Mathematics 7–10 · Year 8
Random samples from a known population: how sample size changes the spread of sample means
Statistics and probability
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The idea
Samples drawn at random from the same population give different means, and the means of larger samples cluster more closely around the population mean, so the spread of the sample means shrinks as the sample size grows.
What you need
- 100 counters numbered 1 to 100 in an opaque bag per group (the population, mean 50.5)
- Recording table for samples of size 5, 10 and 30
- Calculator or spreadsheet
- Access to the CensusAtSchool New Zealand data explorer for a second, real population
How to do it
- Compute the population mean of the numbers 1 to 100 (50.5) before sampling.
- Draw 5 counters without looking, record their mean, and return them. Repeat until the group has 10 sample means of size 5.
- Repeat with samples of size 10 and then size 30, giving 10 sample means at each size.
- Pool the class and draw three dot plots of sample means (n = 5, 10, 30) on the same scale; mark 50.5.
- Compute the range of the sample means at each size (largest minus smallest) and compare the three.
- Repeat with a real variable from the CensusAtSchool data explorer (for example height for one year level), drawing random samples of 10 and 30 and comparing the spread of sample means.
What you should see
The population 1 to 100 has standard deviation 28.87. Sampling without replacement, the standard deviation of the sample mean is 12.65 for n = 5, 8.70 for n = 10 and 4.43 for n = 30 (28.87/sqrt(n) times the finite-population factor sqrt((100 − n)/99), a reference for the teacher: standard deviation is Stage 5 content). So about two thirds of sample means lie within one standard deviation of 50.5: about 38 to 63 for size 5, 42 to 59 for size 10 and 46 to 55 for size 30, and the ranges the learners compute shrink in about the same ratio, near 12.65 ÷ 4.43 = 2.9 from size 5 to size 30. The learner knows it worked when the dot plot for n = 30 is visibly tighter than for n = 5 by a factor of about 3 and all three are centred near 50.5; individual samples remain unpredictable.
What changes
- What you change
- sample size
- What you measure
- spread of the sample means
- What you keep the same
- same population of 100 counters
- counters mixed and drawn without looking
- counters returned before the next sample
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A sample of 30 from 100 is only slightly better than a sample of 5.
- A sample mean far from 50.5 shows the draw was not random.
- Doubling the sample size halves the spread of sample means.
Safety card
Hazards
No hazard is listed.
Controls
No control is listed.
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 4 Data analysis; covers census and sample, the range as a measure of variation and conclusions drawn from sample data, while the effect of sample size on sampling variation is ACARA AC9M8ST03 content with no Stage 4 NSW content point; page read 2026-09-22MA4-DAT-C-02
- Australian Curriculum v9AC9M8ST02AC9M8ST03AC9M8ST04
Sources
The pages the author read to write this activity.