Cumulative Frequency and Box Plots [H]

This lesson covers cumulative frequency diagrams and box plots — both Higher tier topics on the AQA GCSE Mathematics specification. These powerful tools allow you to find the median, quartiles, and interquartile range from grouped data, and to compare distributions visually. Mastering these techniques can secure valuable marks in the exam.

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What Is Cumulative Frequency?

Cumulative frequency is a running total of the frequencies. Each cumulative frequency value tells you how many data values are less than or equal to the upper boundary of that class.

How to Calculate Cumulative Frequency

Simply add each frequency to the total of all previous frequencies.

Worked Example

The table shows the heights (in cm) of 60 plants.

Height (h cm)	Frequency	Cumulative Frequency
0 < h ≤ 10	4	4
10 < h ≤ 20	10	4 + 10 = 14
20 < h ≤ 30	18	14 + 18 = 32
30 < h ≤ 40	16	32 + 16 = 48
40 < h ≤ 50	9	48 + 9 = 57
50 < h ≤ 60	3	57 + 3 = 60

The final cumulative frequency should equal the total number of data values (60 in this case).

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Drawing a Cumulative Frequency Diagram

A cumulative frequency diagram (also called a cumulative frequency curve or ogive) is plotted as follows:

1. Plot each cumulative frequency against the upper boundary of its class interval.
2. The first point is plotted at the lower boundary of the first class with a cumulative frequency of 0.
3. Join the points with a smooth curve (not straight lines between points).
4. Label both axes clearly: the horizontal axis shows the data values, the vertical axis shows the cumulative frequency.

For the plant heights example, you would plot:

Upper Boundary	Cumulative Frequency
0	0
10	4
20	14
30	32
40	48
50	57
60	60

Exam Tip: Always plot cumulative frequency at the upper boundary of each class, not at the midpoint. Start with a point at (lower boundary, 0). Join with a smooth S-shaped curve. Using straight line segments instead of a curve is acceptable but less precise.

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Reading the Median and Quartiles from a Cumulative Frequency Diagram

The key values you can read from a cumulative frequency diagram are:

Measure	Position on Cumulative Frequency Axis
Median (Q2)	n / 2 (where n is the total frequency)
Lower Quartile (Q1)	n / 4
Upper Quartile (Q3)	3n / 4

Worked Example

Using the plant heights data (n = 60):

• Median position: 60 / 2 = 30. Find 30 on the cumulative frequency axis, draw a horizontal line to the curve, then draw a vertical line down to the horizontal axis. Read off the value: approximately 29 cm.

• Lower quartile (Q1) position: 60 / 4 = 15. Find 15 on the cumulative frequency axis, read across and down: approximately 21 cm.

• Upper quartile (Q3) position: 3 x 60 / 4 = 45. Find 45 on the cumulative frequency axis, read across and down: approximately 38 cm.

Interquartile Range (IQR)

The interquartile range measures the spread of the middle 50% of the data.

IQR = Q3 - Q1

IQR = 38 - 21 = 17 cm

graph TD
    A[Cumulative Frequency Diagram] --> B[Find n/4 on CF axis]
    A --> C[Find n/2 on CF axis]
    A --> D[Find 3n/4 on CF axis]
    B --> E[Read across to curve, down to x-axis]
    C --> F[Read across to curve, down to x-axis]
    D --> G[Read across to curve, down to x-axis]
    E --> H[Q1 - Lower Quartile]
    F --> I[Q2 - Median]
    G --> J[Q3 - Upper Quartile]
    H --> K[IQR = Q3 - Q1]
    J --> K

Exam Tip: Show your working on the cumulative frequency diagram by drawing clear dashed lines from the cumulative frequency axis across to the curve and down to the horizontal axis. Examiners look for these lines to award method marks even if your reading is slightly inaccurate.

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Box Plots (Box-and-Whisker Diagrams)

A box plot is a diagram that shows the spread of a data set using five key values:

1. Minimum value (smallest value, excluding outliers)
2. Lower quartile (Q1)
3. Median (Q2)
4. Upper quartile (Q3)
5. Maximum value (largest value, excluding outliers)

Drawing a Box Plot

A box plot is drawn on a number line:

• The box extends from Q1 to Q3.
• A vertical line inside the box marks the median.
• Whiskers extend from the box to the minimum and maximum values.

Worked Example

The test scores of a class are summarised as follows:

Measure	Value
Minimum	22
Q1	35
Median	52
Q3	68
Maximum	89

To draw the box plot:

1. Draw a number line with an appropriate scale (e.g. 0 to 100).
2. Draw a box from 35 (Q1) to 68 (Q3).
3. Draw a vertical line inside the box at 52 (median).
4. Draw a whisker from 35 to 22 (minimum).
5. Draw a whisker from 68 to 89 (maximum).

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Reading a Box Plot

From a box plot, you can identify:

• Median — the line inside the box.
• Q1 and Q3 — the left and right edges of the box.
• IQR — the width of the box (Q3 - Q1).
• Range — the distance from the end of the left whisker to the end of the right whisker (maximum - minimum).
• The skew of the data (see below).

Skewness

If...	The distribution is...
The median is in the centre of the box	Symmetrical
The median is closer to Q1 (left side of box)	Positively skewed (tail to the right)
The median is closer to Q3 (right side of box)	Negatively skewed (tail to the left)

Exam Tip: When interpreting a box plot, always state the median, IQR, and range. If comparing two box plots, comment on both an average (median) and the spread (IQR or range). See Lesson 9 for more on comparing distributions.

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Drawing a Box Plot from a Cumulative Frequency Diagram

This is a common exam question. You will:

1. Read the median, Q1, and Q3 from the cumulative frequency diagram (as described above).
2. Identify the minimum and maximum values from the data.
3. Draw the box plot on a number line using the same scale as the cumulative frequency diagram's horizontal axis.

Using the plant heights example:

Measure	Value
Minimum	0
Q1	21
Median	29
Q3	38
Maximum	60

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Outliers on Box Plots [H]

An outlier on a box plot is sometimes defined as a value that is more than 1.5 x IQR beyond Q1 or Q3.