Lesson 4: 7.4 The Sampling Distribution of a Sample Mean

If we take a sample of 25 students, why is the average height of that group much more likely to be near the "true" average than any one individual student's height?

What happens to the "spread" of our possible sample means as we increase the number of people in each sample?

If the original population is Normal, what can we automatically say about the shape of the sampling distribution?

Sample mean

Population mean

Standard deviation of the sampling distribution

Standard error (introductory concept)

HSS-IC.B.4: Use data from a sample survey to estimate a population mean or proportion.

The SAT often asks how the "reliability" or "precision" of a mean changes with sample size. Students need to know that as n increases, the variability of the sample mean decreases. You might see a question asking which of two studies (one with n=50, one with n=500) is more likely to yield a sample mean close to the true population mean.

This section reinforces that the sample mean is an unbiased estimator of the population mean.
However, while the center stays the same, the "standard deviation of the mean" is much smaller than the population standard deviation.

The Problem: The height of young women follows a Normal distribution with mu = 64.5 inches and sigma = 2.5 inches.

Task 1: If you choose one woman at random, what is the probability she is taller than 66.5 inches?

Task 2: If you choose a random sample of 10 women, what is the probability that their average height is taller than 66.5 inches?

Dividing by n vs. square root of n: Students often forget the square root in the denominator.

Normal vs. Not Normal: Students often think the sampling distribution is always Normal. In this section, emphasize that it’s only Normal because the population was Normal to begin with.

Students confuse the variation of individuals with the variation of averages.

Support: "The Averages are Boring" Rule. Explain that individuals can be extreme (very tall or very short), but averages of groups tend to be "boring" (closer to the middle). This helps them visualize why the sampling distribution is much skinnier than the population distribution.

Extension: Ask students to find the sample size n required to cut the standard deviation of the mean in half. (They should discover they need 4 times the data, not 2).

Teacher assigns examples from textbook and other resources.

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