Sampling Distribution

Introduction to Sampling Distributions

Coming back to the example above, let's say you randomly select and sample \(100\) senior students and calculate the average GPA from this sample. This average GPA would not be the same as the mean GPA of all senior students in Atlanta. It could be lower or higher, but it would most likely not be exactly equal to the population mean.

If you select a second sample of \(100\) senior students, the average GPA for this sample would most likely differ from the mean of your first one. Thus, random samples selected would produce different mean values. Despite this variety of values, when many sample means are obtained, you can plot these collected means on a graph, and then this can provide an estimated mean of the entire population. This process explains the concept of creating sampling distributions of the mean.

Definition of Sampling Distributions

A value that is calculated by taking information from a sample is called a statistic. Statistics allows you to estimate data of an entire population. As you saw in the example above, different random samples can give different values for a statistic; this difference is called sampling variability (or sampling error). This sampling variability can be reduced by increasing the sample size.

Conditions for Sampling Distributions

To ensure that the sampling distribution truly estimates the entire population, you must make sure that these two criteria are checked:

1. Randomization condition: the most important condition necessary for creating a sampling distribution is that your data comes from samples randomly selected.

2. Independence (\(10\%\) condition): the sampled values must be independent one from another. Achieving this condition is the same as considering sample sizes no larger than \(10\%\) of the entire population.

Types of Sampling Distributions

There are 3 types of sampling distributions:

1. Sampling distribution of proportions

2. Sampling distribution of means

3. T-distribution

Sampling Distribution of Proportions

It is used to estimate a population proportion. It calculates the proportion of success, or chance, that a specific event will occur. The mean from each group of the sample proportion is a representation of the estimated proportion of success of the entire population.

Sampling Distribution of Means

It entails calculating the means of all sample groups from a selected population. Then, the average of the means of all the samples is an estimated mean of the entire population.

T-distribution

It is focused on a small population. It is used to measure the mean of the population and other statistical measurements such as confidence intervals, linear regression, and statistical differences. Since this distribution uses \(t\)-scores to calculate probabilities, it is out of the scope of this article.

Formula for Sampling Distributions

The sample proportion, denoted by \(\widehat{p}\), is calculated by counting how many successes are in the sample (success means that an individual possesses the characteristic of interest) and dividing it by the total sample size \(n\)

\[\widehat{p}=\frac{\text{number of successes in the sample}}{n}.\]

The sample mean, denoted by \(\overline{x}\), is calculated by adding up all the values obtained from the sample and dividing by the total sample size \(n\). The idea is the same as finding the average for a set of data. The formula is

\[\overline{x}=\frac{x_1+x_2+...+x_n}{n},\]

where \(\overline{x}\) is the sample mean, \(x_i\) is each one of the values of the sample, and \(n\) is the sample size.

Mean and Standard Deviation of Sampling Distributions

All probability distributions have characteristics that distinguish them. Sampling distributions are no exception, knowing the mean and standard deviation can give you a lot of information about the shape of the distribution.

Mean and Standard Deviation of the Sample Proportion

Let \(p\) be the proportion of success in a population and \(\widehat{p}\) the sample proportion, that is, the proportion of success in a random sample of size \(n\), then the sampling distribution of \(\widehat{p}\) has mean and standard deviation given by \[\mu_\widehat{p}=p\,\text{ and }\, \sigma_\widehat{p}=\sqrt{\frac{p(1-p)}{n}}.\]

Moreover, if \[np\geq 10\,\text{ and }\, n(1-p)\geq 10,\] then, the sampling distribution of \(\widehat{p}\) is similar to a normal distribution.

Mean and Standard Deviation of the Sample Mean

Let \(\mu\) be the mean and \(\sigma\) the standard deviation of the population. Let \(\overline{x}\) be the sample mean of a random sample of size \(n\), then the sampling distribution of \(\overline{x}\) has mean and standard deviation given by \[\mu_\overline{x}=\mu\,\text{ and }\, \sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}.\]

The standard deviation of the sampling distribution of means is also known as the standard error of the mean (SEM).

If the sample size \(n\) is large enough (according to the Central Limit Theorem, \(n\geq 30\) is enough) then, the sampling distribution of \(\overline{x}\) is similar to a normal distribution.

Examples of Sampling Distributions

Let's see an example using sampling distributions.

Let's see one extra example.