Table of Contents
Sampling Distributions
A sampling distribution is the probability distribution of a statistic (like the mean) computed from repeated random samples of the same size from a population. The sampling distribution of the mean tells us how sample means vary from sample to sample.
Understanding sampling distributions is fundamental to inferential statistics. They form the basis for hypothesis testing, confidence intervals, and understanding how reliable our sample statistics are as estimates of population parameters.
Formulas
The standard error decreases as sample size increases, meaning larger samples produce more precise estimates of the population mean.
Central Limit Theorem
| Sample Size | Standard Error (if σ=15) | 95% CI Width |
|---|---|---|
| 9 | 5.00 | ±9.80 |
| 25 | 3.00 | ±5.88 |
| 100 | 1.50 | ±2.94 |
| 400 | 0.75 | ±1.47 |
Frequently Asked Questions
What is the Central Limit Theorem?
The CLT states that regardless of the population distribution shape, the sampling distribution of the mean approaches a normal distribution as sample size increases. Generally, n ≥ 30 is considered sufficient for a good approximation.
What is the standard error?
The standard error is the standard deviation of the sampling distribution. It measures how much sample means typically deviate from the population mean. It equals σ/√n, so it decreases with larger samples.
Why does sample size matter?
Larger samples produce more precise estimates (smaller standard error). Quadrupling the sample size halves the standard error. This is why larger studies are more reliable than smaller ones.