Sampling distribution is a theoretical idea which form the basis for estimating the population parameters from sample statistics. The behavior of sampling distribution is stated by the central limit theorem. This helps in arriving at conclusions on Population behavior by working on the computed sample statistic.

Sampling distribution is the probability distribution of sample statistics like mean, median, variance etc.This means statistics calculated from repeated samples of same size form a probability distribution which are described using parameters, just as population parameters describe the Population data.

The expected value or the mean of the sampling distribution is denoted with a subscript indicating the statistic involved in the distribution.

For example: $\mu _{\overline{x}}$ denotes the mean of the sampling distribution.

The standard deviation of the sampling distributions are called the standard error and it is denoted with the subscript indicating the statistic involved in the distribution. $\sigma _{\overline{x}}$ denotes the standard error of the sampling distribution of means.

While $\mu _{\overline{x}}$ = μ the population mean (unbiased estimator), the standard error $\sigma _{\overline{x}}$ = $\frac{\sigma }{\sqrt{n}}$ when the population size is large and sample size = n.
Based on Central Limit theorem which states that the mean of sampling distribution of a statistic is equal to the corresponding parameter in question, the mean of a sampling distribution is used as a single value to estimate the population parameter and is called the point estimate of the parameter.

Depending upon confidence level of the sampling done (Probability of the samples being a true representative of the population) interval estimates for the true parameter are found applying statistical methods.

The interval estimate comes closer to the mean of the sampling distribution as the sample size increases.
Suppose random samples of size n are selected from a population which has proportion of an attribute of interest = p.
The sampling distribution of sample proportions $\hat{p}$ is approximately normal and has the following properties.
  1. The mean of the sample proportions is equal to the true population proportion. This means $\mu _{p}$ = p.
  2. The standard deviation of sample proportions is given by $\sigma _{p}$ = $\sqrt{\frac{p(1-p)}{n}}$

The Sampling distribution of proportions approach normality as the sample size is increased. The above property is stated by the Central Limit theorem for sample proportions. Z scores and hence the associated probabilities can be computed for the observed $\hat{p}$ values.

If the proportion found from a random sample of size n is taken as $\hat{p}$, then it can be used as a point estimate to the true proportion and using the standard error $\sigma _{p}$ interval estimates of the true proportion can also be found.

The sampling distribution of mean and proportions are approximately normal for large samples that is when the sample size n ≥ 30. Hence the graph of sampling distribution are normal curves with mean = μ the population mean and the standard deviation is the standard error which is $\frac{\sigma }{\sqrt{n}}$ when the distribution of the population is known to be normal with standard deviation σ. When the population distribution is not given as normal then the standard error can be taken as $\frac{s}{\sqrt{n}}$ where s is the sample standard deviation.

When the sample size is small, the student's t distribution represents the sampling distribution. The t distribution curves are bell shaped symmetric curves which are defined by the concept of degrees of freedom.

Chi- square distributions another type of graphs based on degrees of freedom represent the sampling distributions of variance.

Solved Example

Question: The average annual salary of employees of a large chain of departmental stores is stated to be 45,000 dollars with a standard deviation of 12,000 dollars. A random sample of 36 employees are selected and their salaries noted. Assuming the distribution salaries is normally distributed, find the probability that the average salary of the sample will be between 40,000 and 44,000 dollars.
The population distribution can be assumed to be normal.  Hence according to central limit theorem the sampling distribution of means is approximately normal with $\mu _{\overline{x}}$ = 45,000 and standard error $\sigma _{\overline{x}}$ = $\frac{12000}{\sqrt{36}}$ = 2000

The z score the distribution is calculated using the formula

z = $\frac{x-\mu _{\overline{x}}}{\sigma \overline{x}}$

The z scores for x = 40,000 and x = 44,000 are

z1 = $\frac{40000-45000}{2000}$ = -2.5       and z2 = $\frac{44000-45000}{2000}$ = -0.5

Hence P(40,000 ≤ x ≤ 44,000) = P(-2.5 ≤ z ≤ -0.5) = 0.3023.

The area under the normal curve representing the probability is shown below.
Sampling Distribution Example