line does not have to be in the middle of the box. Put it at its actual value on the scale.

5. Comparing Using Box Plots

Box plots are useful because you can compare two data sets quickly.

  • A higher median means the typical value is higher.
  • A larger IQR means the middle 50% is more spread out.
  • A larger range means the whole data set is more spread out.
Example

Comparing two box plots

  1. Class A has median 46 and IQR 18. Class B has median 52 and IQR 9.

Two box plots can be compared by looking at their medians and IQRs on the same scale.

  1. Compare the medians: Class B has the higher typical mark.

  2. Compare the IQRs: Class B is more consistent because its middle 50% is less spread out.

  3. A good comparison sentence would be: Class B generally scored higher and the marks were less spread out.

Exam technique

In the exam

  1. Add frequencies carefully and check the final cumulative frequency equals the total.

  2. Plot upper boundaries, not midpoints, and include the starting point with cumulative frequency 0.

  3. When estimating quartiles, start on the cumulative frequency axis, read across to the curve, then down to the data axis.

Self review

Check yourself

  • If the total frequency is 80, what cumulative frequency values would you use to estimate Q1Q_1Q1​, the median and Q3Q_3Q3​?

  • Why do cumulative frequency graphs use upper boundaries?

  • What does a larger IQR tell you when comparing two box plots?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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