Overview
What you’ll learn
- Organise raw data into tally charts and two-way tables, and read information back out of them.
- Calculate mean, median, mode and range from lists and frequency tables, and compare two data sets using both average and spread.
- Estimate the mean of grouped data with midpoints, name the modal class, and construct pie charts with correct angles (Extended for grouped work).
- Use scatter diagrams and lines of best fit for prediction, and read medians, quartiles and frequencies off cumulative frequency curves and histograms (Extended).
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 9.1Collect, classify and tabulate statistical data, including tally charts and two-way tables.
- 9.2Read, interpret and compare sets of data from tables and diagrams, recognising the limitations of conclusions. (Extended also compares measures of spread.)
- 9.3Calculate the mean, median, mode and range of ungrouped data, and the mean, median and mode from an ungrouped frequency table.
- 9.4Draw and interpret bar charts (composite and dual), pie charts, pictograms, stem-and-leaf diagrams (ordered, with a key) and simple frequency distributions.
- 9.5Draw and interpret scatter diagrams; describe positive, negative or zero correlation; draw and use a line of best fit by eye across the whole data set.
Extended — Extended candidates only
- 9.3Calculate an estimate of the mean of grouped data using class midpoints, and identify the modal class; use quartiles and the interquartile range. (Extended)
- 9.6Draw and interpret cumulative frequency diagrams, reading estimates of the median, quartiles, interquartile range and percentiles from the curve. (Extended)
- 9.7Draw and interpret histograms, including those with unequal class intervals using frequency density. (Extended)
In 30 seconds
- 1Mean = total ÷ number of values; median = middle value after sorting; mode = most frequent; range = largest − smallest.
- 2From a frequency table: total = Σ(value × frequency), then mean = total ÷ Σfrequency.
- 3Grouped data only gives an estimated mean: use each class midpoint × frequency (Extended).
- 4Pie chart angles: angle = (frequency ÷ total) × 360° — and all angles must add to 360°.
- 5Stem-and-leaf keeps every original value; the key (e.g. 2 | 1 means 21) is compulsory.
- 6Correlation describes the pattern: rising → positive, falling → negative, no pattern → zero. A line of best fit predicts inside the data range reliably (Extended reads beyond with care).
- 7Interquartile range = upper quartile − lower quartile; read both off the cumulative frequency curve at ¾n and ¼n (Extended).
- 8For unequal class widths, frequency = frequency density × class width — comparing bar heights instead of densities is wrong (Extended).
Notes · 1 of 4
Collecting and classifying data
Tally charts and two-way tables organise the mess.
Raw data arrives unordered; a tally chart counts it in fives, and a two-way table cross-splits it by two categories — for example transport type against year group. Two-way tables are read with simple subtraction: if 80 pupils travel and 30 walk (18 of them boys), then 12 girls walk. Filling a missing cell always uses a row or column total, not a guess.
When comparing two data sets, quote both an average (usually the median or mean) and a measure of spread (the range, or the interquartile range at Extended). “Class A scored higher on average, but class B’s scores were more consistent” is a complete comparison; quoting the mean alone is not. Conclusions from small or biased samples get stated with their limits — a survey of one class cannot speak for a whole school.
Notes · 2 of 4
Averages and spread
Mean, median, mode and range — from lists, tables and groups.
The mean shares the total equally: add every value, divide by how many. The median is the middle value after sorting — for an even count, the mean of the middle two. The mode is the most frequent value (a list can have two, or none). The range measures spread: largest minus smallest. For the list 4, 7, 4, 9, 6, 4, 8: mean = 42 ÷ 7 = 6, median = 6 (the fourth of seven sorted values), mode = 4, range = 9 − 4 = 5.
A frequency table saves listing: multiply each value by its frequency, add, then divide by the total frequency. The median position from a table is found by walking through the running total until half the total frequency is reached. Extended: grouped data hides the individual values, so the mean is estimated with class midpoints — Σ(midpoint × frequency) ÷ Σfrequency — and the modal class is the class with the highest frequency, not a single value.
Notes · 3 of 4
Charts and diagrams
Pie charts, bar charts and stem-and-leaf.
A pie chart turns frequencies into angles: angle = (frequency ÷ total) × 360°, and every sector’s angles must total 360°. Bar charts compare categories with heights (composite bars stack parts; dual bars place two sets side by side — label which is which). A stem-and-leaf diagram keeps every original value visible: stems carry the tens, leaves the units, in ascending order, always with a key such as 2 | 1 means 21. The median and range can be read straight off the ordered leaves.
Reading questions go both ways: building a chart from data, or recovering the data from the chart. For a pie chart, one known frequency with its angle fixes the scale — people per degree — and every other sector follows. Check the scale on any pictogram too: one symbol may stand for several units.
Notes · 4 of 4
Scatter diagrams, cumulative frequency and histograms
Prediction, quartiles and honest bar heights.
A scatter diagram plots pairs of values; the overall drift is the correlation — up-right (positive: revision time against marks), down-right (negative: screen time against marks), or none. The line of best fit is drawn by eye through the middle of the points (balancing them above and below), and predictions are made by reading the line at a given x — reliable inside the plotted range, risky outside it.
Extended. A cumulative frequency diagram plots running totals against upper class boundaries; the curve always rises to the total. Read the median at half the total frequency, the quartiles at a quarter and three-quarters, and subtract for the interquartile range — quartiles from a curve are estimates, so small reading tolerances are expected. A histogram with unequal class widths must use frequency density (frequency ÷ class width) for bar heights, or wide classes dominate unfairly; recover a frequency as density × width.
Revise
Key terms
The words examiners expect you to use precisely.
- Mean
- Total of the values divided by how many there are.42 ÷ 7 = 6 for 4, 7, 4, 9, 6, 4, 8.
- Median
- The middle value once the data is sorted.6 for the seven values above.
- Mode
- The most frequent value (or values).4 appears three times.
- Range
- Largest value minus smallest.9 − 4 = 5.
- Pie chart sector angle
- (frequency ÷ total) × 360°.24 of 60 people → 144°.
- Two-way table
- A grid crossing two categories with row and column totals.Transport type against year group.
Show 2 more key termsShow fewer key terms
- Interquartile range (Extended)
- Upper quartile minus lower quartile.A spread measure that ignores outliers.
- Frequency density (Extended)
- Frequency ÷ class width; a histogram’s bar height.12 items over a width of 10 → 1.2.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Dividing by the wrong count when finding a mean.
The divisor is the number of values (or total frequency), not the number of distinct values.
Instead: For a frequency table, divide Σ(value × frequency) by Σfrequency.
Mistake: Finding the median before sorting the list.
The median is a position in the ordered data.
Instead: Sort first, then take the middle (or mean of the middle two).
Show 10 more mistakesShow fewer mistakes
Mistake: Using class boundaries instead of midpoints for a grouped mean.
The mean needs a representative value per class.
Instead: Midpoint × frequency, summed, divided by the total frequency.
Mistake: Plotting frequencies as pie chart degrees.
Degrees come from the fraction of 360°, not the raw count.
Instead: Angle = (frequency ÷ total) × 360°.
Mistake: Comparing two data sets by average alone.
Equal averages can hide very different spreads.
Instead: Quote an average and a spread (range, or IQR at Extended).
Mistake: Reading a histogram with unequal widths as if the bars were frequencies.
Taller-but-narrower bars can hold fewer items.
Instead: Height = frequency density; frequency = density × width.
Mistake: Extrapolating a line of best fit far outside the data.
The trend is only established where data exists.
Instead: Predict confidently inside the plotted range; flag outside predictions as unreliable.
Mistake: Stating a single modal value for grouped data.
Grouped data has classes, not exact values.
Instead: Name the modal class: “5 < h ⩽ 10”, not 8.
Mistake: Claiming a correlation proves one variable causes the other.
Association is not causation; a hidden third factor may drive both.
Instead: Describe the pattern and its strength; keep conclusions tentative.
Mistake: Dividing the total by n − 1 out of calculator habit.
The mean of a listed data set divides by how many values there are.
Instead: Five values totalling 70 have mean 70 ÷ 5 = 14.
Mistake: Naming the middle class of a grouped table as the median.
The median sits at position n⁄2 in the cumulative count, which may not be the middle row.
Instead: Cumulative frequencies decide: 80 values median at the 40th, found in whichever class that reaches.
Mistake: Confusing a data count with a pie-chart angle.
15 people out of 60 is a quarter, which is 90°, not 15°.
Instead: Angle = (count ÷ total) × 360°.
Check yourself
Quick quiz
24 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Calculate the mean of a list
The mean of 3, 7, 8, 10 and 12 is
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Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 18 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Topic 18: two-way tables, mean, median, mode and range from lists and frequency tables, pie chart angles, stem-and-leaf diagrams, scatter diagrams and lines of best fit, estimated means of grouped data, cumulative frequency quartiles and histograms.
Mark scheme
PROTopic 18 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
When should I use the median instead of the mean?
When the data is **skewed** or contains outliers. One billionaire in a village drags the mean income up while the median barely moves, so the median describes the “typical” person better. If a question shows a very lopsided list, it is usually hinting that the mean is misleading.
Why is the grouped-data mean only an “estimate”?
Grouping destroys the individual values — we only know how many fall in each class. Replacing each class with its **midpoint** assumes the values spread evenly inside it, which is rarely exactly true. The result is close, but the honest label is an estimate (and exam answers should say so).
How do I read quartiles from a cumulative frequency curve?
Find the total frequency n, then read the curve at n⁄4 (lower quartile), n⁄2 (median) and 3n⁄4 (upper quartile): go up from the cumulative frequency axis to the curve, then across to the value axis. Curve readings are estimates — draw clean guide lines and accept that exam mark schemes allow a small tolerance.
What is the difference between a histogram and a bar chart?
A bar chart compares **categories** with gaps between equal-width bars. A histogram groups **continuous** data, bars touch, and — when class widths are unequal — the bar **height** is the frequency density (frequency ÷ width), not the frequency itself. That keeps area proportional to frequency.
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