1. EBOP Statistics Inquiry Group
Sampling Resource
How big is a useful sample?
Is there a model for sample variation?
This resource was developed to answer the common questions of “How big is a useful sample?” and “Is there a model for this variation?”. The first question can be answered by looking at samples of different sizes and noticing when the variation is starting to stabilize or level out. The second can be answered by modelling the width of the variation.
The EBOP Statistics Group was formed under the SSA Contract 2106.

2. Use of this Resource
Demonstrate using iNZight and then let students repeat the experience. By doing they may understand that as sample size increases the sampling variation decreases.
Measuring this variation simply with a ruler on the screen will give data to model on Excel and plot against sample size. The trend is a power curve and always of the form x^- 0.5 or thereabouts.
This is a very convincing experimental demonstration.
As explained above.
Modeling the data many times as above has always given the power index in the range [-0.47, -0.53].

3. The Data
Use iNZight VIT Tool called Sampling Variation and import a large file such as Taupo Trout This file has weight of a fish as one variable and is a record of the 1099 fish caught in the International Trout Fishing Tournament held on Lake Taupo.
In Lake Taupo only fish longer than 45cm are allowed to be caught so the weight correspondingly is never below about 1.2kg.
There is more about this data on and on Census at Schools.

4. Example of iNZight Screen
This is a screen shot including the control showing sample size selected and the resulting distribution of sample means. Do not measure here as it has been reduced in size to fit the screen.

5. Sample Size 2
Record the width against 2.
1000 samples of size 2 where taken and the sample means were plotted as a vertical blue line. Measure the width of the distribution using a ruler of most of the variation. This might need agreement of what to measure!

6. Sample Size 3
Repeat for each of the following. Record in a table. Try and be consistent about what you measure.

7. Sample Size 4

8. Sample Size 5

9. Sample Size 7

10. Sample Size 10

11. Sample Size 15

12. Sample Size 20

13. Sample Size 30

14. Sample Size 50

15. Sample Size 100

16. Sample Size 250

17. My Measuring
The arrow shows where I measured the colour as it changed to a darker blue and more lines. This is a measure of the variation in the sample mean.

18. My Table of Results Sample Size 2 3 4 5 7 10 15 20 30 50 100 250
Measure (mm) 86 75 70 63 55 48 36 23 16
These mm measures came from the screen of a MacBook Pro.
Results will vary depending on the screen size.
This will not affect the outcome however! Why?

19. Outcome • Excel Model
What do you see? Record all the ideas.

20. What I see? and Where is it?
The is very close to -0.5 or 1/√n, where n is sample size.
The sampling variation keeps decreasing as sample size increases.

21. What I see? and Where is it?
The shape of the graph falls quite sharply at first as the sample size increases from 2 to 20 or so.
By 30 or so the slope of the graph is leveling out. The slope continues to level but is now not changing much for an increase in sample size. It will continue to decrease however.

22. What does it mean? Sample variation seems to be proportional to 1/√n.
A small sample size also generates a lot of variation between samples. This is because data items being selected are quite often very different.
A large sample selects data elements that are pretty much all the same so there will be less variation between samples.

23. What does it mean?
A sensible sample size starts at about 20 to 30 or so. The cost in time and expense is thereby kept to a minimum as well.
Not a lot of sense can be attached to statements about the data from small sample. The variation in the data only supports very broad and probably meaningless statements.
For the 1999 Taupo Trout data a sample of size 30 will give a good estimate of the mean of the population. There are about 1.5 million catchable trout over 45cm in length in the Lake Taupo.

24. An actual random sample
For the 1999 Taupo Trout data a sample of size about 30 will give a good estimate of the mean of the population. The mean in this year was about 1.8kg. A good size fish.

25. So…answers please. How big is a useful sample?
Is there a model for sample variation?

26. My answers… How big is a useful sample?
From about 20 to 50, 30 usually works. A population with a greater variation might require a larger sample size.
Is there a model for sample variation?
Sample variation is proportional to 1/√n. Quadrupling the sample size will halve the sampling variation.

27. A 6kg Trout • 1999 Winner
Actual result from 1994
Let’s go fishing!

28. Thanks Chris Wild Auckland University iNZight Team
Sandra Cathcart, SSA Team Solutions
Census at School