Random samples from the same population give different results. That is sampling variation, not a mistake. Bigger samples vary less, so their results are more reliable.
Two polls of the same election, taken the same week, give 52% and 48% for the same party. Is one of them wrong?
How much can you trust one sample? This lesson takes many samples from the same school and compares them.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Two samples
TryTry it on the board and compare the result with your prediction.
Many samples
Each dot under the school is one sample's result. The orange dot is the latest.
Random samples from the same population almost always differ a little. This is sampling variation: each sample happens to contain different people, not a mistake.
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Take at least 10 samples of 20. How far apart are the lowest and highest results? Compare with your prediction.
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Variation is normal
Read Mia and Jack's results. Could both samples be fair?
Five samples of 20 gave these proportions who play sport: 0.45, 0.6, 0.55, 0.7, 0.5. What is the range of the results?
Mia and Jack compare their samples. Click the line where the reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
Change the size
Here is how to describe sampling variation.
Worked exampleBefore each step, say what you think comes next.
10 samples of 20:0.40 to 0.75The spread of results.
10 samples of 100:0.52 to 0.63Bigger samples: a narrower spread.
Both centred near 0.57Fair samples scatter around the population's value.
Bigger sample⇒more reliable estimateThe pattern.
Need a hint?
Move n up to 100 or more, then take 10 samples.
Need a hint?
Take 10 samples of 10, look at the spread, then make n 50 or more and do it again.
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Make the samples 100 or bigger and take at least 10. How does the spread of results compare with samples of 20?
Use the board. This ticks when you get there.
Now the samples measure mean hours of sleep. Take at least 10 samples of 50 or more. Do means vary less with bigger samples too?
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Variation
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
How big is big enough?
Write the smallest size that worked for you, and why you might not always choose a bigger one.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Try a size, take 10 samples, and read the spread in the panel.
Another hint
If the results are too spread out, make n bigger (that clears the results) and try again, or press Start again.
Find a sample size where the results of 10 samples all lie within 0.15 of each other.
Use the board. This ticks when you get there.
For teachers
Students take many random samples of a simulated school, see their results vary, and find that bigger samples vary less, for proportions and means (AC9M8ST03, nodes 1 and 2). About 40 minutes.
I can explain why two fair samples give different results.
I can describe how the spread of sample results changes with sample size.
I can choose a sample size that is big enough for a job.
Before class
Do the Run board yourself first. Students need sampling methods and describing a sample (last two lessons), and relative frequency from Chance.
The one idea: random samples differ by chance (sampling variation), and bigger samples differ less. The common mistake is deciding one of two different samples must be wrong (Mia and Jack).
The board plots each sample's result under the school, with the school's value dashed. Changing n clears the results so each spread belongs to one size. With 200 in the school, samples of 150 hardly vary (they hold most of the school).
Wider: polls' margin of error (about ±3% for 1000 people), and ensemble weather forecasts.
Running the lesson
Predict: ask what could make two fair samples differ.
Run: ask where the results sit compared with the dashed line.
Investigate: ask whether Mia or Jack chose unfairly.
Modify: ask what happened to the spread when n grew.
Make: ask why not always take the biggest sample.
The hard part (on purpose)
Seeing variation as normal, not an error. Ask: could both samples have been chosen fairly?
When students get stuck
A student thinks a different result means a mistake. Ask: Did the two samples hold the same students? Fix: Different students, different results: sampling variation.
A student thinks bigger samples are always closer. Ask: Is every big sample closer, or are they closer on the whole? Fix: Bigger samples vary less on the whole; any one can still be a little off.
A student confuses variation with bias. Ask: Do the results scatter around the dashed line, or sit to one side? Fix: Variation scatters around the truth; bias pushes to one side.
Stretch and support
Stretch: Record the range of 10 sample results for n = 10, 20, 40, 80 and 150. Plot range against n. What pattern do you see?
Support: A table to record n and the lowest and highest of 10 results, for three sizes.
Where it leads
This covers AC9M8ST03.
Where it leads: investigations with samples, where the uncertainty is reported (next lesson), and Year 9 and 10 statistics.
What each step is for
two samples: Committing to whether two fair samples agree shows whether they expect variation.
many samples: Taking many samples of one size and plotting their results shows the spread of sampling variation.
variation is normal: Read the spread of sample results; differences between fair samples aren't mistakes (Mia and Jack); bigger samples vary less.
change the size: Growing the sample and repeating shows the spread of results shrinks, for proportions and for means.
variation: Reading variation and judging reliability, mixed; treating a difference between fair samples as an error is the planted slip; reporting is the review.
how big is big enough?: Finding a sample size whose results stay within a target band connects size to reliability.
Now you can
Explain why two fair samples give different results.
Describe how the spread of sample results changes with sample size.
Choose a sample size that is big enough for a job.