A sampling distribution is a probability model for a statistic. That is, it is a function that describes the random behavior and properties of a random statistic like the sample mean or sample proportion.

Each population distribution is described by a PMF/PDF, depending on whether the random variable it describes is discrete or continuous. The named distributions are indexed by a parameter or multiple parameters that determine the center, shape, and spread of the probability distribution. This information is important to evaluating the sampling distribution.

From the population distribution, or data observed from the population distribution, we can evaluate the sampling distribution in three different ways.

**Exact Sampling Distributions:**
In some cases, mathematical statisticians can derive the exact distribution of the sample mean. When available these provide the most accurate probability calculations around the sample mean.

**Central Limit Theorem Approximations:**
Under mild sample size assumptions and finite means and variances, the sampling distribution for the sample mean can be well approximated with a Gaussian (Normal) distribution.

**Resampling Distributions:**
Under the assumption that the sample is representative of the population and other mild assumptions, the sampling distribution of the sample mean and many other statistics can be assessed by sampling from the sample with replacement and calculating the statistic of interest on those new samples. The idea is that doing so mimics sampling from the population so the calculated statistics mimic what repeated studies or experiments would look like.

Step 1: To use this app, go to the 'Probability Calculator' tab.

Step 2: Next, you must select the named population distribution and specify the necessary parameters, or select 'Resampling' and upload a .csv file of your data.

Step 3: You can specify a probability of interest to calculate by selecting the appropriate expression, and specifying the necessary inputs.

Step 4: The resulting probability calculation is calculated and plotted and provided as output.

Please contact us if you have any questions at datascience@colgate.edu.

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