Central Limit Theorem & Sampling Distributions Lab
Sample from various population distributions and watch the sampling distribution of the mean converge to a normal curve. Explore how sample size affects standard error and the speed of convergence to normality.
Guided Experiment: Effect of Sample Size on the Sampling Distribution
How does increasing the sample size (n) affect the shape and spread of the sampling distribution of the mean? At what n does the distribution become approximately normal?
Write your hypothesis in the Lab Report panel, then click Next.
Population histogram will appear here
Sampling distribution will appear here
Controls
Same seed produces identical results
Sampling Results
Adjust the parameters and click "Draw Samples" to see the sampling distribution.
Data Table
(0 rows)| # | Trial | Population | n | k | Pop μ | Pop σ | Mean of Means | SE (obs) | SE (theory) |
|---|
Reference Guide
Central Limit Theorem
The Central Limit Theorem states that, regardless of the population distribution, the sampling distribution of the sample mean approaches a normal distribution as the sample size n increases.
This holds for any population with finite mean and variance. The more skewed the population, the larger n needs to be for the approximation to hold well.
Standard Error
The standard error of the mean measures how much sample means vary from sample to sample. It decreases as sample size increases.
Doubling the sample size reduces the standard error by a factor of approximately 1.41 (the square root of 2). To halve the SE, you need to quadruple the sample size.
Sampling Distribution
A sampling distribution is the distribution of a statistic (like the mean) computed from many repeated samples of the same size from the same population.
The z-score tells you how many standard errors a sample mean is from the population mean. By the CLT, this z-score follows a standard normal distribution for large n.
Conditions & Applications
The CLT requires independent random samples and a population with finite variance. The rule of thumb is n greater than or equal to 30 for most populations, though symmetric distributions converge faster.
The CLT underpins confidence intervals, hypothesis tests, and many statistical methods. It explains why the normal distribution appears so frequently in practice.