Central Limit Theorem Calculator
Compute z-scores and confidence intervals using the central limit theorem.
Table of Contents
How to Use
- Enter the population mean and standard deviation for your process.
- Specify the sample size and the observed sample mean.
- Click calculate to obtain the sampling distribution metrics, tail probabilities, and a 95% confidence interval.
Central Limit Theorem Overview
The central limit theorem states that the sampling distribution of the mean approaches a normal distribution as the sample size grows, regardless of the population’s original distribution (given finite variance).
This allows researchers to use normal probability techniques for inference about means, even when the underlying data are not perfectly normal.
When to Apply the CLT
- Sample size n ≥ 30 is a common rule of thumb for skewed populations.
- For symmetric or normal populations, smaller sample sizes can still work.
- Ensure observations are independent and drawn from the same distribution.
- Use the standard error σ / √n to measure how the sample mean varies.
Frequently Asked Questions
- What if the population standard deviation is unknown?
- If σ is unknown, substitute the sample standard deviation and consider using the t-distribution, especially for small sample sizes.
- Does the CLT work for small sample sizes?
- For highly skewed populations, larger samples (n ≥ 30) are recommended. For approximately normal populations, even small samples can produce accurate normal approximations.
- How is the two-tailed probability used?
- The two-tailed probability indicates how extreme the observed sample mean is on either side of the population mean. It is often used for two-sided hypothesis tests.
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