Sampling Distribution

What is a Sampling Distribution?

A sampling distribution can be defined as a probability distributionA Probability DistributionProbability distribution is the calculation that shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required. It is a mathematical function that gives results as per the possible more using statistics by first choosing a particular population and then making use of random samples which are drawn from the population, i.e., it basically targets at the spreading of the frequencies related to the spread of various outcomes or results which can possibly take place for the particular chosen population.



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Example of Sampling Distribution

  1. Assuming that a researcher is conducting a study on the weights of the inhabitants of a particular town and he has five observations or samples, i.e., 70kg, 75kg, 85kg, 80kg, and 65kg. The town is generally considered to be having a normal distribution and maintains a standard deviation of 5kg in the aspect of weight measures. Thus the mean can be calculated as (70+75+85+80+65)/5 = 75 kg.
  2. Also, we assume that the population size is huge; thus, to go to the second step, we will divide the number of observations or samples by 1, i.e., 1/5 = 0.20. Now we need to take the square root of 0.20, which comes to 0.45. The square root is then multiplied by the standard deviation, i.e., 0.45*5 = 2.25kg. Thus standard error obtained is 2.25kg, and the mean obtained was 75kg. These two factors can be used to describe the distribution.

Types of Sampling Distribution

#1 – Sampling Distribution of Mean

  • This can be defined as the probabilistic spread of all the means of samples chosen on a random basis of a fixed size from a particular population. When samples have opted from a normal population, the spread of the mean obtained will also be normal to the mean and the standard deviation.
  • If the population is not normal to still, the distribution of the means will tend to become closer to the normal distribution provided that the sample size is quite large.

#2 – Sampling Distribution of Proportion

This is primarily associated with the statistics involved in attributes. Here the role of binomial distribution comes into play. Generally, it responds to the laws of the binomial distributionBinomial DistributionThe Binomial Distribution Formula calculates the probability of achieving a specific number of successes in a given number of trials. nCx represents the number of successes, while (1-p) n-x represents the number of more, but as the sample size increases, it usually becomes normal distribution again.

#3 – Student’s T-Distribution

This type of distribution is used when the standard deviation of the population is unknown to the researcher or when the size of the sample is very small. This type of distribution is very symmetrical and fulfills the condition of standard normal variate. As the sample size increases, even T distributionT DistributionThe formula to calculate T distribution is T=x¯−μ/s√N. Where x̄ is the sample mean, μ is the population mean, s is the standard deviation, N is the size of the given more tends to become very close to normal distribution.

#4 – F Distribution

#5 – Chi-Square Formula Distribution

This type of distribution is used when the data set involves dealing with values that include adding up the squares. The set of squared quantities belonging to the variance of samples is added, and thus a distribution spread is made, which we call as chi-square distribution.


  • This is important because it simplifies the path to statistical inference. Moreover, it allows analytical considerations to be focussed upon a static distribution rather than the mixed probabilistic spread of each chosen sample unit.
  • Elimination of variability present in the statistic is done by using this distribution.
  • It provides us with an answer about the probable outcomes which are most likely to happen.
  • They play a key role in inferential statistical studies, which means they play a major role in making inferences regarding the entire population.


  • This is key in statistics because they act as a major guideline to statistical inference. They basically guide the researcher, academicians, or statisticians about the spread of the frequencies, signaling a range of varied probable outcomes that could be further tagged to the entire population.
  • The prime factor involved here is the mean of the sample and the standard error, which, if estimates, help us calculate the sampling distribution too. There are various types of distribution techniques, and based on the scenario and data set, each is applied.

This has been a guide to what is Sampling Distribution & its Definition. Here we discuss the types of the sampling distribution, importance, and how to calculate along with examples. You can learn more about from the following articles –

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