Statistical Charts for Categorical Data

Understanding Categorical Data

When you ask your friends what their favourite chocolate bar is, you cannot calculate a mathematical "average" chocolate bar. This type of information is known as qualitative data, which describes a quality or characteristic and is usually expressed in words (like car makes or favourite colours).

When this data is divided into distinct, non-overlapping groups where each piece of data belongs to exactly one group, it is called categorical data. While mostly qualitative, categorical data can occasionally be numerical if the numbers act purely as labels, such as bus route numbers or phone models. Because you cannot calculate a meaningful mean for this type of data, the mode (the most frequent category) is the only average that can be used.

Constructing Frequency Tables from Raw Data

Before you can draw a chart, you need to organise your raw data (data in its original, unorganised form) into a frequency table. A frequency table summarises large data sets by showing the frequency, which is the total count of how many times a specific category occurs.

When constructing a table from raw data, a tally system acts as an essential intermediate step to ensure no data point is missed.

Worked Example: Constructing a Frequency Table

A survey asked 20 students for their favourite sport. The raw data is: Football, Netball, Football, Rugby, Netball, Football, Football, Netball, Rugby, Football, Netball, Tennis, Football, Football, Netball, Rugby, Netball, Football, Tennis, Netball.

Step 1: List the distinct categories in the first column.

Step 2: Go through the raw data one by one and record a tally using "five-bar gate" notation (four vertical lines and a diagonal fifth).

Step 3: Count the tallies to find the frequency ( $f$ ) for each category.

Step 4: Sum the frequencies to ensure they match the total number of students surveyed ( $\sum f = 20$ ).

Sport	Tally	Frequency ( $f$ )
Football
Netball
Rugby
Tennis
Total		20

Interpreting Pictograms

A pictogram is a visual representation where frequency is shown by the number of symbols. There are no axes, but every pictogram must have a key that defines the mathematical value of one full symbol.

Symbols must be consistently sized, shaped, and aligned horizontally or vertically so they can be visually compared like bars in a chart. When dealing with smaller frequencies, fractional symbols are used.

Worked Example: Pictogram with Partial Symbols

A school fair recorded the number of cakes sold using a pictogram. Key: One full circle ( $\large \bigcirc$ ) = 12 cakes. Therefore, a half circle = 6 cakes, and a quarter circle = 3 cakes.

Chocolate: 5 full circles. Calculate: $5 \times 12 = 60$ cakes.
Vanilla: 3 full circles + 1 quarter circle. Calculate: $(3 \times 12) + 3 = 39$ cakes.
Lemon: 1 full circle + 1 half circle. Calculate: $12 + 6 = 18$ cakes.

To verify the total frequency, sum the values: $60 + 39 + 18 = 117$ total cakes sold.

Constructing Bar Charts

A bar chart is a visual representation where the height or length of bars is directly proportional to the frequency. When drawing bar charts for categorical data in your AQA exam, there are strict construction rules you must follow to secure marks.

Equal width bars: Every bar must be exactly the same width.
Equal gaps: Consistent spacing between every single bar is mandatory. This visually distinguishes it as categorical data rather than continuous data (which uses a histogram).
Axes: The vertical axis is for frequency and must feature a linear scale starting at zero, while the horizontal axis lists the discrete categories.

Calculating Sector Angles for Pie Charts

A pie chart represents categorical data as a proportion of a whole, rather than showing absolute frequencies. A full circle always represents $360^\circ$ and the total frequency ( $n$ ). To construct a pie chart accurately, you must calculate the sector angle for each category, which is directly proportional to its frequency.

Formula for Sector Angle:

A = \left( \frac{\text{Frequency}}{\text{Total Frequency}} \right) \times 360

Alternatively, you can calculate the multiplier (the number of degrees representing one unit of frequency) by dividing $360$ by the Total Frequency.

Worked Example: Calculating Pie Chart Angles

Data shows the methods of transport for 60 employees. Calculate the sector angle for each method of transport.

Step 1: Identify the Total Frequency. Here, it is 60 employees.

Step 2: Calculate the multiplier.

Multiplier = $360 \div 60 = 6^\circ$ per employee.

Step 3: Multiply each individual frequency by $6^\circ$ to find the angle.

Method	Frequency ( $f$ )	Calculation ( $f \times 6^\circ$ )	Sector Angle
Car	18	$18 \times 6^\circ$	$108^\circ$
Bus	12	$12 \times 6^\circ$	$72^\circ$
Walk	23	$23 \times 6^\circ$	$138^\circ$
Bike	7	$7 \times 6^\circ$	$42^\circ$
Total	60		$360^\circ$

Step 4: Verify your calculations by checking that the sector angles sum to exactly $360^\circ$ .

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Exam Tips

Common Mistake
Students often misinterpret a quarter symbol in a pictogram as automatically representing '1 unit'. You must always divide the key value by 4 (e.g., if the key is 12, a quarter symbol is 3).
2
In AQA exams, it is mandatory to leave equal gaps between bars in a bar chart for categorical data; failing to do so will lose you construction marks.
3
When calculating pie chart angles, if your angles do not sum to exactly 360°, you have made a calculation error and should immediately recheck your multiplier.
4
When tallying raw data, draw a single line through each value in the original list as you record it to ensure you do not miscount or skip any data points.
5
If a pie chart question states 'Not to Scale', do not attempt to measure the angles with a protractor; you must use calculation or ratio methods to find the missing frequencies.

Key Terms(12)

Qualitative data: Data describing a quality or characteristic, typically expressed in words and impossible to measure numerically.
Categorical data: Data divided into non-overlapping groups where each piece of data belongs to exactly one category.
Mode: The most frequent value or category in a data set; the only average suitable for categorical or qualitative data.
Raw data: Data in its original, unorganised form exactly as it was collected.
Frequency table: A table used to summarise large sets of data, displaying categories alongside their corresponding tallies and frequencies.
Frequency: The total number of times a specific data value or category occurs.
Tally: A system of recording data occurrences using marks, typically grouped in fives (four vertical lines and a diagonal fifth).
Pictogram: A visual representation of data where frequency is represented by the proportional number of symbols.
Key: A crucial guide accompanying a pictogram or chart that defines the numerical value or meaning of specific symbols, colours, or patterns.
Bar chart: A visual representation using equal-width bars where the height or length is proportional to the frequency, separated by equal gaps for categorical data.
Sector angle: The central angle in a pie chart representing a category's proportional slice of the total 360 degrees.
Multiplier: The number of degrees representing one unit of frequency in a pie chart, found by calculating 360 divided by the total frequency.

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Key Terms(12)

Qualitative data: Data describing a quality or characteristic, typically expressed in words and impossible to measure numerically.
Categorical data: Data divided into non-overlapping groups where each piece of data belongs to exactly one category.
Mode: The most frequent value or category in a data set; the only average suitable for categorical or qualitative data.
Raw data: Data in its original, unorganised form exactly as it was collected.
Frequency table: A table used to summarise large sets of data, displaying categories alongside their corresponding tallies and frequencies.
Frequency: The total number of times a specific data value or category occurs.
Tally: A system of recording data occurrences using marks, typically grouped in fives (four vertical lines and a diagonal fifth).
Pictogram: A visual representation of data where frequency is represented by the proportional number of symbols.
Key: A crucial guide accompanying a pictogram or chart that defines the numerical value or meaning of specific symbols, colours, or patterns.
Bar chart: A visual representation using equal-width bars where the height or length is proportional to the frequency, separated by equal gaps for categorical data.
Sector angle: The central angle in a pie chart representing a category's proportional slice of the total 360 degrees.
Multiplier: The number of degrees representing one unit of frequency in a pie chart, found by calculating 360 divided by the total frequency.

Statistical Charts for Categorical Data

Understanding Categorical Data

Constructing Frequency Tables from Raw Data

When constructing a table from raw data, a tally system acts as an essential intermediate step to ensure no data point is missed.

Worked Example: Constructing a Frequency Table

Step 1: List the distinct categories in the first column.

Step 2: Go through the raw data one by one and record a tally using "five-bar gate" notation (four vertical lines and a diagonal fifth).

Step 3: Count the tallies to find the frequency ( $f$ ) for each category.

Step 4: Sum the frequencies to ensure they match the total number of students surveyed ( $\sum f = 20$ ).

Sport	Tally	Frequency ( $f$ )
Football
Netball
Rugby
Tennis
Total		20

Interpreting Pictograms

Worked Example: Pictogram with Partial Symbols

A school fair recorded the number of cakes sold using a pictogram. Key: One full circle ( $\large \bigcirc$ ) = 12 cakes. Therefore, a half circle = 6 cakes, and a quarter circle = 3 cakes.

Chocolate: 5 full circles. Calculate: $5 \times 12 = 60$ cakes.
Vanilla: 3 full circles + 1 quarter circle. Calculate: $(3 \times 12) + 3 = 39$ cakes.
Lemon: 1 full circle + 1 half circle. Calculate: $12 + 6 = 18$ cakes.

To verify the total frequency, sum the values: $60 + 39 + 18 = 117$ total cakes sold.

Constructing Bar Charts

Equal width bars: Every bar must be exactly the same width.
Equal gaps: Consistent spacing between every single bar is mandatory. This visually distinguishes it as categorical data rather than continuous data (which uses a histogram).
Axes: The vertical axis is for frequency and must feature a linear scale starting at zero, while the horizontal axis lists the discrete categories.

Calculating Sector Angles for Pie Charts

Formula for Sector Angle:

A = \left( \frac{\text{Frequency}}{\text{Total Frequency}} \right) \times 360

Alternatively, you can calculate the multiplier (the number of degrees representing one unit of frequency) by dividing $360$ by the Total Frequency.

Worked Example: Calculating Pie Chart Angles

Data shows the methods of transport for 60 employees. Calculate the sector angle for each method of transport.

Step 1: Identify the Total Frequency. Here, it is 60 employees.

Step 2: Calculate the multiplier.

Multiplier = $360 \div 60 = 6^\circ$ per employee.

Step 3: Multiply each individual frequency by $6^\circ$ to find the angle.

Method	Frequency ( $f$ )	Calculation ( $f \times 6^\circ$ )	Sector Angle
Car	18	$18 \times 6^\circ$	$108^\circ$
Bus	12	$12 \times 6^\circ$	$72^\circ$
Walk	23	$23 \times 6^\circ$	$138^\circ$
Bike	7	$7 \times 6^\circ$	$42^\circ$
Total	60		$360^\circ$

Step 4: Verify your calculations by checking that the sector angles sum to exactly $360^\circ$ .

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often misinterpret a quarter symbol in a pictogram as automatically representing '1 unit'. You must always divide the key value by 4 (e.g., if the key is 12, a quarter symbol is 3).
2
In AQA exams, it is mandatory to leave equal gaps between bars in a bar chart for categorical data; failing to do so will lose you construction marks.
3
When calculating pie chart angles, if your angles do not sum to exactly 360°, you have made a calculation error and should immediately recheck your multiplier.
4
When tallying raw data, draw a single line through each value in the original list as you record it to ensure you do not miscount or skip any data points.
5
If a pie chart question states 'Not to Scale', do not attempt to measure the angles with a protractor; you must use calculation or ratio methods to find the missing frequencies.

Key Terms(12)

Qualitative data: Data describing a quality or characteristic, typically expressed in words and impossible to measure numerically.
Categorical data: Data divided into non-overlapping groups where each piece of data belongs to exactly one category.
Mode: The most frequent value or category in a data set; the only average suitable for categorical or qualitative data.
Raw data: Data in its original, unorganised form exactly as it was collected.
Frequency table: A table used to summarise large sets of data, displaying categories alongside their corresponding tallies and frequencies.
Frequency: The total number of times a specific data value or category occurs.
Tally: A system of recording data occurrences using marks, typically grouped in fives (four vertical lines and a diagonal fifth).
Pictogram: A visual representation of data where frequency is represented by the proportional number of symbols.
Key: A crucial guide accompanying a pictogram or chart that defines the numerical value or meaning of specific symbols, colours, or patterns.
Bar chart: A visual representation using equal-width bars where the height or length is proportional to the frequency, separated by equal gaps for categorical data.
Sector angle: The central angle in a pie chart representing a category's proportional slice of the total 360 degrees.
Multiplier: The number of degrees representing one unit of frequency in a pie chart, found by calculating 360 divided by the total frequency.

Previous Subtopic: Population InferencePopulation Inference Next NoteCharts for Discrete and Time Series Data

Back to Statistical Charts

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(12)

Qualitative data: Data describing a quality or characteristic, typically expressed in words and impossible to measure numerically.
Categorical data: Data divided into non-overlapping groups where each piece of data belongs to exactly one category.
Mode: The most frequent value or category in a data set; the only average suitable for categorical or qualitative data.
Raw data: Data in its original, unorganised form exactly as it was collected.
Frequency table: A table used to summarise large sets of data, displaying categories alongside their corresponding tallies and frequencies.
Frequency: The total number of times a specific data value or category occurs.
Tally: A system of recording data occurrences using marks, typically grouped in fives (four vertical lines and a diagonal fifth).
Pictogram: A visual representation of data where frequency is represented by the proportional number of symbols.
Key: A crucial guide accompanying a pictogram or chart that defines the numerical value or meaning of specific symbols, colours, or patterns.
Bar chart: A visual representation using equal-width bars where the height or length is proportional to the frequency, separated by equal gaps for categorical data.
Sector angle: The central angle in a pie chart representing a category's proportional slice of the total 360 degrees.
Multiplier: The number of degrees representing one unit of frequency in a pie chart, found by calculating 360 divided by the total frequency.