Formula of standard deviation of the sampling distribution of the sample mean

A sampling distribution is defined as the probability-based distribution of specific statistics. Its formula helps calculate the sample’s means, range, standard deviation, and variance. 

For a sample size of more than 30, the sampling distribution formula is given below –

µ͞x =µ and σ͞x =σ / √n

Here,

• The mean of the sample and population represented by µ͞x and µ.
• The standard deviation of the sample and population is represented as σ͞x and σ.
• The sample size of more than 30 is represented as n.

Table of contents

Explanation

One can calculate the formula for Sampling DistributionSampling DistributionA sampling distribution is a probability distribution using statistics by first choosing a particular population and then using random samples drawn from the population. It targets the spreading of the frequencies related to the spread of various outcomes or results which can take place for the particular chosen population.read more by using the following steps:

1. Firstly, find the count of the sample having a similar size of n from the bigger population having the value of N.
2. Next, segregate the samples in the form of a list and determine the mean of each sample.
3. Next, prepare the frequency distributionu0022Frequencyu0022Frequencyread more of the sample mean as determined in Step 2.
4. Next, determine the probability distribution of the determined sample means after determining the frequency distribution in Step 3.

You are free to use this image on your website, templates, etc., Please provide us with an attribution linkHow to Provide Attribution?Article Link to be Hyperlinked
For eg:
Source: Sampling Distribution Formula (wallstreetmojo.com)

Examples of Sampling Distribution Formula (with Excel Template)

Let’s see some simple to advanced practical examples of the sampling distribution equation to understand it better.

You can download this Sampling Distribution Formula Excel Template here – 

Example #1

Take the example of the female population. The sample size is 100, with a mean weight of 65 kg and a standard deviation of 20 kg. Help the researcher determine the mean and standard deviation of the sample size of 100 females.

Solution

Use the below-given data for the calculation of the sampling distribution.

The sample’s mean is equivalent to the population’s mean since the sample size is more than 30.

The calculation of the standard deviation of the sample size is as follows:

• =20/√100

The standard deviation of sample size will be:

• σ͞x =2

Therefore, the standard deviation of the sample is 2, and the sample’s mean is 65 kg.

Example #2

Let us take the example of taxes paid by vehicles. In California, the average tax paid is $12,225, with a standard deviation of $5,000. They made observations on the sample size of 400 trucks and trailers combined. Help the transport department determine the sample’s mean and standard deviation.

Solution

Use the below-given data for the calculation of the sampling distribution.

The calculation of the standard deviation of the sample size is as follows:

• = $5,000 / √400

The standard Deviation of the Sample SizeSample SizeThe sample size formula depicts the relevant population range on which an experiment or survey is conducted. It is measured using the population size, the critical value of normal distribution at the required confidence level, sample proportion and margin of error.read more will be –

• σ͞x =$250

Therefore, the standard deviation of the sample, as assessed by the transport department, is $250, and the sample’s mean is $12,225.

Example #3

Let us take the example of the following data displayed below:

Help the researcher determine the mean and standard deviation of the sample.

Determine the mean of the sample as displayed below: –

• =20*0.67

Mean will be –

• =13.33

Total Mean

• =13.33+7+10
• Total Mean =30.33

Determine the variance of the sample as displayed below: –

• =20^2*0.67
• =266.66667

Variance 

Total Variance

• = 713.67

The calculation of standard deviation of the sample size is as follows:

• σ͞x = √ 713.67 – 30.33

The standard Deviation will be –

• σ͞x = 26.141

Therefore, the standard deviation of the sample assessed by the researcher is 26.141, and the sample’s mean is 30.33.

Relevance and Use

Many entities utilize the sampling distribution for research. It could be analysts, researchers, and statisticians. When the population is large, such methodology helps formulate a smaller sample, which one could then utilize to determine average means and standard deviations. The average means can be plotted on the graph to arrive at a uniform distributionUniform DistributionUniform Distribution is a probability distribution type where every probable outcome has the same possibility of occurrence & it is further categorized into Continuous & Discrete Distribution. This is represented as a straight horizontal line. read more relating to the population. If the researcher increases the sample size, the probability of the graph reaching normal distribution is enhanced.

It helps in the major simplification of the inferences taken up in statistics. It further helps deduce analytical contemplation by determining the frequency of the probability distributionProbability DistributionProbability distribution could be defined as the table or equations showing respective probabilities of different possible outcomes of a defined event or scenario. In simple words, its calculation shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required.read more of sample means. Finally, the sampling distribution forms the base for several statistical concepts that the researchers may use to facilitate their hypothesis.

Recommended Articles

This article is a guide to Sampling Distribution Formula. Here, we discuss calculating the sampling distribution of standard deviation along with practical examples and a downloadable Excel sheet. You can learn more from the following articles: –

• Attribute Sampling
• What is Conditional Probability?
• Poisson Distribution in Excel
• Log Normal Distribution DefinitionLog Normal Distribution DefinitionA lognormal distribution is a continuous distribution of random variables whose logarithms are distributed normally. In other words, the lognormal distribution is generated by the function of x, where x (random variable) is supposed to be normally distributed.read more
• Exponential DistributionExponential DistributionExponential distribution refers to the continuous and constant probability distribution which is actually used to model the time period that a person needs to wait before the given event happens. This distribution is a continuous counterpart of a geometric distribution that is instead distinct.

  What is the formula for the standard deviation of the sampling distribution?

  The standard deviation of the sampling distribution of means equals the standard deviation of the population divided by the square root of the sample size.

  What is the formula in finding the standard deviation of the sampling distribution of the mean when population variance is known?

  Since the mean is 1/N times the sum, the variance of the sampling distribution of the mean would be 1/N2 times the variance of the sum, which equals σ2/N. The standard error of the mean is the standard deviation of the sampling distribution of the mean.

  What is the mean and standard deviation of the sampling distribution of sample means?

  The Sampling Distribution of the Sample Mean. If repeated random samples of a given size n are taken from a population of values for a quantitative variable, where the population mean is μ (mu) and the population standard deviation is σ (sigma) then the mean of all sample means (x-bars) is population mean μ (mu).

  Which is the formula of the mean of the sampling distribution of the sample means?

  The mean of the sampling distribution of the mean formula The formula is μM = μ, where μM is the mean of the sampling distribution of the mean.