Lesson 1 ~35 min Data Analysis C · Path +85 XP

Designing a Statistical Inquiry

Everything that can go wrong with a statistical investigation is decided before any data is collected. This lesson is about the part most people skip: turning a vague question into something that can actually be answered.

Today's hook: "Do students who sleep more get better marks?" sounds like a question you could investigate. But sleep more than what? Better marks in which subject? Measured how, and over what period? Until those are settled, no amount of data collection will produce an answer, because there is no question yet.

0/5QUESTS

Think First

warm-up

Take the question "is our canteen good value?" and try to write down exactly what you would measure. You will find you need several decisions: value compared with what, judged by whom, and over which items. Write down three decisions you had to make, and note that a different person could reasonably decide each one differently.

Record your answer in your workbook.

The Big Idea

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A statistical inquiry begins by converting a question of interest, which is usually vague, into an aim and a hypothesis, which are precise. The test of a good aim is simple: could two people, working separately from your written aim alone, collect comparable data? If not, the aim is not finished.

Question of interest → aim → hypothesis → variables → plan → data

Each arrow narrows something. The question says what you are curious about; the aim says exactly what will be measured, on whom, and when; the hypothesis states what you expect to find, in a form that the data can contradict. That last point matters: a hypothesis nothing could disprove is not a hypothesis, it is an opinion.

The two-people test

If two people cannot collect comparable data from your aim, it is not precise enough.

A hypothesis can be wrong

State it so the data could contradict it. If nothing could, it says nothing.

Decide before you collect

Changing what you measure after seeing some data invalidates the comparison.

What You'll Master

objectives

Know

The stages of a statistical inquiry, from question of interest to reported conclusion
The difference between an aim and a hypothesis
What makes a variable categorical, discrete or continuous

Understand

Why a hypothesis must be falsifiable to be useful
Why the design decisions must be fixed before any data is collected

Can Do

Turn a vague question of interest into a precise aim
Write a hypothesis the data could contradict
Identify the variables in an inquiry and classify each one

Words You Need

vocabulary

Question of interestThe everyday question that motivates an inquiry. Usually too vague to investigate as stated.

AimA precise statement of what will be measured, on whom, and over what period.

HypothesisA prediction, stated so that the data collected could contradict it.

VariableAnything that is measured or recorded and can differ between subjects.

Categorical variableOne whose values are labels, such as favourite subject or transport method.

Numerical variableOne whose values are numbers. Discrete if counted, continuous if measured.

From Question to Aim

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A question of interest is where you start, not what you investigate. Compare these three versions of the same curiosity.

Version	Statement	Problem
Question	Do students who sleep more do better at school?	Nothing here can be measured
Better	Is there a relationship between hours of sleep and test marks?	Which students, which test, over what period?
Aim	To investigate the relationship between average nightly sleep over one week and the mark achieved in the Year 10 mathematics half-yearly examination, for students in Year 10 at this school	None: this can be collected

Apply the two-people test to the final version. Two students given only that sentence would measure the same thing on the same group. That is the standard an aim has to meet.

Writing a Hypothesis That Can Be Wrong

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A hypothesis states what you expect, in a form the data could contradict.

Not a hypothesis: "Sleep affects marks in some way." Any result whatsoever is consistent with this, so collecting data cannot teach you anything.

A hypothesis: "Students who average more than eight hours of sleep per night will achieve a higher mean mark than students who average fewer than seven." Now a result where the two groups are equal, or where the second group scores higher, would contradict it.

Being contradictable is the whole point. An investigation is only informative if it had some chance of coming out the other way, and a hypothesis that nothing could disprove guarantees the investigation is uninformative before it starts.

Note

A hypothesis being contradicted is a successful investigation, not a failed one. You have learned something. Rewriting the hypothesis afterwards to match the data is not.

Identifying and Classifying the Variables

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Every inquiry has at least one variable, and naming them forces you to decide how each will be recorded.

Categorical variables take labels: favourite subject, method of travel to school, yes or no. These are counted into groups; you cannot average them.

Numerical discrete variables are counted: number of siblings, number of pets, goals scored. Values are whole numbers with gaps between them.

Numerical continuous variables are measured: height, time taken, hours of sleep. Between any two values there is another, and the precision is set by your instrument.

The classification is not bookkeeping. It decides what you may calculate and what you may draw: a mean is meaningless for a categorical variable, and a histogram needs a continuous one. Choosing the wrong display is usually a classification error made earlier.

The Plan, Written Before Any Data

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A plan records five decisions, and records them in advance.

Decision	Example
Who is the population?	All Year 10 students at this school
How will the sample be chosen?	Random selection from the roll, 40 students
What exactly is measured?	Sleep, self-reported to the nearest half hour, for seven nights
How is it recorded?	A shared spreadsheet, one row per student, columns for each night
What could go wrong?	Self-reported sleep may be over-estimated; some students may not complete all seven nights

The last row is the one most often left out, and the most valuable. Naming a weakness in advance lets you design around it, and lets you say honestly, in the report, what your conclusion cannot support.

Common Pitfalls

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Writing an aim that only the author could act on: "to see if sleep matters".

Fix: apply the two-people test. If a classmate could not collect comparable data from your aim alone, add what is missing.

Writing a hypothesis no result could contradict.

Fix: before collecting anything, write down what result WOULD contradict your hypothesis. If you cannot, rewrite the hypothesis.

Deciding what to measure after glancing at some of the data.

Fix: fix the design first. Choosing the comparison once you can see which one looks interesting is how a genuine-looking result gets manufactured from noise.

Watch Me Solve It · 3 examples

Watch Me Solve It · Turning a question into an aim

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PROBLEM

A student asks: "Is the school canteen too expensive?" Convert this into an aim that meets the two-people test, and state the variables.

1

Name what is vague

"Too expensive" compared with what, for which items, and judged against whose prices? Three separate decisions are hidden in three words.
2

Choose a comparison

Compare the canteen price of an item with the price of the same item at the nearest supermarket. That makes "expensive" measurable rather than a matter of opinion.
3

Fix the scope

Restrict to the twenty most frequently sold items, priced during one week, so that the comparison is finite and repeatable.
4

Write the aim and name the variables

Aim: to compare the price of the twenty most frequently sold canteen items with the price of the same items at the nearest supermarket, during the first week of term. Variables: item name (categorical), canteen price (continuous), supermarket price (continuous).

AnswerAn aim naming the items, the comparison and the period, with three variables identified

Watch Me Solve It · Making a hypothesis falsifiable

+15 XP per step

PROBLEM

A student writes the hypothesis: "Year 10 students use their phones a lot." Explain why this cannot be tested, and rewrite it.

1

Ask what result would contradict it

There is none. Any amount of phone use could be described as "a lot" or "not a lot" after the fact, so the data can never disagree with the statement.
2

Find the missing element

There is no threshold and no comparison. A testable statement needs one or the other.
3

Rewrite with a threshold

"The mean daily phone use of Year 10 students at this school exceeds three hours." A mean of 2.4 hours would now contradict it.
4

Check it can fail

State the contradicting result explicitly before collecting: a sample mean at or below three hours. Being able to write that sentence is the test.

Answer"The mean daily phone use of Year 10 students at this school exceeds three hours"

Watch Me Solve It · Classifying the variables

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PROBLEM

An inquiry records, for each student: their year group, how many siblings they have, their height, and whether they walk to school. Classify each variable and state one display suited to it.

1

Year group

Categorical. The values are labels, and although they look like numbers, averaging them is meaningless. A column graph of counts suits it.
2

Number of siblings

Numerical discrete. It is counted, so the values are whole numbers with gaps. A column graph or a dot plot suits it.
3

Height

Numerical continuous. It is measured, and between any two heights there is another. A histogram suits it, since the values group into intervals.
4

Walks to school

Categorical, with two values. A two-way table or a simple column graph suits it, and it is often used to split the other variables into groups for comparison.

AnswerCategorical, discrete, continuous, categorical

Brain Trainer · 4 problems

Brain Trainer · Sharpen each one

4 problems

Four quick problems. Work each one, then reveal the answer.

1 Is "do people like our uniform?" an aim?

Which people, and what counts as "like"? Nothing here can be measured.No, it is a question of interest
2 Classify the variable "time taken to run 100 m".

It is measured, not counted, and between any two times there is another.Numerical continuous
3 Is "our team will probably do well" a hypothesis?

No result could contradict it.No, it is not falsifiable
4 Which stage names what could go wrong?

It belongs in the plan, written before any data is collected.The plan

Complete in your workbook.

Multiple Choice · 5 questions

MC1

Aim or question

+10 XP

Which of these is a properly written aim?

MC2

Falsifiability

+10 XP

A hypothesis is useful only if:

MC3

Classifying a variable

+10 XP

The number of text messages a student sent yesterday is:

MC4

Why the plan comes first

+10 XP

Deciding which comparison to make AFTER looking at some of the data is a problem because:

MC5

The most-skipped step

+10 XP

Naming, in advance, what could go wrong with your data collection is valuable mainly because:

Short Answer · 3 questions

Design an inquiry

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SHORT ANSWER

A student is curious whether people who travel to school by car arrive less tired than people who walk.
(a) Write an aim that passes the two-people test.
(b) Write a falsifiable hypothesis, and state what result would contradict it.
(c) Name three variables in your design and classify each.

Write your working in your book.

Repair a poor design

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SHORT ANSWER

A student writes:
"Aim: to see if music helps studying. Hypothesis: music makes a difference. I will ask some friends."
(a) Identify three separate problems.
(b) Rewrite the aim and hypothesis.
(c) State one thing that could go wrong with the data collection, and how you would design around it.

Write your working in your book.

Explain a design principle

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SHORT ANSWER

Explain, in your own words, why each of these rules exists. Give a concrete example of what goes wrong when each is broken.
(a) A hypothesis must be able to be contradicted.
(b) The design must be fixed before any data is collected.
(c) The limitations must be stated in the report.

Write your working in your book.

Stretch Challenge · Two designs, one question

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CHALLENGE

The question is: does a school breakfast programme improve attendance?
(a) Design A compares attendance at schools that have a breakfast programme with attendance at schools that do not. State two reasons why a difference found this way might not be caused by the programme.
(b) Design B measures attendance at one school for a term before the programme starts and a term after it starts. State two reasons why a difference found this way might not be caused by the programme.
(c) Describe a third design that addresses the main weakness of both, and state what it still cannot rule out.

Quick Review

recap

Aim

What, on whom, over what period

Hypothesis

A prediction the data could contradict

Variables

Categorical, discrete or continuous

Plan first

Including what could go wrong

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Data Analysis C (Path) index · Maths subject page