Types of Distributions

There are 3 main types of sampling distributions are:

• Sampling Distribution of Mean
• Sampling Distribution of Proportion
• T-Distribution

Sampling Distribution of Mean

Mean is the most common type of sampling distribution.

It focuses on calculating the mean or rather the average of every sample group chosen from the population and plotting the data points. The graph shows a normal distribution where the center is the mean of the sampling distribution, which represents the mean of the entire population.

We take many random samples of a given size n from a population with mean µ and standard deviation σ. Some sample means will be above the population mean µ and some will be below, making up the sampling distribution.

For any population with mean µ and standard deviation σ:

• Mean, or center of the sampling distribution of x̄, is equal to the population mean, µ.

[Tex]µ_{x^{-}}[/Tex] = µ

There is no tendency for a sample mean to fall systematically above or below µ, even if the distribution of the raw data is skewed. Thus, the mean of the sampling distribution is an unbiased estimate of the population mean µ.

• Standard deviation of the sampling distribution is σ/√n, where n is the sample size.

[Tex]σ_x[/Tex] = σ/√n

• Standard deviation of the sampling distribution measures how much the sample statistic varies from sample to sample. It is smaller than the standard deviation of the population by a factor of √n. Averages are less variable than individual observations.
  

  sampling distribution of standard deviation

Sampling Distribution of Proportion

Sampling distribution of proportion focuses on proportions in a population. Here, you select samples and calculate their corresponding proportions. The means of the sample proportions from each group represent the proportion of the entire population.

Formula for the sampling distribution of a proportion (often denoted as p̂) is:

p̂ = x/n

where:

• p̂ is Sample Proportion
• x is Number of “successes” or occurrences of Event of Interest in Sample
• n is Sample Size

This formula calculates the proportion of occurrences of a certain event (e.g., success, positive outcome) within a sample.

T-Distribution

Sampling distribution involves a small population or a population about which you don’t know much. It is used to estimate the mean of the population and other statistics such as confidence intervals, statistical differences and linear regression. T-distribution uses a t-score to evaluate data that wouldn’t be appropriate for a normal distribution.

Formula for the t-score, denoted as t, is:

t = [x – μ] / [s /√(n)]

where:

• x is Sample Mean
• μ is Population Mean (or an estimate of it)
• s is Sample Standard Deviation
• n is Sample Size

This formula calculates the difference between the sample mean and the population mean, scaled by the standard error of the sample mean. The t-score helps to assess whether the observed difference between the sample and population means is statistically significant.

Sampling distribution is essential in various aspects of real life. Sampling distributions are important for inferential statistics. A sampling distribution represents the distribution of a statistic, like the mean or standard deviation, which is calculated from multiple samples of a population. It shows how these statistics vary across different samples drawn from the same population.

In this article, we will discuss the Sampling Distribution in detail and its types along with examples and go through some practice questions too.