Montana Math MT.PLUS.DATA.DES.3.b
The Standard
- MT.PLUS.DATA.DES.3
- Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic by: Simulating a sampling distribution of sample means from a population with a known distribution, observing the effect of the sample size on the variability, and Demonstrating that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size.
Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic by: Demonstrating that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size.
2026 Montana Mathematics Content Standards · Data Reasoning
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students take repeated random samples of different sizes from the same population and record each sample mean. They compare the standard deviation of those means with the population standard deviation divided by the square root of the sample size.
What Mastery Looks Like
- Given a population standard deviation of 8, a student predicts that sample means for sizes 4 and 16 will have standard deviations of 4 and 2. They compare those predictions with a simulation and explain why the results may be close but not exact.
Common Misconceptions
- Students may use the population standard deviation as the spread of sample means, or divide by sample size instead of its square root. They may expect simulated results to match the prediction exactly or apply the rule to individual observations.
How to Assess It
- Exit ticket: A population has a standard deviation of 12. Find the expected standard deviation of sample means for sizes 9 and 36, and say which simulated distribution should be narrower.
Lesson moves
Ways to Teach It
Give pairs 0 and 10 cards; have them draw with replacement at sizes 4 and 16, pool means, and calculate both standard deviations.
Ask: Why does quadrupling sample size halve the expected spread of sample means, rather than cut it to a quarter?
Put sample-size cards 1, 4, 9, and 16 on desks; give a population SD of 12 and have teams match the expected spreads.
Tell students bottle fills have a known SD of 4 mL; have them calculate expected spread of averages from 4 versus 16 bottles.
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