This calculator helps statisticians and students understand the concept of sample distributions according to the Central Limit Theorem. It is designed to illustrate how sample means approach a normal distribution as the sample size increases.
All calculations are based on standard statistical methods for the Central Limit Theorem. For detailed methodologies, refer to authoritative statistical texts.
The formula for finding the standard error of the mean is: \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
Example: If the population mean is 50, the population standard deviation is 10, and the sample size is 30, the standard error can be calculated as follows:
\(\sigma_{\bar{x}} = \frac{10}{\sqrt{30}} \approx 1.83\)
The Central Limit Theorem states that the distribution of the sample means approximates a normal distribution as the sample size becomes larger, regardless of the population's distribution.
It allows statisticians to make inferences about population parameters based on sample statistics.
The sample mean is the average value of a sample, calculated as the sum of the observations divided by the number of observations.
The standard error measures the variability of the sampling distribution of a statistic, often the mean.
As the sample size increases, the standard error decreases, leading to a more precise estimate of the population mean.