Mathematics 11–12 · Years 11–12

Estimating 30 seconds: z-scores and the empirical rule on a class dataset

The normal distribution: Normally distributed datasets (Mathematics Standard 2, Year 12); The normal distribution: Calculating z-scores (Mathematics Standard 2, Year 12); Random variables: The normal distribution (Mathematics Advanced, Year 12); Data analysis: Measures of centre and spread (Mathematics Standard, Year 11)

Practical, model not builtLow risk

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The idea

A class of time estimates gives a real dataset whose mean, median and mode can be compared, whose values convert to z-scores, and whose spread can be tested against the 68-95-99.7 rule.

What you need

A stopwatch per pair (the estimator must not see it)
A recording sheet and a spreadsheet

How to do it

The timer says start; the estimator, eyes closed and without counting aloud, says stop when they judge 30 seconds have passed; record the true elapsed time to 0.1 s. Swap roles; each learner makes one estimate for the class dataset.
Pool at least 30 estimates (no names) and draw a histogram with 2 s bins.
Compute the mean, median, mode and population standard deviation and describe the shape.
Convert each estimate to a z-score with z = (x - mu) / sigma and count how many lie between -1 and 1, -2 and 2, and -3 and 3.
Compare the counts with 68, 95 and 99.7 per cent of the class and decide whether a normal model is reasonable.
Use a z-score to compare one learner's estimate with their score on a second task of a different scale, such as a 60-second estimate.

What you should see

Under a normal model, 68.3, 95.4 and 99.7 per cent of values lie within 1, 2 and 3 standard deviations of the mean. For a class of 30 that means about 20 estimates within one standard deviation; because a class is a small sample, any count from 17 to 24 is consistent with a normal model, and about 29 lie within two standard deviations. For the model's test inputs mu = 30.4 s and sigma = 3.2 s, an estimate of 36.0 s has z = 1.75. The z-scores always have mean 0 and standard deviation 1, because they are computed from the class's own mean and standard deviation, so that fact checks only the arithmetic and says nothing about the normal model. The learner knows it worked when the counts within 1, 2 and 3 standard deviations are compared with these ranges rather than with exact percentages, and the decision about the normal model rests on those counts and on how closely the mean, median and mode agree.

What changes

What you change: the learner making the estimate
What you measure: estimated duration of 30 seconds
What you keep the same: eyes closed, no counting aloud
same start signal
same room and noise level

Common misconceptions

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

Exactly 68 per cent of any class must lie within one standard deviation; a sample of 30 varies about that value.
A z-score is a percentage; it counts standard deviations above or below the mean.
Any single-peaked histogram is normal; the mean, median and mode must also nearly coincide and the spread must follow the empirical rule.

Safety card

Low riskLearners carry it out

Hazards

No hazard is listed.

Controls

No control is listed.

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 Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area The normal distribution; 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 2026-09-22MST-12-S2-10
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
Mathematics Standard 11–12 Syllabus (2024), Year 11 focus area Data analysis; 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 2026-09-22MST-11-08
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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