Sampling Distribution of the Mean — Foundation of Inference
Foundations of Statistics
Why Sample Means Behave Predictably
The sampling distribution of the mean explains why repeated samples give consistent results and underpins all confidence intervals and hypothesis tests. Understanding this concept is the key to mastering statistical inference.
- Clinical Trials — Determining whether drug effects are real or due to sampling variation
- Polling — Estimating margin of error for survey results
- Manufacturing — Quality control through process monitoring and control charts
The sampling distribution transforms individual randomness into collective predictability.
Core Concepts
The sampling distribution of the mean describes how the sample mean varies across all possible samples of size .
Central Limit Theorem
Formal Properties of
Worked Example
Suppose the heights of adult males in a city are normally distributed with cm and cm. A researcher samples men.
Step 1. The sampling distribution of is exactly:
Step 2. The standard error is cm.
Step 3. Probability the sample mean exceeds 177 cm:
Step 4. Even though individual heights have , the sample mean of 64 observations has . The sampling distribution is 8 times narrower than the population distribution — a direct consequence of averaging.