Summary
Highlights
A sampling distribution is the distribution of the values one could get for a statistic (like the mean) when repeatedly taking samples from a population. Population parameters are truths about the entire population, such as the population mean or standard deviation. Since these parameters are often unknown or difficult to calculate, we use samples to estimate them. However, each sample will likely yield a slightly different statistic, and a sampling distribution helps understand the frequency of these different statistical outcomes.
To make the concept concrete, an example is set up with a population of three balls numbered 1, 2, and 3. The population mean is 2. The goal is to estimate this population mean by taking samples of two balls at a time, with replacement. For each sample, the sample mean is calculated.
A table is created to list all possible combinations of two balls (with replacement) and their corresponding sample means. There are 3x3 = 9 possible samples. The sample means vary, for example, from 1 (picking 1,1) to 3 (picking 3,3). This demonstrates how different samples lead to different estimates of the population mean.
The frequencies of these possible sample means are then plotted to visualize the sampling distribution. For instance, a sample mean of 1 appears once (1/9 probability), 1.5 twice (2/9 probability), 2 three times (3/9 probability), 2.5 twice (2/9 probability), and 3 once (1/9 probability). This plot represents the sampling distribution for the sample mean when the sample size (n) is 2.