Statistical charts: pie charts and scatter graphs
Calculating pie chart sector angles and percentages from a frequency table, and describing correlation on a scatter graph
About three lessons of 45 to 60 minutes
2 very different chart types, for 2 very different questions
A pie chart answers 'what SHARE of the whole does each category take up?', a survey of favourite sports splits a full circle (360 degrees, or 100%) proportionally between football, cricket, tennis and swimming fans. A scatter graph answers a completely different question: 'is there a RELATIONSHIP between 2 different numerical variables?', like height and shoe size, or hours of revision and exam score.
Both charts turn raw data into an instant visual answer, but building a pie chart's sectors and reading a scatter graph's trend both rely on real calculation underneath the picture, exact angle and percentage arithmetic for a pie chart, and a genuine best-fit LINE (found by calculation, never by eye) for a scatter graph.
- A survey of 36 students' favourite sporteach sport's sector angle = (that sport's count / 36) x 360 degrees
- A pie chart slice worth 25%takes up exactly 1 quarter of the circle, 90 degrees
- Height vs shoe size for a class of studentsa scatter graph typically shows positive correlation: taller students tend to have bigger feet
- Hours spent watching TV vs test scoreoften shows negative correlation: more TV time, lower scores, on average
What students will be able to do
Students will calculate the sector angle and percentage for each category of a pie chart from a frequency table, and describe the correlation shown by a scatter graph (positive, negative or none), using the line of best fit to estimate values.
- I can calculate a pie chart sector's angle from a frequency table, using angle = (frequency / total) x 360.
- I can calculate a pie chart category's percentage from a frequency table, using percentage = (frequency / total) x 100.
- I can describe a scatter graph's correlation as positive, negative or none, based on the overall trend of the points.
- I can use a scatter graph's line of best fit to estimate a value.
Standards this unit teaches
- KS3 Maths: StatisticsUK National Curriculum (England)Statistics
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Statistics" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data" (the pie-chart half; bar charts and pictograms already exist site-wide as generic, non-KS3-tagged content, so are not rebuilt here); "describe simple mathematical relationships between 2 variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Frequency
- how many times a particular category or value occurs in a data set
- Sector
- a 'slice' of a pie chart, a wedge-shaped region representing 1 category
- Bivariate data
- data made up of 2 different variables measured on the same items, such as height AND weight for the same group of people
- Correlation
- how closely 2 variables move together: positive (both increase together), negative (one increases as the other decreases), or none (no consistent pattern)
- Line of best fit
- a single straight line drawn as close as possible to all the points on a scatter graph, used to estimate values and describe the overall trend
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Pie chart sector angles
ConcreteA full circle is 360 degrees, and the WHOLE circle represents the WHOLE data set. Each category's share of the circle is worked out proportionally: sector angle = (that category's frequency / the total frequency) x 360.
As a check, the sector angles for all categories should always add up to exactly 360 degrees, since together they make up the whole circle.
A survey of 40 students asked their favourite subject: Maths 10, Science 8, Art 6, PE 16. Find the sector angle for each subject.
- Maths: (10 / 40) x 360 = 90 degrees.
- Science: (8 / 40) x 360 = 72 degrees.
- Art: (6 / 40) x 360 = 54 degrees.
- PE: (16 / 40) x 360 = 144 degrees.
- Check: 90 + 72 + 54 + 144 = 360 degrees.
Answer: Maths: 90 degrees. Science: 72 degrees. Art: 54 degrees. PE: 144 degrees.
- Why must every pie chart's sector angles always add up to exactly 360 degrees?
- A survey of 20 people found 5 prefer tea. Find the sector angle for tea.
2. Pie chart percentages
ConcreteThe same proportional idea works with percentages instead of degrees: percentage = (frequency / total) x 100. A pie chart is often labelled with percentages instead of (or alongside) angles, since percentages are usually easier for most people to interpret at a glance.
In the same survey (Maths 10, Science 8, Art 6, PE 16, total 40), find the percentage for each subject.
- Maths: (10 / 40) x 100 = 25%.
- Science: (8 / 40) x 100 = 20%.
- Art: (6 / 40) x 100 = 15%.
- PE: (16 / 40) x 100 = 40%.
- Check: 25 + 20 + 15 + 40 = 100%.
Answer: Maths: 25%. Science: 20%. Art: 15%. PE: 40%.
- A pie chart shows Maths as 90 degrees. What percentage of the circle is that?
3. Scatter graphs: correlation and line of best fit
PictorialA scatter graph plots 1 variable against another for the same set of items, and the overall pattern (or lack of one) tells you about the relationship between them. If the points generally trend UPWARD left to right, that is POSITIVE correlation. If they trend DOWNWARD, that is NEGATIVE correlation. If there is no consistent up-or-down pattern, there is NO correlation.
A line of best fit is a single straight line drawn to follow the overall trend as closely as possible, used to estimate a y value for a given x (or vice versa) even for x values not directly in the data.
A scatter graph plots (1, 9), (2, 7), (3, 5), (4, 3). Describe the correlation.
- As x increases (1 to 4), y consistently decreases (9 to 3).
Answer: Negative correlation.
- What would a scatter graph with NO correlation look like, compared to one with strong positive correlation?
Common misconceptions and how to address them
MisconceptionA bigger sector frequency always means a bigger sector angle number in degrees than in percent (confusing the 2 scales).
Why it happens: Since both angle and percentage are calculated from the same frequency-over-total fraction, students sometimes expect the 2 resulting numbers to match, rather than realising they are scaled to 2 DIFFERENT totals (360 vs 100).
How to address it: Calculate both side by side for the same category (e.g. 25% is 90 degrees, not 25 degrees), making the different scaling totals (360 for angles, 100 for percentages) explicit.
MisconceptionCorrelation means 1 variable directly CAUSES the other to change.
Why it happens: 'They move together' is easily misread as 'one makes the other happen', when in fact a scatter graph only shows an association, not a cause.
How to address it: Give a concrete counterexample: ice cream sales and drowning incidents both rise in summer (positive correlation), but ice cream does not CAUSE drowning; a 3rd factor (hot weather) affects both.
MisconceptionThe line of best fit must pass exactly through as many data points as possible.
Why it happens: Students think of the line as 'connecting the most dots', similar to a dot-to-dot puzzle, rather than as a calculated best overall summary of the trend.
How to address it: Show that the real line of best fit (calculated by least squares) usually passes through NONE of the individual points exactly, it balances the distances to ALL of them at once, rather than hitting the most points.
Guided practice (with answers)
1. A survey of 60 people found 15 prefer cats. Find the sector angle for cats.
Answer: 90 degrees, because (15/60) x 360 = 90.
2. A survey of 60 people found 15 prefer cats. Find the percentage that prefer cats.
Answer: 25%, because (15/60) x 100 = 25.
3. A survey of 24 students found 6 walk to school. Find the sector angle for walking.
Answer: 90 degrees, because (6/24) x 360 = 90.
4. A scatter graph plots (1, 2), (2, 4), (3, 6), (4, 8). Describe the correlation.
Answer: Positive correlation, because y consistently increases as x increases.
5. A scatter graph plots (1, 10), (2, 3), (3, 15), (4, 6). Describe the correlation.
Answer: No clear correlation, because y goes up and down with no consistent pattern.
6. Why can't a pie chart have sector angles that add up to more than 360 degrees?
Answer: Because 360 degrees is the whole circle, representing the entire data set; the sectors can only divide up that fixed total, never exceed it.
Independent practice worksheets
Practise pie chart angle/percentage calculations and scatter graph correlation with computed, never-wrong answer keys.
Differentiation
- For pie charts, always write the fraction (frequency / total) first, before multiplying by 360 or 100, so the underlying proportion is visible.
- Use a totals-check every time (do the angles add to 360? do the percentages add to 100?) to self-verify.
- For scatter graphs, physically trace a finger along the general trend before naming the correlation type.
- Start with clearly positive and clearly negative examples before introducing ambiguous 'no correlation' data sets.
- Given a pie chart's sector angle, work backward to find that category's frequency (needing the total).
- Investigate why a pie chart with only 2 categories always has supplementary-looking sectors (their angles add to 360, and if one is p%, the other is (100-p)%).
- Estimate a value using a scatter graph's line of best fit for an x value OUTSIDE the plotted data range, and discuss why that estimate (extrapolation) is less reliable than one within the range.
- Research a real correlation-is-not-causation example and explain the likely 3rd factor connecting the 2 variables.
Assessment: exit ticket
A three-question exit ticket sampling pie chart angles, percentages and scatter graph correlation.
1. A survey of 50 people found 20 prefer coffee. Find the sector angle for coffee.
Answer: 144 degrees, because (20/50) x 360 = 144.
2. A survey of 50 people found 20 prefer coffee. Find the percentage that prefer coffee.
Answer: 40%, because (20/50) x 100 = 40.
3. A scatter graph plots (1, 8), (2, 6), (3, 4), (4, 2). Describe the correlation.
Answer: Negative correlation, because y consistently decreases as x increases.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 pie chart angles (section 1), Lesson 2 pie chart percentages (section 2), Lesson 3 scatter graphs (section 3) plus the exit ticket.
- Pie charts and scatter graphs are combined into 1 unit deliberately, following the same 'group closely related bullets' judgment call earlier batches used (e.g. batch 6 combining 3 angle-relation types into 1 unit): both sit under the KS3 Statistics strand's 'construct and interpret charts' family of bullets, and combining them keeps the unit count for this closing batch manageable.
- Bar charts and pictograms (also named in the same curriculum bullet as pie charts) are deliberately NOT rebuilt here: the site already has extensive generic (non-KS3-tagged) bar chart and pictograph content (lib/content_data.ts and others), so building KS3-specific duplicates would be thin, repetitive content rather than a genuine gap.
- The pie-chart section ships WITHOUT a drawn pie-chart figure: no figure kind in components/MathFigures.tsx or components/StandardFigures.tsx currently draws a pie/sector chart (the closest, scatterPlot and functionGraph, are Cartesian-plane figures, not circular). Every pie-chart item is still fully computed (the angle/percentage arithmetic, the genuinely assessed skill), just described via a frequency table instead of a rendered circle, following the same precedent batch 6 set for its angle-facts unit.
- The scatterPlot figure in section 3 reuses the site's real least-squares line-of-best-fit calculation (components/StandardFigures.tsx's ScatterPlot, the same engine every /us/standards scatter-graph page uses); the matching worksheet's own line-of-best-fit items are constructed from points placed EXACTLY on a chosen line, so the least-squares fit reproduces that exact line and the worksheet's stated estimate is always precisely correct, never an approximation.