Collecting & Representing Data

SPEC S1, S2 • STATISTICS

Frequency tables, bar charts, pie charts, pictograms — choosing the right picture for the right kind of data.

Why Choose the Right Chart?

Every chart type is designed for a specific kind of data. Pick the wrong one and the chart is either wrong by convention or genuinely misleading.

The chart types in this lesson

Frequency table — organise raw counts by category.

Bar chart — for categorical data (colour, sport).

Vertical line chart — for ungrouped discrete numerical data (goals, shoe size).

Pie chart — for proportions of a whole.

Pictogram — like a bar chart but with symbols and a key.

Line graph — for time-series data (year, month, day).

MCE approach

Method — identify the data type and pick the chart. Calculate — angles for pie charts, heights for bars, symbol counts for pictograms. Evaluate — check totals and check that the chart answers the question.

Discrete vs Continuous Data

The first question on any data-handling task: what kind of data is this?

Discrete data

Can only take specific, separate values. You count it.

Examples: number of siblings, shoe size, number of goals scored.

Continuous data

Can take any value in a range. You measure it.

Examples: height, mass, time, temperature.

Why it matters

Discrete data goes on a bar chart (gaps between bars). Continuous data goes on a histogram (no gaps — covered in Lesson 6.04). Get the data type right before you pick the chart.

Data Sources & Frequency Tables

Primary data

Collected first-hand by the investigator — surveys, experiments, measurements you took yourself. More reliable but more work.

Secondary data

Sourced from elsewhere — books, websites, government statistics, another researcher's study. Quick but depends on the source.

Sample vs population

Surveys usually use a sample (a smaller group) to infer facts about the whole population. A bigger and more representative sample gives a better estimate — but a sample is never a guarantee. This is the core limitation of sampling.

Frequency table — the basic organiser

Categories down the left, counts on the right. A tally column is optional — each group of five is four vertical marks with a diagonal strike (shown below).

Favourite fruitTallyFrequency

Total—60

Bar Charts

The AQA specification uses bar charts for categorical data — labels like colour, sport, or fruit that have no natural numerical order. Each category gets its own bar.

The four rules

Bars all have equal width.

Gaps between bars (this is what separates a bar chart from a histogram).

The vertical axis shows frequency — label it.

Label the horizontal axis and give the chart a title.

Vertical Line Charts

When the data is ungrouped discrete numerical — where the categories are numbers like 0, 1, 2, 3 — the AQA specification asks for a vertical line chart. It's like a bar chart, but each "bar" is just a thin line.

When to use

Data values are whole numbers with no overlap — number of goals scored, shoe size, number of pets.

The horizontal axis is a numerical scale, not a list of labels.

Each frequency is shown by the height of a single vertical line, often with a dot on top.

Frequencies: 3, 5, 7, 3, 2 — totalling 20 matches.

Bar chart or vertical line chart?

Categorical (words like "red, blue, green") → bar chart. Discrete numerical (whole numbers like 0, 1, 2, 3) → vertical line chart. In practice AQA often accepts a bar chart for discrete numerical data too, but the vertical line chart is the formally correct choice per the spec.

Composite & Dual Bar Charts

Sometimes you want to compare two groups. Two common styles appear in AQA papers:

Composite (stacked)

Each bar adds up to 100% (or the group total). Different colours stack on top of each other — good for showing how a whole is split between categories.

Dual (side-by-side)

Two bars side by side for each category, one colour per group. Good for direct like-for-like comparisons of frequencies.

Composite Bar Chart — Drink Sales by Year

Reading composite bars

On a percentage composite bar, the height of each coloured section equals its percentage. To find a frequency, apply that percentage to the year's total.

Pie Charts — The Formula

A pie chart uses the whole circle — all 360° — to represent the total. Each category gets a sector whose angle is proportional to its frequency.

Sector angle

sector angle = (frequency ÷ total) × 360°

Four equal sectors: each 90°, summing to 360°.

Non-negotiable

All sector angles must sum to 360°. Not 180°. Not 100°. Always 360°.

Useful shortcut

If total = 360, each frequency is equal to its angle. If total = 60, each unit of frequency is worth 6° (360 ÷ 60).

Worked Example — Drawing a Pie Chart01

In a survey of 60 people's favourite fruit: apples 20, bananas 15, cherries 10, other 15. Work out the sector angle for each fruit and sketch the pie chart. [3 marks]

MMethod. Each angle = (frequency ÷ 60) × 360°. Since 360 ÷ 60 = 6, every unit of frequency is worth 6°.

Calculate.

•

Apples: (20 ÷ 60) × 360 = 120°

•

Bananas: (15 ÷ 60) × 360 = 90°

•

Cherries: (10 ÷ 60) × 360 = 60°

•

Other: (15 ÷ 60) × 360 = 90°

Evaluate.

360°

Reading From a Pie Chart

You're given the total and a sector's angle, and asked for the frequency. Just rearrange the formula.

Reverse formula

frequency = (angle ÷ 360°) × total

Or think of it as a fraction

The sector angle divided by 360° is the fraction of the total that category takes up. Multiply the total by that fraction.

Common AQA trap

Questions often give you the angle for one category and ask about a different category. Identify what's asked for first — don't use the wrong angle.

Worked Example — Reading a Pie Chart02

A pie chart shows the favourite sport of 240 students. The football sector is 150°, cricket is 90°, and the remaining 120° is split equally between rugby and tennis. How many students chose rugby? [3 marks]

MMethod. Find the rugby angle, then apply frequency = (angle ÷ 360°) × total with total = 240.

Calculate.

•

Remaining angle = 360 − 150 − 90 = 120°

•

Rugby angle = 120 ÷ 2 = 60°

•

Rugby frequency = (60 ÷ 360) × 240 = (1 ÷ 6) × 240 = 40

EEvaluate. Check: football (100) + cricket (60) + rugby (40) + tennis (40) = 240 ✓. Answer: 40 students chose rugby.

Pictograms

A pictogram uses a symbol to represent a fixed number of items. The only thing that makes a pictogram work is the key.

The three rules

Every pictogram must include a key (e.g. "○ = 4 books").

Partial symbols represent a fraction of the key value (half a symbol = half the key).

All symbols should be the same size to avoid misleading the reader.

Wednesday shows 1½ symbols → 1.5 × 4 = 6 books. Thursday shows 2½ symbols → 10 books.

Line Graphs for Time Series

When the horizontal axis is time — days, weeks, years — a line graph is usually the right choice. Plot the points and join them with straight lines.

When to use

Data recorded at regular time intervals (daily temperature, monthly sales).

You want to see a trend over time — rising, falling, seasonal.

Do not use for categorical data (e.g. favourite fruit) — there's no natural order.

Choose the Right Chart

Before you draw anything, identify the data type. This table is your decision guide.

Data typeBest chartKey feature

Categorical (e.g. colour, sport)Bar chart or pictogramEqual-width bars with gaps; symbols with a key

Ungrouped discrete numerical (e.g. shoe size, goals)Vertical line chartThin vertical lines on a numerical x-axis

Proportions of a wholePie chartAngles sum to 360°

Time series (over days, months, years)Line graphPoints joined by straight lines

Grouped continuous (e.g. height)Histogram (Lesson 6.04)No gaps; frequency density at Higher

Comparing two datasetsDual bar chart or composite bar chartTwo bars per category, or stacked to 100%

Exam technique

If a question gives you a total and percentages or angles, it's almost always a pie-chart or composite-bar-chart problem. Find the quantity you need by multiplying the total by the fraction.

Worked Example — Composite Bar Chart03

A composite bar chart shows the percentage of drinks sold by a café in 2023. Coffee was 42%, tea was 35%, juice was 15% and other was 8%. The café sold 2400 drinks that year. Work out how many more coffees than juices were sold. [3 marks]

MMethod. Each percentage of the 2400 total gives the frequency for that drink. Find coffee and juice frequencies, then subtract.

Calculate.

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Coffee: 42% of 2400 = 0.42 × 2400 = 1008

•

Juice: 15% of 2400 = 0.15 × 2400 = 360

•

Difference: 1008 − 360 = 648

EEvaluate. Quick check — the percentages sum to 42 + 35 + 15 + 8 = 100% ✓. Answer: 648 more coffees than juices.

Examiner Alert — Bar Chart vs Histogram

Common confusion

Bar charts and histograms look similar but are not the same chart. Using the wrong one is a classic AQA trap.

Bar chartHistogram

Data typeDiscrete / categoricalGrouped continuous

Gaps between bars?Yes — alwaysNo — bars touch

Bar widthEqualCan vary (class width)

Vertical axisFrequencyFrequency density (at Higher)

Frequency shown byHeight of barArea of bar

What to write instead

Before drawing, state the data type to yourself. "This is discrete, so bar chart." Or "This is grouped continuous, so histogram." That one-line check prevents the error.

Examiner Alert — Pie Angles Sum to 360°

The most fundamental error

Every pie chart has sector angles that sum to exactly 360°. Not 180°. Not 90°. Not 100%. 360°.

Where students go wrong

Using 180° in the formula because they've confused it with angles on a straight line.

Using 100 instead of 360° because they're thinking about percentages.

Forgetting to check that their angles sum correctly at the end.

Always finish with the check

After computing the last sector angle, add them all up. If they don't make 360°, something is wrong — go back and find the arithmetic error before moving on.

Another reliable trap

A sector angle of the pie chart being given as the smallest sector visually but the largest numerical percentage, or vice versa — diagrams are often "not drawn accurately". Always trust the numbers, not the picture.

Practice Question

A pie chart shows how 180 people travelled to work. The sector angles for bus and walking are 80° and 40° respectively. The remaining sector represents those who travelled by car or bike, and it is split between them in the ratio 2 : 1 (car : bike).

(a) How many people travelled by walking? [2 marks]

(b) How many people travelled by car? [3 marks]

MMethod. For (a), use frequency = (angle ÷ 360°) × total with total = 180. For (b), find the remaining angle, split it in ratio 2 : 1, then apply the same formula.

Calculate.

•

(a) Walking = (40 ÷ 360) × 180 = (1 ÷ 9) × 180 = 20 people

•

(b) Remaining angle = 360 − 80 − 40 = 240°

•

(b) Car angle = (2 ÷ 3) × 240 = 160°

•

(b) Car frequency = (160 ÷ 360) × 180 = (4 ÷ 9) × 180 = 80 people

EEvaluate. Check angles: 80 + 40 + 160 + 80 = 360° ✓. Check frequencies: bus (40) + walking (20) + car (80) + bike (40) = 180 ✓.

Summary — MCE Checkpoint

Key formulas

sector angle = (frequency ÷ total) × 360°

frequency = (angle ÷ 360°) × total

Chart chooser

Bar chart — categorical data; equal widths, gaps between bars.

Vertical line chart — ungrouped discrete numerical data (e.g. goals).

Pie chart — proportions of a whole; angles sum to 360°.

Pictogram — must have a key; partial symbols = partial key values.

Line graph — time-series data only.

Composite bar — stacked to show percentage splits.

Examiner traps

Pie angles sum to 360° , not 180°.

Bar chart ≠ histogram — discrete vs continuous.

Read the right sector — questions often mix the angle given with a different quantity asked.

A pictogram without a key is meaningless — always include one.

✓ You should now be able to

Identify discrete vs continuous and primary vs secondary data.

Describe a sample and recognise the limitations of sampling.

Construct frequency tables (with tallies), bar charts, vertical line charts and pictograms with labels and keys.

Compute sector angles for a pie chart and verify they sum to 360°.

Read a frequency from a pie chart using the reverse formula.

Extract frequencies from composite bar charts using percentage totals.

Choose the right chart type for the data given.