KS3 · Maths

Planning and drawing conclusions from a statistical inquiry

Ask a room full of gym-goers how much people exercise and you get a perfectly accurate answer, just not about the group you wanted.

Maths · Statistical inquiry

Is this decision sound?

Every investigation is a pile of decisions: who to ask, what to calculate, what to claim. Pick a decision, then pick the box that fits it best.

Still to sort

Sound decision (0)

Nothing here is likely to make the answer wrong.

Where the line is: A sound decision fits the group being studied and claims no more than the data can show.

Biased sample (0)

The people asked don't represent the group you want to know about.

Where the line is: Bias is about WHO was asked. If the people are fine but the average or the claim is the problem, it belongs in another box.

Unsuitable measure (0)

The average doesn't make sense for this data or this job.

Where the line is: The measure is the problem. The people asked could be perfectly fine.

Over-generalised conclusion (0)

The claim reaches further than the sample or the data can support.

Where the line is: The data may be fine. It's the claim that has run too far ahead of it.

8 of 8 still to sort.

Eight decisions from eight investigations. Sort each one, then read why it belongs there.

Maths · The data handling cycle

Put the investigation in order

the order an investigation usually runs in

1 · First step6 · Last step

  1. Draw conclusions and present the data to illustrate them

  2. Organise and present the data, for example in a table or graph

  3. Reach outcomes, decisions or actions that address the original problem

  4. Start with a question or problem to investigate

  5. Analyse it with methods suited to the problem, such as averages, the range or a graph

  6. Collect data, being careful not to cause unnecessary bias

Maths · Collecting data

Which tool does which job?

Tick every cell you think is a yes, then check. The question you're investigating decides the tool.

Questionnaire
Interview
Tally
Scales, stopwatch or ruler
Existing data set

Population and sample

PopulationvsSample

These two get mixed up all the time. Start with who counts as the population.

Focus

Who counts?

Population

The whole group your investigation intends to study, for example all the pupils in one school.

Sample

A subset of the population: only some of that group.

The insight

The investigation defines the population. It isn't necessarily everyone in a country or the world.

When do you use it?

Population

Collecting from every member is often impractical.

Sample

Used instead, when collecting from the whole population is impractical.

What are we assuming?

Population

Its summaries are what you really want to know about.

Sample

Its summaries are assumed to be approximately similar to the population's.

How big?

Population

As big as the whole group.

Sample

No right or wrong size, only pros and cons. Bigger is harder to collect and ends up pretty much the population. Too small may not give enough data for a reliable conclusion.

Maths · Reading data

Tickets sold at the school fair

Tap a bar to read its value. Look at the bar heights first, then read the scale on the left.

505254565860MonTueWedThuFriDayTickets sold

Notice where the scale starts: at 50, not 0. Tap a bar to see its value.

Mean

57.4

Median

58

Mode

55

Range

5

Watch out: Monday (60) looks about twice as tall as Tuesday (55), but 60 is only a little bigger than 55. Start the axis at 0 and the bars look almost the same height. A scatter graph is different: starting both axes at zero can squash the points into a small space, so a scale that starts above zero can show the shape of the data without distorting it.

Maths · Drawing conclusions

What does a summary really tell you?

Two towns record their maximum temperature every day for a summer. Town A has the higher median maximum temperature and the smaller range.

Which is closest to what you think this shows?
How sure are you?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Planning and judging a statistical investigation

What you need to know

  • An investigation usually runs question, collect, organise and present, analyse, conclude, then reach an outcome that addresses the original problem.
  • Your question decides what data you need: integers like ages, decimals like weights, or frequencies, which are counts.
  • Match the tool to the data: questionnaire for short questions, interview for detail, tally for counting, a stopwatch or ruler for measuring.
  • Have a goDani says: “To find everyone's height, just ask them. It's quicker than measuring.” Which tool beats asking, and what could go wrong with asking?

    A ruler measures it directly; people may not know their height or may not answer honestly.

    Measuring tools collect lengths directly, while asking leans on memory and honesty, which is the trap in Dani's quicker plan.

  • The population is the whole group you're studying; a sample is a subset, used when collecting from everyone is impractical.
  • A sample that doesn't represent the population causes bias, so sample from every part of it, or at least watch for bias.
  • Have a goZara asks 40 pupils at her school about lunch, then announces: “My population is everyone in the UK!” What is her population really, and what are the 40?

    Her population is all the pupils at her school; the 40 are a sample of it.

    The population is whatever group the investigation intends to study, not necessarily everyone in a country or the world.

  • Summaries like the mean, median, mode and range sum up a data set, but the mean isn't always the best average.
  • A bar chart axis that doesn't start at zero can distort comparisons, because bars grow from the axis start.
  • Have a goA bar chart's vertical axis starts at 30. One bar shows 45 and another shows 40. Will the 45 bar look about the same height as the 40 bar, or noticeably taller?

    Noticeably taller: about 1.5 times as tall (15 units above the axis start against 10).

    Bars are drawn up from where the axis starts, so a start above zero makes a small difference look big.

  • Pick the graph for the job: bar or pie charts for categories, line graphs for change over time, scatter graphs for relationships.
  • Conclusions only hold for the sample, or a population it fully represents, so don't stretch one context to cover all.
  • Use summaries, graphs and context together: averages can hide details, and a good conclusion can lead to action.

The big picture

A statistical conclusion is only as good as the choices behind it: who you ask, how you collect the data, which summary and graph you pick, and how far you let the conclusion reach.

Key points

1The population is the group your investigation is about; a sample is the part of it you actually collect data from.
2A biased sample, an unsuitable average and a stretched conclusion are three different flaws. Work out which one you're looking at.
3Match the average to the data: categories get a mode, and a few extreme values can drag the mean.
4Check the scale before you trust a graph, and choose the graph that suits the job.
5Draw conclusions from summaries, graphs and context together, and only as far as the sample reaches.

Worked example

Problem

Maya's school council wants to know how pupils at their school usually travel to school. Plan Maya's investigation from question to outcome, then say what her results allow her to conclude. Of the 100 pupils she sampled, 34 walk, 29 take the bus, 22 come by car and 15 cycle.

⚠ Watch out

Treating a result as if it speaks for everyone. A survey of children tells you what those children like. It doesn't tell you what all ages like. It's an easy trap because the numbers really are accurate. They just only describe the people who were actually asked.

🧠

Memory hook

Accurate answer, wrong group. Before you trust a result, ask: who was asked, which measure and graph, and how far does the claim reach?

✓

Check yourself

Pick any survey result you've seen this week. Can you say who was asked, how the data was collected, and who the conclusion really covers?

Flashcards

(15)
What are the steps of the data handling cycle?
Usually: start with a question or problem; collect data, avoiding unnecessary bias; organise and present it; analyse it with suitable methods; draw conclusions; reach outcomes, decisions or actions that address the original problem.
What kinds of data can an investigation collect?
Integers (such as ages or scores), decimals (such as distances or weights) or frequencies (counts, such as the number of people who travel by bus).
Which tool suits which job?
Questionnaire: a series of short questions. Interview: longer, more detailed answers. Tally: counting while you collect. Scales, stopwatch or ruler: weights, times or lengths.
Why time people solving a puzzle rather than ask them how long it took?
Timing measures it directly. If you ask, people may not have timed themselves, or may not answer honestly.
When might you use data somebody else has already collected?
When it's available, for example large data sets from gov.uk or the Office for National Statistics. It can be quicker, easier or more accurate, or the only practical option.
What is the difference between a population and a sample?
The population is the whole group the investigation intends to study (such as all pupils in one school). A sample is a subset of it.
How big should a sample be?
There's no right or wrong answer, only pros and cons. Bigger is harder to collect and ends up pretty much the population. Too small may not give enough data for a reliable conclusion.
What is bias in a sample, and how do you avoid it?
Bias comes from a sample that doesn't represent the population. Avoid it by sampling from every year group, or surveying where both gym members and non-members go. At least be aware that bias could be there.
What does a statistical summary do, and what is the range?
It sums up a data set in a small amount of information, using averages (mean, median, mode) and spread (the range: highest value minus lowest value). That makes comparing easier than reading a long list.
Why isn't the mean always the best average?
You can't calculate it for non-numerical data like colours (use the mode). A few extremely high values can drag it above what most of the data shows. And the best average depends on the job: a shoe shop needs the mode.
Must a bar chart's vertical axis start at zero?
There's no fixed rule, but starting above zero can distort how bars compare. On a scatter graph, starting above zero can make the shape of the data clearer without distorting it.
Which graph suits which job?
Bar or pie charts for frequencies of categories. Comparative bar charts for two groups with roughly equal totals. Pie charts for proportions when totals differ. Line graphs for change over time. Scatter graphs for whether a relationship exists.
What are pictograms and infographics good and bad at?
They give an instant sense of context, but their pictures aren't always to scale, so a bar chart may be better when accuracy matters. Graphs can also be chosen to emphasise a particular conclusion.
Who does a conclusion from a sample apply to?
Only the sample itself, or a population fully represented by it, and it's limited by any bias. A conclusion from one context doesn't necessarily apply to all contexts.
Why not rely on averages alone?
They give an impression and point you at what could be important, but details can be overlooked. Use summaries, graphs and context together.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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