Sample sizes (PDF, 39 KB)
*Analysing Data for Interpretation progression, 4th–5th steps *

## The purpose of the activity

In this activity, the learners develop an understanding of how sample size influences the accuracy of a survey.

## The teaching points

- The learners understand the concept of variation, including: each sample gives different results; smaller samples tend to be more variable than larger samples; larger samples tend to be more similar to each other and to the population data.
- The learners understand that a random sample is a sample of a population where each member of the population has an equal chance of being in the sample.
- Discuss with the learners relevant or authentic situations where the size of the sample influences the accuracy of the results (quality checks, election polls).

## Resources

## The guided teaching and learning sequence

1. The context for this activity relates to the question “How many glasses of water do New Zealand tertiary learners drink in a day?” Discuss with the learners what they know about the benefits of drinking water and how many glasses of water they drink each day. Ask:

“About how much water is it recommended we drink each day?”

“What do we mean by a glass?”

“Do other drinks count?”

“Is it important we all agree on what to include as glasses of water?”

2. Record on the board the number of glasses of water drunk by people in the class the previous day. Here is an example set of responses from 12 people: 4, 4, 2, 6, 1, 0, 1, 6, 8, 2, 1, 9

3. Illustrate these results on a dot plot:

“What can you say about this information?” (More than half drink 4 or less glasses of water a day; only 2 people (17%) drink 8 or more glasses a day)

“What is the mean?”

“What is the median?”

“What is the range?”

“Can we use the results of this survey to make comments about New Zealand tertiary learners more generally?”

4. Ask the learners how we could improve the survey to answer the question: “How many glasses of water do New Zealand tertiary students drink in a day?”

The learners’ responses may bring up a number of issues, including:

- The class is not representative of all tertiary learners.
- The sample was not random.
- The sample size is only 12, which is not large enough.
- The previous day is not random and may have been affected by weather or activity.

5. The size of the sample raises the question of how large a sample should be. To investigate this further, explain to the learners that they are going to compare samples taken from a ‘population’. Have the learners work in groups or pairs and give each group or pair a bag of 100 glasses of water cards. Tell them that the bag contains 100 people’s responses to the glass of water survey. Ask them to take 10 cards from the bag and record the results on a dot plot.

6. Tell them to repeat this four more times, returning the 10 cards to the bag between each sample. Give the learners the dot plot template (Appendix C) so that the dot plots are recorded under each other for comparison. Ask the groups to consider their five dot plots. Ask:

“Do all your graphs (dot plots) have the same shape?”

“What do you think the next sample of 10 will be?”

“Can you confidently say how much water you think the population of 100 people drink?”

7. Encourage the learners to notice that each sample is different from all the others and that there is considerable variation in smallsized samples. Encourage the learners to use statistical language when they are describing the dot plots by writing key words (‘sample’, ‘population’, ‘sampling distribution’, ‘outliers’, ‘spread’, ‘range’, ‘skew’) on the board for the learners to refer to.

8. Ask the learners to take 30 cards from the bag and record the results on a dot plot. Display the dot plots from the various groups of learners on the board. Ask the learners to make statements about the graphs. To encourage more discussion about the shape of the graph, you could show the following sketches of graph shapes and ask the learners which drawing is the best shape for their data.

“Which is the closest shape to your data?”

“Which graph is least like your data?”

Encourage the learners to explore their reasoning.

9. Ask the learners to display their data on a bar chart, using the model from Appendix A. Discuss the fact that the use of bar charts (with percentages) rather than dot plots (with frequencies) allows samples of different sizes to be compared more easily.

10. Display the graph of the population (Appendix A: Glasses of water graph) and ask the learners to reflect on how close their samples of 30 and 10 were to representing the population.

### Number of glasses of water per day

11. Ask:

“Which of the samples (10 or 30) were closest to representing the population?” (Generally smaller samples tend to be more variable than larger samples. This means that small samples are often not representative of the population.) “Were all the samples of 30 equally close at representing the population?” “Why or why not?” (Larger samples tend to be more similar to each other and to the population data; however, they will still be variable.)

## Follow-up activities

Ask the learners to look for reports of surveys in local newspapers. Ask them to consider if they have enough information to be confident about whether the survey is representative of the population, considering the following questions:

“What is the size of the sample?”

“How was the sample selected?”

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