Sampling, bias and outliers

Nearly every mark lost on this topic comes from one belief: that a bigger sample is a better sample. It is not. A badly chosen sample of 5000 is worse than a well chosen sample of 50, because it is confidently wrong.

Start with one value out of place

Nine students recorded how long they read for. Eight of them are fixed. Move the ninth, and watch the two markers.

The ninth student read for 34 min

–mean

–median

–gap between them

Push it out to 95 and back. Nothing is stored and nothing is sent.

The mean uses every value, so one extreme number drags it. The median only cares about position, so with these nine values it can only ever land on one of three numbers however far you push the ninth. That is the entire reason a median is reported alongside a mean.

Push one value out to 95 and the mean chases it while the median barely moves. That is the whole argument for quoting both.

Population and sample

The population is every member of the group your question is about. If the question is about Year 12 at your school, the population is those students, not everyone in the country. A population does not have to be large. It has to be all of them.

A sample is the part you actually collected from. The moment you take one, everything you say about the population becomes an estimate.

The five techniques

Simple random. Everyone is numbered and numbers are drawn at random. Every person, and every possible group, has an equal chance.
Systematic. A random start, then every nth name down an ordered list.
Convenience. Whoever was easy to reach. Fast, and almost always biased.
Quota. A fixed number from each group, chosen however you like inside the group.
Stratified. Split into groups, then a random sample from each, in proportion to its size.

Quota and stratified are not the same. Both fix how many come from each group. Only stratified chooses randomly inside the group. Somebody standing in the playground asking until they have twenty of each is doing quota sampling, and it is biased by who happened to be standing there.

Worked example: a stratified sample

A school has 300 students in Year 11, 200 in Year 12 and 100 in Year 13. A stratified sample of 60 is taken. How many come from Year 12?

total= 300 + 200 + 100 = 600 the whole population first Year 12= 200600 = 13 what fraction of the school it is sample= 13 × 60 = 20 the same fraction of the sample

So 20 students are chosen at random from Year 12. That last phrase is what makes it stratified rather than quota.

Outliers

A value is treated as an outlier if it falls below Q₁ − 1.5 × IQR, or above Q₃ + 1.5 × IQR, where IQR = Q₃ − Q₁.

Worked example: testing a value

For a set of data, Q₁ = 24 and Q₃ = 36. Is a value of 58 an outlier?

IQR= 36 − 24 = 12 always the interquartile range first upper= 36 + 1.5 × 12 one and a half IQRs above Q₃ = 36 + 18 = 54 58 > 54, so it is an outlier by this rule. 52 would have failed and still looked large, so compare rather than eyeball

What the rule does not say. It does not say the value is a mistake, and it does not give you permission to delete it. A correct extreme value is information. Removing an awkward number quietly changes your results and hides that you changed them.

Your turn

1. A school has 450 students in Year 11, 300 in Year 12 and 150 in Year 13. A stratified sample of 60 is taken. How many come from Year 11?

2. For a set of data, Q₁ = 18 and Q₃ = 30. Find the value above which a result is treated as an outlier.

3. A student surveys 500 people outside one Bangkok shopping centre and writes: "Because the sample is large, it is representative of the country." What is the mistake?

500 is not a large enough sample for a country. A large sample from one place is still only representative of that place. The student should have used a systematic sample instead. Shopping centre data is continuous, so it cannot be used.

4. A student's height is recorded as 17.2 cm. Every other height is between 150 cm and 190 cm. What should happen?

Delete it, because it is an outlier. Keep it unchanged, because removing data is never allowed. Investigate it. It is almost certainly 172 cm mistyped, and should be corrected. Replace it with the mean of the other heights.

Where the marks go

Stratified sample sizes are nearly free marks, and students lose them by rounding badly. Work with the fraction, not a decimal.

On outliers, the calculation is one mark and the judgement is the other. An answer that says "58 is an outlier so I removed it" scores worse than one that says "58 is an outlier, so I checked it, and it is genuine".

On bias, examiners want the mechanism: who was more likely to be asked and who could not be asked at all. "The sample is too small" earns nothing, and it is the most common answer given.

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