Graphical representation of data

Q: Where do I plot cumulative frequency points?

Always plot cumulative frequency against the upper class boundary of each interval, not the midpoint or lower boundary, to ensure your diagram is accurate for estimation.

Edexcel International GCSE Mathematics A· 6.1· 25 min read

1. Foundation Tier: Core Statistical Diagrams★★☆☆☆⏱ 7 min

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All Foundation tier students must be able to construct and interpret four core diagram types: pictograms, bar charts, pie charts, and two-way tables. All require basic tabulation of raw or grouped data before drawing.

📘 Definition

Pie Chart

A circular diagram where sector angles are proportional to the frequency of each category, with total angle equal to 360°.

Example:

A category with frequency 10 out of total 50 has a sector angle of (10/50) × 360 = 72°.

📐 Worked Example

The table shows favourite sports of 60 Year 11 students: Football = 25, Netball = 15, Tennis = 10, Other = 10. Calculate the sector angle for each category to draw a pie chart.

1
Calculate total frequency = 60, so 1 unit of frequency = 360 ÷ 60 = 6° per person.
2
Football angle: 25 × 6 = 150°
3
Netball angle: 15 × 6 = 90°
4
Tennis angle: 10 × 6 = 60°
5
Other angle: 10 × 6 = 60°
6
Check sum: 150 + 90 + 60 + 60 = 360°, so values are correct.

2. Higher Tier: Histograms with Unequal Class Intervals★★★★☆Higher only⏱ 8 min

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Higher tier students must construct and interpret histograms for continuous data with unequal class intervals. Unlike bar charts, histogram bar height represents frequency density, and bar area represents frequency.

📘 Definition

Frequency Density

FD = \frac{f}{w}

Measure of frequency per unit of class width, used for histograms with unequal intervals, where f = frequency of interval and w = class width.

📐 Worked Example

The grouped data shows test marks out of 100: 0-20: f=8, 20-40: f=12, 40-65: f=15, 65-100: f=10. Calculate frequency density for each interval to draw a histogram.

1
Calculate class widths: 0-20 = 20, 20-40 = 20, 40-65 = 25, 65-100 = 35.
2
$FD_{0-20} = \frac{8}{20} = 0.4$
3
$FD_{20-40} = \frac{12}{20} = 0.6$
4
$FD_{40-65} = \frac{15}{25} = 0.6$
5
$FD_{65-100} = \frac{10}{35} \approx 0.29$

3. Higher Tier: Cumulative Frequency Diagrams★★★☆☆Higher only⏱ 6 min

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Cumulative frequency (CF) diagrams show the running total of frequencies up to the upper class boundary of each interval. They are used to estimate how many data points fall below a given value.

📘 Definition

Cumulative Frequency

Sum of frequencies of all intervals up to and including the current interval, plotted against the upper class boundary of the interval.

📐 Worked Example

The same test mark data: 0-20: f=8, 20-40: f=12, 40-65: f=15, 65-100: f=10. Calculate cumulative frequencies and state the coordinates to plot for the CF diagram.

1
List upper class boundaries for each interval: 20, 40, 65, 100.
2
CF at 20 = 8
3
CF at 40 = 8 + 12 = 20
4
CF at 65 = 20 + 15 = 35
5
CF at 100 = 35 + 10 = 45
6
Coordinates to plot: (20, 8), (40, 20), (65, 35), (100, 45)

4. Interpreting Statistical Diagrams for Exam Questions★★★☆☆⏱ 4 min

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All students will be asked to interpret diagrams to answer context-based questions, such as calculating the number of people in a category from a pie chart, or estimating frequency from a histogram area.

📐 Worked Example

A pie chart sector for 'travel to school by bus' has an angle of 120°, and the total number of students surveyed is 180. How many students travel by bus?

1
Fraction of students who take bus = 120 ÷ 360 = 1/3
2
Number of students = 180 × (1/3) = 60

5. Common Pitfalls

Wrong move:

Drawing gaps between bars on a histogram

Why:

Histograms represent continuous data, gaps imply discrete categories, leading to lost marks.

Correct move:

Leave no gaps between bars on histograms, only on bar charts for discrete data.

Wrong move:

Plotting cumulative frequency at midpoint of class intervals

Why:

Cumulative frequency counts all values up to the upper boundary, midpoint plotting leads to incorrect estimates.

Correct move:

Always plot CF against the upper class boundary of each interval.

Wrong move:

Using frequency instead of frequency density for histogram y-axis

Why:

For unequal class intervals, frequency height misrepresents the actual count, as area is the measure of frequency.

Correct move:

Label histogram y-axis as Frequency Density, calculate FD = frequency ÷ class width.

Wrong move:

Calculating pie chart angles as (frequency × 180) instead of 360

Why:

Total angle of a circle is 360°, using 180 gives incorrect sector sizes.

Correct move:

Sector angle = (frequency / total frequency) × 360, always check total angles add up to 360°.

Wrong move:

Forgetting to include a key/legend for pictograms or multi-category bar charts

Why:

Unlabelled diagrams are impossible to interpret, leading to lost method marks.

Correct move:

Add a clear key, labels for both axes, and a title for all diagrams you draw.

6. Quick Reference Cheatsheet

Diagram Type	Tier	Key Rule	Formula (if applicable)
Pictogram	Foundation	Use consistent symbol scale, add key	None
Bar Chart	Foundation	Label axes, gaps between bars for discrete data	None
Pie Chart	Foundation	Sector angles sum to 360°	Angle = (f ÷ total f) × 360°
Two-way Table	Foundation	Row/column totals match overall total	None
Histogram (unequal intervals)	Higher	Area of bar = frequency, no gaps between bars	Frequency Density = f ÷ class width
Cumulative Frequency Diagram	Higher	Plot at upper class boundary	CF = running total of frequencies

7. Frequently Asked

Do histograms have gaps between bars?

No. Histograms represent continuous data, so there are no gaps between bars, unlike bar charts which are used for discrete categorical data and include gaps between bars.

Where do I plot cumulative frequency points?

Always plot cumulative frequency against the upper class boundary of each interval, not the midpoint or lower boundary, to ensure your diagram is accurate for estimation.

What's Next

Now that you have mastered graphical representation of data, you can move on to analysing data using measures of central tendency and spread, covered in the next section of the Edexcel IGCSE Maths A statistics syllabus. You will use the diagrams you learned to draw here to support interpretation of numerical statistics, and apply these skills to context-based exam questions that combine both diagram construction and data analysis. This topic appears on almost every exam paper, carrying between 3 and 8 marks per question, so regular practice of past paper questions is highly recommended.