How far can a sample mean stray?: Year 11 Extended sampling, correlation and regression

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

The mean of 4, 7, 10 and 15 is what?
Answer
9, because 36 ÷ 4 = 9.
From Year 11 Standard: a stratified sample of 40 is taken from 600 girls and 400 boys. How many girls are in it?
Answer
24, because 600 ÷ 1000 × 40 = 24.
From Year 11 Standard: a survey asks only the students who are leaving the canteen what they think of the canteen. What is wrong with it?
Answer
It leans: students who avoid the canteen are never asked, so the sample is biased whatever its size.

Words used in this lesson

sample mean: the mean of the values in one sample. For example: the mean of the bags picked by the farmers chosen.
population mean: the mean of every value in the population. For example: the mean of all eight farmers' bags.
stratum: one group within a population, sampled separately. For example: the farmers of one village.
in proportion: each group takes the same share of the sample as it has of the population. For example: a group with 300 of the 500 people supplies 30 of a sample of 50.
spread: how far apart the lowest and highest values are. For example: the gap between the lowest and highest possible sample mean.

Eight farmers, one buyer

A coffee cooperative has eight farmers. This morning they picked 2, 4, 6, 8, 8, 10, 12 and 14 bags, a mean of 8 bags each. A buyer cannot weigh every farmer's pile, so she will pick some farmers at random and use their mean.

Predict first: if she picks 4 farmers, what is the lowest her sample mean could be, and what is the highest? Then try 1, 2, 4 and 8 farmers on the first figure, and count how many different samples there are each time.

2lowest possible sample mean

14highest possible sample mean

8different samples

2samples with a mean within 1 of 8

Samples of 1: 8 possible samples, with sample means from 2 to 14. 2 of them are within 1 of the true mean 8.

The figure needs JavaScript. In words: eight farmers picked 2, 4, 6, 8, 8, 10, 12 and 14 bags, a mean of 8. Taking one farmer gives 8 possible samples with means from 2 to 14, 2 of them within 1 of 8. Taking two gives 28 samples with means from 3 to 13, 12 of them within 1 of 8. Taking four gives 70 samples with means from 5 to 11, 42 of them within 1 of 8. Taking all eight gives one sample, with mean 8.

Show me the steps

Sample of 1: each farmer on their own. 8 samples, with means from 2 to 14, and only 2 of them (the two farmers who picked 8) are within 1 of the true mean 8.
Sample of 2: 28 samples. The lowest mean is (2 + 4) ÷ 2 = 3 and the highest is (12 + 14) ÷ 2 = 13. 12 of the 28 are within 1 of 8.
Sample of 4: 70 samples. The lowest mean is (2 + 4 + 6 + 8) ÷ 4 = 5 and the highest is (8 + 10 + 12 + 14) ÷ 4 = 11. 42 of the 70 are within 1 of 8.
Sample of 8: everyone is in the sample, so there is 1 sample and its mean is exactly 8.

A sample mean wobbles; the population mean does not.

Different random samples of the same size give different sample means. List every possible sample and you can see the whole spread, from the lowest sample mean to the highest.
The larger the sample, the narrower that spread, until the sample is the whole population and there is one answer.
A stratified sample in proportion takes each group in its share, so no sample can be all one group. When the groups differ a lot, this narrows the spread of the sample means a great deal.

Now take twelve values that fall into two groups, a low group A and a high group B, and compare plans for a sample of four. Predict: will taking 1 from A and 3 from B give sample means centred on the population mean, or off it?

Taken from stratum A (the rest from B) 2

17lowest possible sample mean

43highest possible sample mean

30centre of all the possible sample means

495different samples

All four at random from the twelve: 495 possible samples, with sample means from 17 to 43, centred on 30.

The figure needs JavaScript. In words: stratum A is 14, 16, 18, 20, 22, 24 (mean 19) and stratum B is 36, 38, 40, 42, 44, 46 (mean 41), so the population mean is 30. Drawing four at random from all twelve gives 495 samples with means from 17 to 43, centred on 30. Taking 2 from each stratum gives 225 samples with means from 26 to 34, centred on 30. Taking 1 from A and 3 from B gives 120 samples with means from 32 to 39, centred on 35.5, which is above 30. Taking 3 from A and 1 from B gives means from 21 to 28, centred on 24.5.

Show me the steps

All four at random from the twelve: 495 samples, means from 17 to 43, centred on 30. The centre is right, but the range is wide, because a sample may be all stratum A or all stratum B.
2 from A and 2 from B, the split in proportion to the strata (both have 6): 225 samples, means from 26 to 34, centred on 30. The range is much narrower and the centre is still right.
1 from A and 3 from B: 120 samples, means from 32 to 39, centred on 35.5. Narrow, but centred away from 30: the sample over-represents B.
3 from A and 1 from B: means from 21 to 28, centred on 24.5. Narrow, centred away from 30 in the other direction.

“Narrow” is not the same as “right”. A plan can squeeze the sample means into a small range and still put the middle of that range in the wrong place. Two questions to ask of any plan: how wide is the spread of what it can give, and is that spread centred on the truth? Size and stratifying help the first; only a fair design helps the second. A fixed number from each group that is not in proportion is narrow but off-centre: a lean, not a wobble.

Worked examples

A population is 5, 9, 12 and 20. Find every sample of size 2 and its mean. What is the lowest and highest, and what is the mean of all the sample means?

Choosing 2 from 4 gives 6 samples: (5, 9), (5, 12), (5, 20), (9, 12), (9, 20), (12, 20).
Their means are 7, 8.5, 12.5, 10.5, 14.5 and 16. The lowest is 7 and the highest is 16.
The six means add to 69, so their mean is 69 ÷ 6 = 11.5, which is the mean of 5, 9, 12 and 20 (46 ÷ 4 = 11.5). Averaged over every sample, a random sample mean is right.

Stratum A is 4, 6 and 8 and stratum B is 20, 22 and 24. Compare a random sample of 2 from all six with a stratified sample of 1 from each, in terms of the lowest and highest sample mean.

Random from all six: the lowest pair is 4 and 6, mean 5; the highest pair is 22 and 24, mean 23. Range 5 to 23.
Stratified, one from each: the lowest is (4 + 20) ÷ 2 = 12 and the highest is (8 + 24) ÷ 2 = 16. Range 12 to 16.
The population mean is 14, and 12 to 16 sits right around it. Fixing one from each group removes the samples that are all low or all high.

Your go. The first step is given. A population is 3, 6, 9 and 12. List every sample of size 3 and find the lowest and highest sample mean.

Choosing 3 from 4 is the same as leaving one value out, so there are 4 samples.
Work out each sample's total and its mean, then pick the lowest and highest.

Check the answer

Leaving out 12, 9, 6 and 3 in turn gives totals 18, 21, 24 and 27 and means 6, 7, 8 and 9. The lowest is 6 and the highest is 9, a narrower range than for samples of size 2 (4.5 to 10.5). A common slip is to divide by 4, the population size, instead of 3, the sample size.

Check before you go on

1. The mean of a whole population is 50. A random sample of 3 from it has a mean of 56. Which statement is best?

The population mean must really be 56, because the sample was drawn at random. Nothing has gone wrong: this sample is one of many, and a gap of 6 from 50 can happen by chance. The sampling must have been biased, because a random sample's mean always equals the population mean. The sample tells us nothing, because only a sample much larger than 3 has a mean worth reporting.

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A population is 3, 6, 9, 12, 15 and 21. What is the highest mean a sample of 2 can have?

3. Region A has 300 farms with a mean yield of 20 bags. Region B has 100 farms with a mean yield of 40 bags. What is the mean yield over all 400 farms?

Level 2 · Building

4. A market has 9 stalls. How many different samples of 2 stalls can be chosen? (The order does not matter.)

5. Stratum A is 6, 8, 10 and 12. Stratum B is 30, 32, 34 and 36. A stratified sample takes 2 from A and 2 from B. What is the lowest mean it can have?

Level 3 · Stretch

6. A population of 5 values has a mean of 10. All 10 possible samples of size 3 are listed and the mean of each is found. What is the total of those 10 sample means?

7. To estimate the mean monthly spend of 6000 families, plan P takes 30 families at random from all 6000. Plan Q sorts the families into three districts and takes families from each district in proportion to its share, 30 in all. The three districts turn out to have almost the same mean spend and the same spread. Which statement is right?

Plan Q is much better, because stratifying always narrows the spread of the sample means a great deal. Plan Q always gives exactly the population mean, because each district is represented. Plan P is biased, because it ignores the districts. The two plans give much the same spread: the districts are alike, so fixing the mix changes little.

Something to think about

A researcher says: “I took a stratified sample in proportion, so my sample mean is certainly the population mean.” What is wrong with that, and what is true?

One way to think about it

A stratified sample in proportion has a much narrower set of possible sample means, and they centre on the population mean, but any one sample is still one of many and rarely hits the population mean exactly. The method makes a poor sample less likely; it does not remove the wobble.

Exit ticket

8. Stratum A is 6, 8, 10, 12, 14 and 16 (mean 11) and stratum B is 30 and 32 (mean 31), so the population mean is 16. A sample of 4 always takes 2 from A and 2 from B. Over all the samples this plan can give, where do the sample means centre?

On 16, because every sample is drawn at random within each stratum. Around 11, which is below 16, because A has the most values. Around 21, which is above 16, because half of every sample comes from B. Around 31, because the two values from B pull the whole sample to them.