1. What is the Central Limit Theorem?

The Central Limit Theorem (CLT) states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough, regardless of the population distribution.

2. How Does the Calculator Work?

The calculator uses the Central Limit Theorem formula:

Where:

• \( SE \) — Standard Error of the mean
• \( SD \) — Standard Deviation of the population
• \( n \) — Sample size

Explanation: The standard error quantifies how much the sample mean varies from the true population mean. As sample size increases, standard error decreases.

3. Importance of Standard Error

Details: Standard error is crucial for constructing confidence intervals and conducting hypothesis tests. It measures the precision of the sample mean as an estimate of the population mean.

4. Using the Calculator

Tips: Enter the population standard deviation and sample size. Both values must be positive numbers (SD > 0, n ≥ 1).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between SD and SE?
A: SD measures variability in the population, while SE measures how far the sample mean is likely to be from the true population mean.

Q2: How large should n be for CLT to apply?
A: Typically n ≥ 30 is sufficient, but for very non-normal distributions, larger samples may be needed.

Q3: Can I use this for proportions?
A: Yes, but the formula changes to SE = sqrt(p(1-p)/n) where p is the proportion.

Q4: What if my population SD is unknown?
A: You can use the sample SD as an estimate, but this introduces additional uncertainty.

Q5: Does CLT apply to non-random samples?
A: No, CLT assumes random sampling from the population.