Mathematics 11–12 · Year 12

Means of dice: the sampling distribution of the mean and the central limit theorem

The binomial distribution and sampling distribution of the mean: Sampling distribution of the mean and the central limit theorem (Mathematics Extension 1, Year 12); Random variables: Discrete random variables (Mathematics Advanced, Year 12)

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

A single die is flat, but the mean of ten dice is bunched around 3.5 with a spread that shrinks with the square root of the sample size, and its distribution looks normal although the population does not.

What you need

10 six-sided dice per pair
A tally sheet and a spreadsheet

How to do it

Compute the population mean and variance of one fair die from its probability distribution.
Roll one die 50 times and draw the histogram of the results.
Roll 10 dice together, record their mean, and repeat 50 times; draw the histogram of the 50 means.
Pool the class's means of 10 dice and compute their mean and standard deviation; compare with mu and sigma / sqrt(10).
Combine three sets of 10 to form means of 30 dice and compare their spread with sigma / sqrt(30).
Use the central limit theorem to estimate the probability that a mean of 30 dice lies between 3.2 and 3.8, decide whether means of exactly 3.2 and 3.8 count, and check the estimate against the pooled data.

What you should see

One die has mu = 3.5, variance 35/12 = 2.917 and sigma = 1.708. The standard deviation of the sample mean is 1.708 for n = 1, 0.764 for n = 5, 0.540 for n = 10 and 0.312 for n = 30. The histogram of single rolls is flat, while the histogram of means of 10 is single-peaked and symmetric about 3.5. By the central limit theorem, without a continuity correction, P(3.2 < mean of 30 < 3.8) is about 0.664 (z = plus or minus 0.96). The mean of 30 dice moves in steps of 1/30 and can equal 3.2 or 3.8 exactly (totals 96 and 114), so the exact probability is 0.635 with those end points excluded and 0.689 with them included; a continuity correction brings the normal estimate for the strict inequality to 0.636. The learner knows it worked when the pooled means of 10 have a standard deviation close to 0.54 and the proportion of means of 30 strictly between 3.2 and 3.8 is close to 0.635 (0.689 if the end points are counted).

What changes

What you change: sample size n (number of dice averaged)
What you measure: spread and shape of the distribution of sample means
What you keep the same: fair dice
same rolling method
number of samples per size

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

Averaging more dice does not change the spread of the mean; its standard deviation shrinks by a factor of sqrt(n).
The central limit theorem needs the population to be normal; the die is uniform and the means still become close to normal.

Safety card

Low riskLearners carry it out

Hazards

dice on the floor

Controls

roll inside a tray

Note

No chemicals and no 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 Extension 1 11–12 Syllabus (2024), Year 12 focus area The binomial distribution and sampling distribution of the mean; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-12-06
Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Random variables; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-07
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities