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Sample Distribution Calculator


Table of Content


Our sample distribution calculator determines the sample mean, through related parameters like a population mean, standard deviation, and sample size. It derives the probability distribution of sample statistics that lie within a specified range.

What is Sampling Distribution?

The probability distribution of statistics that is obtained from a large number of samples drawn from a specific population is known as the sampling distribution.

It gives a view of the range of possible outcomes for statistics like mean or mode of variables. In statistics, sampling distribution depends on various factors, including:

  • Underlying distribution of the population
  • The statistic being considered
  • The sampling procedure employed
  • The sample size is used

How Sample Distribution Calculator Function?

This probability of sample mean calculator determines the probability for certain values like mean or mode from a random population sample. It works by using the central limit theorem that gives an overview of how the shape of the sampling distribution will be normalized as the sample size increases.

Provide your Values:

  • Put the population mean, standard deviation, and sample size 
  • Select the type of Probability
  • Insert the values for Probability

Result Summary:

Our mean of sampling distribution Calculator determines the sample size required based on a given set of constraints.

Mean of Sampling Distribution: Get the mean of sampling distribution provides a central measure.
Graphical Representation: Visualize the sampling distribution through graphs to enhance user understanding.

Sampling Normal Distribution Formula:

Obtain a certain value from the random sample of a population for statistics by using our user-friendly sample distribution calculator. The formula that is given below is used by the tool to conduct hypothesis tests, calculate confidence intervals, and make other statistical inferences.

$$ \dfrac{\bar X - μ}{σ} $$


μ _ Mean of the population 
σ _ Standard deviation 
X̄ _ Sample mean 

How to Calculate Sample Distribution for Normal Probability?

Our sample distribution calculator predicts the probability of obtaining a certain value for statistics. Let us examine an example that describes the purpose of this sample distribution for a normal probability in a better way. 


What is the probability for the sample means where the normal distribution values are?

  • Population Mean (μ) = 23
  • Population Standard Deviation (σ) = 6.5
  • Sample Size (n) = 12
  • Probability Event = ?


Here, we are required to find an $$ Pr(11.741669750802 \le \bar{X} \le 11.741669750802) $$ for the corresponding z-values. This is calculated as follows:

The objective is to calculate the value of Pr(?=$res?), by whcih the necessary z-values are computed:

$$ Z_{lower} = \dfrac{X_1 - μ}{σ / \sqrt{n}} = \dfrac{11.741669750802 - 23}{6.5 / \sqrt{12}} = -9.593 $$

$$ Z_{upper} = \dfrac{X_2 - μ}{σ / \sqrt{n}} = \dfrac{11.741669750802 - 23}{6.5 / \sqrt{12}} = -9.593 $$

Applying the characteristics of the normal distribution, we can infer that the X N (μ,σ), follows a normal distribution. Therefore, the probability can be computed as follows:

$$ Pr(11.741669750802 \le \bar{X} \le 11.741669750802) = Pr \left(\dfrac{11.741669750802 - 23}{6.5 / \sqrt{12}} \le \dfrac{\bar{X} - 23}{6.5 / \sqrt{12}} \le \dfrac{11.741669750802 - 23}{6.5 / \sqrt{12}} \right) $$

$$ = Pr \left(\dfrac{11.741669750802 - 23}{6.5 / \sqrt{12}} \le Z \le \dfrac{11.741669750802 - 23}{6.5 / \sqrt{12}} \right) $$

$$ = Pr (-9.593 \le Z \le -9.593) $$

$$ = Pr (Z \le -9.593) - Pr(Z \le -9.593) $$

$$ Pr(11.741669750802 \le \bar{X} \le 11.741669750802) $$

Plotting Sample Distribution:


What Is The Central Limit Theorem?

This theorem states that the sample mean distribution is approximately normal even if the population distribution is abnormal or the sample size is large enough.

How Many Types of Sampling Distributions?

There are three types of sampling distribution named as:

  • Sampling distribution of mean 
  • Sampling distribution of a proportion
  • T-distribution

What Is The Importance of Using Sampling Distribution?

Sampling distribution helps you to predict future data by using a sample probability calculator with mean and standard deviation of sample distribution. 

It is important to use a sampling distribution to range large data sets of populations into random subsets. It is also important to manage a foundation for statistical inferencing. 

Why Is Sampling Used To Gather Population Data?

A sampling distribution is a way to analyze large information. As we know population is given in large amounts and due to sheer volume, researchers can not study the whole population. 


Wikipedia: Sampling distribution, Standard error, Examples.

Corporatefinanceinstitute: Sampling Distribution, How Does it Work? Types of Sampling Distribution, Practical Example, Importance of Using a Sampling Distribution, Central limit theorem. 

Investopedia: What Is a Sampling Distribution? How Sampling Distributions Work, Special Considerations, Types of Sampling Distributions, Plotting Sampling Distributions, Why Is Sampling Used to Gather Population Data?

Alan Walker

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