Back to Sampling Distributions

Lesson PreviewLesson 7 of 8 in this unit

5.7 Sampling Distributions for Sample Means

Part of Sampling Distributions in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

Start free

Sampling Distribution of x̄ (Detailed)

For a random sample of size n{n} from a population with mean μ{\mu} and standard deviation σ{\sigma}:

Shape:

  • If population is normal: Xˉ∼N(μ,σ/n){\bar{X} \sim N(\mu, \sigma/\sqrt{n})} exactly (any n{n})
  • If population is not normal: Xˉ≈N(μ,σ/n){\bar{X} \approx N(\mu, \sigma/\sqrt{n})} approximately when n≥30{n \geq 30} (CLT)
Center:
μXˉ=μ(unbiased)\mu_{\bar{X}} = \mu \quad (\text{unbiased})
Standard Error:
σXˉ=σn\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}

Conditions to check:

  1. Independence / 10% condition: n<0.10N{n < 0.10N} (when sampling without replacement)
  2. Normality: Population is normal OR n≥30{n \geq 30} (for CLT)

Summary: Xˉ∼N ⁣(μ, σn){\bar{X} \sim N\!\left(\mu,\ \frac{\sigma}{\sqrt{n}}\right)} when conditions are met.

In this unit

8 lessons

  1. 5.1Introducing Statistics: Why Is My Sample Not Like Yours?Preview lesson
  2. 5.2The Normal Distribution, RevisitedPreview lesson
  3. 5.3The Central Limit TheoremPreview lesson
  4. 5.4Biased and Unbiased Point EstimatesPreview lesson
  5. 5.5Sampling Distributions for Sample ProportionsPreview lesson
  6. 5.6Sampling Distributions for Differences in Sample ProportionsPreview lesson
  7. 5.7Sampling Distributions for Sample MeansThis lesson
  8. 5.8Sampling Distributions for Differences in Sample MeansPreview lesson

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Statistics.

Start free