A sample can only support a claim about the group it was taken from, and only as far as it represents that group. A question asking “is the conclusion valid?” is testing whether you can name the gap between the sample and the claim. This lesson closes statistics and distributions.
What makes a conclusion unsupported?
Four problems turn up again and again:
- The sample is not the population. Asking one class does not tell you about the whole school.
- The sample is biased. The way people were chosen favours one answer.
- The sample is too small for the claim being made.
- The data shows a link, not a cause. Two things moving together does not prove one causes the other.
How to test a conclusion, step by step
- Write down the claim and the group it is about (the population).
- Write down who was actually surveyed (the sample) and how they were chosen.
- Compare the two. Ask who could not have been included.
- Check the size of the sample against the strength of the claim.
- Rewrite the claim so the data does support it, for example by limiting it to the sample.
Worked example
A student stands in the canteen queue at lunch and asks 40 students their favourite food. Twenty-eight say nasi lemak. She concludes, “70% of all Malaysians prefer nasi lemak.”
Step 1, check the arithmetic: 28 ÷ 40 = 0.7, which is 70% of the sample. The calculation is fine.
Step 2, compare sample and population: the claim is about all Malaysians. The sample is 40 students from one school at one time of day.
Step 3, find the gaps: only students were asked, only at lunch, and only at one canteen where nasi lemak may be sold. Adults, other schools and other regions had no chance to be included.
Step 4, conclude: the claim is not supported. The data does support “70% of the 40 students asked in that queue chose nasi lemak”.
To make a stronger claim she would choose students at random from a full school list, across several year groups, and ask them all the same question.
The mistake to watch for
A common slip is to accept the claim because the percentage was calculated correctly.
Mistaken answer: “The conclusion is valid, because 28 ÷ 40 = 70%.”
Correct arithmetic does not make the claim valid. The question is whether the sample can speak for the population.
The correction is to separate two questions: “is the calculation right?” and “does the evidence reach as far as the claim?”. Many wrong answers come from answering only the first one.
Check yourself
Try these, then open each answer.
1. A survey of 50 people at a gym found that 12 exercise daily. A newspaper says, “24% of the town exercises daily.” Is this supported?
Show answer
12 ÷ 50 = 0.24, so 24% of the sample exercise daily. The claim about the town is not supported, because gym members are more likely to exercise than the rest of the town, so the sample is biased.
2. Describe one way to choose a fairer sample of the students in a school of 800.
Show answer
Number every student on the school list and use random numbers to pick the sample, for example 80 students. Every student has an equal chance of being chosen, so no single group is favoured.
3. A scatter graph shows that on days with more traffic jams, more umbrellas are sold. A writer says umbrellas cause traffic jams. Is that supported?
Show answer
No. The graph shows a positive correlation, not a cause. A third factor, rain, probably increases both umbrella sales and traffic jams. The evidence supports “the two tend to rise together” only.
Where this leads next
With all five lessons done, test yourself in the mixed practice set, which includes sample and average questions in the same format as this lesson. The statistics and distribution explorer also reminds you that correlation is not automatically causal. To review the order of study, return to the module overview, and revisit comparing spread and centre if your comparison sentences need work.
Wording a conclusion carefully is a skill that improves quickly when someone reads your answers and points at the exact phrase that overreaches. That is a regular part of online one-to-one Mathematics tuition.