Central Limit Theorem Calculator

1. What is the Central Limit Theorem?

Definition: The central limit theorem (CLT) states that the sampling distribution of the sample mean approximates a normal distribution as the sample size becomes large, regardless of the population's distribution shape, provided certain conditions are met.

Purpose: Enables estimation of population parameters from sample statistics and simplifies statistical inference.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Sample Mean (x̄) = μ: The sample mean equals the population mean.
Sample Standard Deviation (s) = σ / √n: The sample standard deviation is the population standard deviation divided by the square root of the sample size.

Steps:

Enter the population mean (μ), population standard deviation (σ), and sample size (n).
Click "Calculate" to compute the sample mean and sample standard deviation.

3. Why Use the Central Limit Theorem?

Useful for:

Statistical Inference: Estimating population means from large samples.
Quality Control: Analyzing sample data in manufacturing.
Survey Analysis: Generalizing results from sample surveys.

4. Using the Calculator

Example:

μ = 60, σ = 15, n = 100: x̄ = 60, s = 15 / √100 = 1.5.
μ = 100, σ = 20, n = 400: x̄ = 100, s = 20 / √400 = 1.

5. Frequently Asked Questions (FAQ)

Q: What sample size is large enough?
A: Generally, n ≥ 30 is considered sufficient, though it depends on the population distribution.

Q: Does CLT apply to all distributions?
A: Yes, for large sample sizes, though the population should have a finite variance.

Q: What if σ is unknown?
A: The sample standard deviation (s) from the data can be used as an estimate.