Central Limit Theorem Calculator
Enter population parameters to explore the sampling distribution of the sample mean under the central limit theorem (CLT).
Sampling Distribution Summary
Sampling Mean: 50.000
Standard Error: 1.826
Z-score: 1.960
Confidence Interval: (46.422, 53.578)
Margin of Error: 3.578
Probability that sample mean lies within margin around 50.000: 95.00%
Instructions
- Provide the population mean and standard deviation, along with your sample size.
- Choose a confidence level to compute the margin of error and sampling distribution interval.
- Optionally, examine the probability that the sample mean falls within a specified range.
- Use the results to justify sample sizes or interpret sampling error under the CLT.
Formula
Sampling mean = μ
Standard error = σ / √n
Margin of error = z × (σ / √n)
Confidence interval = μ ± margin of error
The central limit theorem states that the sampling distribution of the mean approaches normality as n increases, even when the population is not normal.
Common Questions
What sample size ensures the CLT applies?
Common guidelines suggest n ≥ 30, but fewer observations may suffice if the population is roughly symmetric.
How does confidence level affect the interval?
Higher confidence levels increase the z-score, widening the interval and the margin of error.
Can I use sample standard deviation?
Yes, when population σ is unknown you typically substitute the sample standard deviation and use the t-distribution.
What if data are highly skewed?
Increase sample size or apply transformations. The CLT converges more slowly for heavily skewed distributions.