You may use our central limit theorem (CLT) calculator to compute the sample mean and standard deviation. If you know the population mean, you can get the sample mean. If you know the population standard deviation and sample size, you can discover the sample standard deviation. We also define the central limit theorem and examine the central limit theory formulae that underpin these computations.

While here, spread your knowledge in the field of statistics with our related calculators, such as Coefficient of Variation Calculator, Correlation Coefficient, Midrange Calculator, and check also this interesting Bayes’ Theorem Calculator.

## What is the central limit theorem definition?

The central limit theorem in statistics is concerned with sample distributions. The central limit theorem asserts that as the sample size increases, the sampling distribution approaches a normal distribution, regardless of the population distribution.

Population distribution is made up of the total collection of measurements or data pool. A sampling distribution is a set of sample values gathered from the population distribution through repeated sampling, which can we use to derive conclusions about the entire population.

## Central Limit Theorem Formula

We use the formula for the central limit theorem in probability distribution and sampling techniques. According to the CLT, as the sample size increases, the sample approaches a normal distribution. Because the sample size is more than 30 data points, the fact is actual regardless of the form of the population distribution. The central limit theorem has the following characteristics:

1. The sample mean is the same as the population mean.
2. The estimated standard deviation is the same as the population standard deviation divided by the square root of the sample size.

The following is a formula for the Central Limit Theorem:

\sigma_x = \frac{\sigma}{\sqrt{n}}

Where,

• \sigma = Population Standard Deviation
• \sigma_x= Sample Standard Deviation
• n = Sample size

## How to calculate central limit theorem?

The sample mean is the same as the population mean. Therefore, the estimated standard deviation is the same as the population standard deviation divided by the square root of the sample size.

## How do we find the sample mean?

The sample mean is the mean of the data set’s sample values, whereas the population mean is the mean of all values in the data set. If we know the population mean, the sample mean will be the same as the population mean, as long as the sample size is large enough.

This is because the variance of the sampling distribution of the mean is minimal for high sample sizes, making the sample mean the best point estimate for the population means. For example, if the population mean is 20, simply put 20 for \sigma, and you’re done! According to the central limit theorem calculator, the sample mean (x) is also 20, according to the central limit theorem calculator!

## How do we find the sample standard deviation?

The standard deviation measures how much the data deviates from the mean value. Given the population standard deviation and sample size, the sample standard deviation, s, may be computed using the following central limit theorem formula, which we described previously in the text above.

Where n is the sample size and is the population standard deviation.

Also, the sample set, a subset of the population, must be defined to calculate the sample standard deviation, which depends on the size of the sample chosen.

## Central limit theorem example

Assume we know the population standard deviation,, of people’s ages in a city is 35 years, with a mean age of 60 years, and we’re selecting 49 people at random. Do the following in this CL theorem calculator:

1. As a population mean, type 60 is.
2. Enter 35 as the value for.
3. Enter 49 as n.
4. The calculator gives the following outcomes
• The sample mean equals the population mean:

60 = x

• The sample standard deviation (s) is 5 years, as estimated below:

35 / 49 = 35 / 7 = 5 s = 35 / 49 = 35 / 7 = 5

Because our central limit theorem calculator is omnidirectional, you may get the population standard deviation by inputting the sample standard deviation and sample size!

## Central limit theorem conditions

Specific requirements must be met for the CLT formulations to be relevant –

• Data must be sampled at random.
• The sample values must be distinct from one another. The data is usually devoid of any dependence bias when random sampling is employed.
• A sample size of at least 30 must be in use (Refer to FAQs to learn more).
• The sample size must be no more than 10% of the whole population (Refer to FAQs to learn more).

## How are samples selected in statistics?

There are two different sampling techniques: We use random selection in probability sampling, which allows you to draw solid statistical judgments about the whole group. Non-probability sampling entails making non-random selections based on convenience or other factors easier for data collection.