Sampling Distribution Calculator

Enter the values into the sample distribution calculator and click 'Calculate' to determine sampling distribution probabilities

Population St. Dev. (μ)

Population Mean (σ)

Select Probability (n)

Sample Size

X₁

X₂

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Sampling Distribution Calculator

Use this calculator to find the probability distribution of a sample statistic. This mean of sampling distribution calculator also calculates the characteristics of the sample distribution such as expected value and standard error of the mean along with a graphical representation.

What is Sampling Distribution?

“A sampling distribution refers to the probability distribution of a statistic obtained from many samples drawn from a specific population.”

Variability of a Sampling Distribution

The variability of a sampling distribution depends on the following factors:

n – Number of observations in the sample
N – Number of observations in the population
σ – Standard deviation of the population from which the sample is drawn

For example, to estimate the average age of adults in a country, you would take multiple random samples, calculate the mean age for each sample, and plot these sample means on a graph. This distribution of sample means is called the sampling distribution of the mean. Sampling distributions are always based on many random samples drawn from the same population.

How to Calculate a Sampling Distribution?

When the population standard deviation is not available, the sampling distribution is estimated using sample data. Follow these steps:

Random samples: Select multiple random samples from the population and calculate the mean for each sample. Repeat this process multiple times.
Differences of means: Find the differences between the means of different samples.
Standard error: Determine the standard error using the sample data.
Distribution plot: Plot the sample means or their differences to visualize the distribution.

The standard error indicates how much the results of different samples from the same population may vary. It is calculated using the formula:

Standard Error = σ_x = σ √n

If the sample size is small relative to the population, the standard deviation of the sampling distribution equals the standard error. A normal probability calculator can approximate sampling distributions, especially when the sample size is small compared to the population.