STEM Calculators

Mean and Standard Deviation of X

Statistics • Sampling Distributions

Written by STEM Calculators Team Published Updated
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Inputs

These describe the population distribution.

Needed only if sampling is without replacement and you want the finite population correction.

If sampling is without replacement and n / N > 0.05, the calculator applies the correction factor.

CSV can be either key,value rows (mu, sigma, N, n) or a header row with those columns.

Builds a histogram of simulated sample means x̄.

Ready
Enter μ, σ, and n, then click Calculate.

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Frequently Asked Questions

What are the mean and standard deviation of the sampling distribution of xbar?

For the sample mean, the sampling distribution has mean mu_xbar = mu. Its standard deviation (standard error) is sigma_xbar = sigma / sqrt(n) for sampling with replacement or an effectively infinite population.

What is the finite population correction and when should I use it?

When sampling without replacement from a finite population, the standard error is multiplied by sqrt((N - n) / (N - 1)). This correction is typically applied when the sampling fraction n/N is not small (often when n/N > 0.05).

Why does sigma_xbar get smaller when n increases?

Because sigma_xbar = sigma / sqrt(n), increasing n increases sqrt(n) in the denominator. Larger samples make sample means more concentrated around the population mean.

Do I need to enter the population size N?

N is only needed when you are sampling without replacement and want the finite population correction. If you sample with replacement, or treat the population as effectively infinite, N is not required.

What does the simulation histogram show?

The simulation repeatedly generates sample means and plots their distribution as a histogram. It lets you compare the simulated spread of xbar to the theoretical sampling distribution centered at mu with standard deviation sigma_xbar.

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