Statistical Enquiry and Misleading Graphs
Statistical enquiry is the full cycle of carrying out a survey or investigation: posing a clear question, planning how to collect suitable data, collecting it, often from a sample rather than an entire population, processing and representing it in tables and charts, then interpreting the results to answer the original question. It also covers spotting when a graph misleads the reader, whether deliberately or not, by starting an axis away from zero, using inconsistent scales, using 3D effects, or basing a claim on an unrepresentative sample.
Before you start
Make sure you're comfortable with these topics first:
Method
- Start any statistical enquiry with a clear, answerable question, and decide what data needs to be collected to answer it.
- Decide between using the whole population (every member of the group being studied) or a sample (a smaller, representative part of it) - a sample is used when the population is too large or impractical to survey completely.
- Check a sample is large enough, and chosen fairly without bias, before trusting conclusions drawn from it.
- When reading a graph, check the axis labels, the scale, and where the scale starts, before comparing bars, sectors or points by eye.
- Look out for a vertical axis that does not start at zero - this exaggerates small differences between bars, making them look much larger than they really are.
- Look out for inconsistent scale intervals, missing labels, or images and 3D effects used in place of a simple bar - all of these can distort how big a value looks compared to how big it actually is.
- When interpreting results, only draw conclusions that the data actually supports, and be cautious about generalising from a small or biased sample to a whole population.
Worked example
A company's advertisement shows a bar chart of 'customer satisfaction score out of 10' for two products. The vertical axis is drawn starting at 7 and ending at 8, with Product A's bar at a score of 7.8 and Product B's bar at a score of 7.4, so Product A's bar looks about four times taller than Product B's. Explain why the chart is misleading, and describe how it should be redrawn.
- Find the actual difference between the two scores: 7.8 - 7.4 = 0.4 out of 10, a genuinely small difference.
- Notice the vertical axis only covers the range 7 to 8, a span of just 1, instead of starting at 0.
- Because the whole visible axis only spans 1, the gap of 0.4 fills nearly half of that visible height, making it look enormous compared with the true 0.4-in-10 difference.
- This is why Product A's bar looks about four times taller, even though its score is only about 5 percent higher than Product B's (0.4 out of 7.4).
- To redraw it fairly, the vertical axis should start at 0 and run at least up to 10, the maximum possible score, so that equal amounts on the scale represent equal amounts of satisfaction and the bars' heights look almost the same.
Practice questions
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Q1A head teacher wants to know how all 900 pupils at her school travel to school. She asks 60 pupils, chosen at random from across all year groups. Is this data collected from a population or a sample?Show answer
Answer: A sample (60 out of the 900 pupils, not every pupil)
Q2A researcher wants to find out how much exercise pupils at a school do each week, so he surveys only pupils leaving the school gym after football club. Explain why this sample might give a biased result.Show answer
Answer: Pupils leaving football club are likely to do more exercise than the average pupil at the school, so this sample is not representative of the whole school - the results would probably overestimate how much exercise pupils typically do
Q3A bar chart comparing the height of two buildings has its vertical axis starting at 40 metres instead of 0 metres. Building X, 42 m, looks about six times taller than Building Y, 41 m, on the chart. Explain why this is misleading.Show answer
Answer: The axis does not start at zero, so only the small difference between the two heights, 1 m, is stretched to fill the whole visible chart, making a tiny real difference look huge; in reality Building X is only about 1 m, roughly 2 percent, taller than Building Y
Q4A line graph's horizontal axis shows the years 2020, 2021, 2022, 2025, 2026, spaced out with EQUAL gaps between each label. Explain why this is misleading.Show answer
Answer: The gaps between the years are not equal in real time (three years between 2022 and 2025, but only one year between the other labels), yet they are drawn with equal spacing, which makes the data look like it changes at a steady, even rate when it does not
Q5A magazine claims '90 percent of people prefer Brand A', based on a survey of just 10 people. Explain why this claim should not be trusted.Show answer
Answer: A sample of only 10 people is far too small to represent everyone; asking a different 10 people could easily give a very different result, so the true percentage across the whole population could be very different from 90 percent
Q6A charity claims that donations have 'doubled' this year, showing a bar chart where this year's bar is drawn twice as tall as last year's. Last year's donations were 40,000 pounds and this year's were 52,000 pounds. Use a calculation to show whether the chart is drawn fairly.Show answer
Answer: This year's donations, 52,000 pounds, are 1.3 times last year's, 40,000 pounds - a 30 percent increase, not double - so a bar drawn twice as tall is not a fair representation of the true increase
Q7Put these four stages of a statistical enquiry into the correct order: 'Interpret the results and answer the original question', 'Collect the data', 'Pose a clear question to investigate', 'Choose how to represent the data in tables or charts'.Show answer
Answer: Pose a clear question to investigate; Collect the data; Choose how to represent the data in tables or charts; Interpret the results and answer the original question
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A gym advertises that 'membership numbers have rocketed' using a bar chart with a vertical axis that runs from 500 to 560 members, with no zero shown. January's bar shows 510 members and December's bar shows 555 members. (a) Calculate the actual percentage increase in membership from January to December. (b) Explain why the advert's bar chart is likely to exaggerate this increase to a reader.
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A school surveys pupils' opinions on a new lunch menu by asking the 15 pupils on the School Council. 12 of the 15 say they like the new menu, and the school reports this as '80 percent of pupils like the new menu' in the newsletter, without further explanation. (a) Calculate the percentage of the surveyed pupils who said they like the new menu, and confirm whether 80 percent is correct for this sample. (b) Explain two reasons why the newsletter's claim about 'pupils' in general might still be misleading.
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A company's poster uses a pictogram to show its coffee sales, where 1 cup symbol represents 1,000 cups sold. Monday's sales are shown using 3 cup symbols, but the poster draws Monday's symbol at DOUBLE the width and height of the symbol used for every other day. Tuesday's sales are shown using 3 normal-sized symbols too. Explain why a reader would wrongly conclude that Monday's sales were higher than Tuesday's, even though the data shows they were equal.
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Free printable worksheet
Want more practice on paper? Download the statistical enquiry and misleading graphs worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 38 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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