Mathfolis

Sampling Distribution of a Sample Mean

Unit 5 · Sampling Distributions

What AP Stats asks here

The sampling distribution of xˉ\bar{x} centers at μ\mu with standard error σ/n\sigma/\sqrt{n}. When the population is normal it is normal for any nn; otherwise the CLT kicks in for n≥30n \ge 30. The recurring AP problem: compare P(X>c)P(X > c) for an individual to P(xˉ>c)P(\bar{x} > c) for a sample — averages are far less variable than individuals.

Sampling distribution of xˉ\bar{x}

Center
μxˉ=μ\mu_{\bar{x}} = \mu
Standard error
σxˉ=σn\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}

Normality conditions

Population normal
xˉ normal for any n\bar{x} \text{ normal for any } n
Otherwise (CLT)
n≥30  (larger for severe skew)n \ge 30 \;(\text{larger for severe skew})
10% condition
n<0.10⋅Nn < 0.10 \cdot N
AP Tip: Average vs single. With n = 25, the sample mean has SD 5× smaller than the individual SD — averages are far more stable than individuals.
Caution: When asked about a sample mean, always divide σ by √n in the denominator of z. Using σ alone is the canonical mistake.
Type 1

Describe the sampling distribution

Use the CLT or normal-population guarantee to claim normality, then plug μ,σ/n\mu, \sigma/\sqrt{n} into the distribution.

Example 1
Heights of US adult women are approximately normal with μ=64\mu = 64 in and σ=2.5\sigma = 2.5 in. An SRS of n=25n = 25 women is taken. Which describes the sampling distribution of xˉ\bar{x}? (A) xˉ∼N(64,2.5)\bar{x} \sim N(64, 2.5) (B) xˉ∼N(64,0.5)\bar{x} \sim N(64, 0.5) (C) xˉ∼N(64,0.1)\bar{x} \sim N(64, 0.1) (D) Not normal because n<30n < 30.

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Type 2

Compute probabilities for xˉ\bar{x}

Standardize the threshold with σ/n\sigma/\sqrt{n} in the denominator and use the z-table.

Example 2
For the previous distribution (μ=64\mu = 64, σxˉ=0.5\sigma_{\bar{x}} = 0.5), what is P(xˉ>65)P(\bar{x} > 65)? (A) ≈ 0.345 (B) ≈ 0.158 (C) ≈ 0.023 (D) ≈ 0.50

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Type 3

Individual vs sample mean

Individuals follow the population SD; sample means follow the SD divided by n\sqrt{n}. The same threshold is far rarer for the average than for an individual.

Example 3
Women's heights: μ=64\mu = 64 in, σ=2.5\sigma = 2.5 in (population normal). Compare P(X>65)P(X > 65) for one woman to P(xˉ>65)P(\bar{x} > 65) for an SRS of 25 women. Which choice is correct? (A) Both ≈ 0.50 because the threshold equals one SD above the mean. (B) The individual probability ≈ 0.34, the sample-mean probability ≈ 0.023 — averages are much less variable. (C) The individual probability ≈ 0.023, the sample-mean probability ≈ 0.34. (D) Both are ≈ 0.023 because z = 2 in each case.

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Sampling Distribution of a Sample Mean | AP Statistics — Mathfolis