# Central Limit Theorem Calculator

Enter the population mean, population standard deviation, and sample size to calculate the sample mean and standard deviation using the central limit theorem.

## Results:

### Sample Mean (x̄)

### Sample Standard Deviation (s)

### Steps to Solve Using the Central Limit Theorem

#### Find the Sample Mean

Provided the sample is sufficiently large, the sample mean will be equal to the population mean

x̄ = μ

#### Find the Sample Standard Deviation

The central limit theorem provides the following formula to solve the standard deviation for a sample

s = σ√n

## How to Use the Central Limit Theorem

The central limit theorem for sample means states that as you take larger samples and calculate their means, the sample means form their own normal distribution, which is known as the sampling distribution of the mean. This distribution has the same mean as the original distribution and a variance equaling the original variance divided by the sample size.

Given a known population mean and a sufficiently large sample, the central limit theorem indicates that the sample mean will equal the population mean. This holds true for sample sizes larger than 30.^{[1]}

As you can see in the image above, as the sample size n increases, the values of each sample continue to be distributed around an equal mean.

### How to Find the Sample Mean

You can use the central limit theorem to find the sample mean. The sample mean x̄ is equal to the population mean μ.

x̄ = μ

**For example,** if the population mean μ is equal to 50, then a sample size of 100 will have a mean x̄ equal to 50 as well.

You can use our mean calculator to find the population mean μ.

### How to Find the Sample Standard Deviation

You can also use the central limit theorem to find the standard deviation of a sample s, given a known standard deviation of the population σ.

#### Central Limit Theorem Formula for Sample Standard Deviation

The central limit theorem provides the following formula to calculate the standard deviation of a sample.^{[2]}

s = σ√n

Thus, the central limit theorem states that the standard deviation of a sample s is equal to the standard deviation of the population σ divided by the square root of the sample size n.

**For example,** if the population standard deviation σ is equal to 2, let’s find the standard deviation s of a sample with a size of 100.

s = 2√100

s = 210

s = 0.2

So, this sample will have a standard deviation s equal to **0.2**.

You can use our standard deviation calculator to find the population standard deviation σ.

## How to Find Probabilities Using the Central Limit Theorem

The central limit theorem can also be used to find the probabilities of sample means. Start by using the following formula to find the z-score.

z = x̄ – μ_{x}s

The z-score z is equal to the sample mean x̄ minus μ, which is the average of x and x̄, divided by the sample standard deviation s.

Using the z-score, you can look up the probability of getting this mean using a z table.

## Recommended

## References

- McLeod, S., What is central limit theorem in statistics?,
*Simply Psychology*, 11/25/2019, https://www.simplypsychology.org/central-limit-theorem.html - Illowsky, B., & Dean, S., The Central Limit Theorem for Sample Means (Averages),
*LibreTexts*, 11/5/2021, https://stats.libretexts.org/@go/page/757