Year 11 Standard · Unit 4 Sampling and relationships in data · Lesson 1 of 6

Can a handful of items tell you about all of them?

A sample is a small part of a population, picked to learn about the whole. Two samples from the same population rarely agree, and that gap is the thing to understand before any survey is trusted.

Statistics and ProbabilityNo calculator

Warm-up

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

  1. From Year 8: the mean of 4, 9, 11 and 8 is ?
    Answer8. The total is 32, and 32 ÷ 4 = 8.
  2. From Year 7: 5% of 640 is ?
    Answer32. 10% is 64, so 5% is half of that.
  3. From Year 9: one dot on a scatter graph is far from the trend. Does it change the line of best fit a little, or not at all?
    AnswerA little. Every dot pulls on the line, and a far-away dot pulls more.
Words used in this lesson
population
every item or person the question is about. For example: all the pupils in a school year.
sample
the part of the population that is actually measured. For example: 30 pupils chosen from a year group.
estimate
a value worked out from a sample to stand in for the population value. For example: using a sample mean in place of the population mean.
sample size
the number of items in a sample. For example: the count of readings taken.
tagged
marked so that it can be counted again later. For example: a bird with a small ring fixed to its leg.
variation
the natural difference between one sample and another. For example: the weights of two handfuls of the same flour.

Weighing the harvest

A fruit grower in Oaxaca has a harvest of mangoes too large to weigh one by one. She will weigh a few and use their mean to describe all of them. The forty dots below stand for a crate small enough to know exactly, so we can see how well a few mangoes describe the rest.

Predict first: as she weighs more mangoes, will her sample mean get steadily closer to the true mean? Is there a size at which she can stop being unsure? Test your idea with the first figure, then use the second to compare samples of the same size.

5
212sample mean in grams
12gap from the population mean in grams

Sample of 5: mean 212 g, which is 12 g above the population mean of 200 g.

Show me the steps
  1. Sample of 5: the mean is 212 g, which is 12 g above the population mean of 200 g.
  2. Sample of 10: the mean moves further away, 214.9 g. A bigger sample does not have to be closer every time.
  3. Sample of 20: the mean is 204.1 g, only 4.1 g above 200 g. Samples of 25 and 30 are closer still.
  4. Sample of 40: every mango is in the sample, so the sample mean equals the population mean, 200 g, and the gap is 0.

A sample is used to estimate the population.

  • The population is every item you want to know about. A sample is the part of it you actually measure.
  • A sample mean is an estimate of the population mean. It is usually close, but not equal.
  • Different samples of the same size give different estimates. A larger sample tends to give estimates that are closer together.
  • To estimate a count: (number in the sample with the feature ÷ sample size) × population size.
1
214.9this sample's mean in grams
14.9gap from the population mean in grams

Sample 1: mean 214.9 g, which is 14.9 g above 200 g. The four sample means run from 190.6 g to 214.9 g.

Show me the steps
  1. Sample 1: the mean is 214.9 g, 14.9 g above the population mean of 200 g.
  2. Sample 2: the mean is 193.3 g, now 6.7 g below the population mean.
  3. Sample 3: the mean is 190.6 g, the lowest of the four, 9.4 g below.
  4. Sample 4: the mean is 201.2 g, almost on the population mean. Four honest samples, four different answers.

A sample that differs from the population is not wrong. Some difference is expected, and it has nothing to do with a mistake in the weighing. A sample becomes a problem when the method pushes every result the same way. That is bias, and Lesson 4 deals with it. Also, a bigger sample is not guaranteed to be closer: it is closer on average.

Worked examples

A sample of 80 shirts from a batch of 2500 has 6 with a stitching fault. Estimate the number of faulty shirts in the batch.

  1. The share in the sample is 6 ÷ 80 = 0.075.
  2. Assume the batch has the same share: 0.075 × 2500 = 187.5.
  3. A count is whole, so the estimate is about 188 faulty shirts.

Capture and recapture. A pond holds an unknown number of carp. A warden catches and tags 25, then releases them. Later she catches 20 carp and 4 of them carry a tag. Estimate the number of carp.

  1. The tagged share of the second catch is 4 ÷ 20 = 1 ÷ 5.
  2. If that is also the tagged share of the whole pond, the 25 tagged fish are one fifth of the pond.
  3. So the pond holds about 25 × 5 = 125 carp. As a formula: tagged × second catch ÷ tagged in the second catch.

Your go. The first step is given. A sample of 150 jars from a store of 6000 has 9 jars with a cracked lid. Estimate the number of cracked jars in the store.

  1. The share in the sample is 9 ÷ 150 = 0.06.
  2. Use that share for the whole store.
Check the answer

0.06 × 6000 = 360 cracked jars. A common slip is to divide by the store size instead of multiplying: the share must be applied to the 6000, not the 150.

Check before you go on

1. A health office wants the mean daily walking time of the 2400 residents of one village. It records the walking time of 60 residents. Which statement is correct?

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. Six students are asked how many minutes they spend walking to school: 26, 15, 28, 20, 27, 22. What is the sample mean, in minutes?

3. A sample of 50 light bulbs from a batch of 4000 has 3 faulty bulbs. Estimate the number of faulty bulbs in the batch.

Level 2 · Building

4. A lake survey tags 35 fish and releases them. A later catch of 24 fish includes 6 tagged ones. Estimate the number of fish in the lake.

5. Sample A has 20 values with mean 52. Sample B has 30 values with mean 47. What is the mean of all 50 values?

Level 3 · Stretch

6. 12 values have a mean of 30. One more value is added and the mean of the 13 values becomes 31. What is the new value?

7. In a sample of 200 voters, 58% say they will vote yes. Estimate the number of yes voters in a town of 25 000 voters.

Something to think about

A grower weighs a sample of 20 mangoes and gets a mean of 204 g. A second grower, using the same method, weighs 20 different mangoes and gets 197 g. Whose figure is closer to the truth, and how could you find out without weighing every mango?

One way to think about it

Neither can be called closer without the true mean: both are honest estimates. What can be judged is the method. If every mango had a fair chance of being weighed, neither figure leans, and a gap of 7 g is ordinary sample variation. A larger random sample would give a figure you could trust more, but no single sample can show which of the two was closer.

Exit ticket

8. A school has 1200 students. One sample of 40 has 14 who walk to school, and a second sample of 40 has 11. Which statement is best?