# Central Limit Theorem Calculator

Reviewed by Hanna Pamuła, PhD and Jack Bowater
Last updated: Nov 02, 2022

Our central limit theorem calculator enables you to calculate the sample mean and sample standard deviation. If the population mean (cf. mean calculator) is known, you can use it to find the sample mean, while if the population standard deviation and the sample size are known, then our calculator can help you find the sample standard deviation. We also explain what the central limit theorem is and have a look at the central limit theorem formulas that are behind these calculations.

## What is the central limit theorem definition?

In statistics, the central limit theorem concerns sample distributions. The central limit theorem definition states that the sampling distribution approximates a normal distribution as the sample size becomes larger, irrespective of the shape of the population distribution. You can visit our normal distribution calculator for more on the topic.

A population distribution comprises the complete set of measurements or the entire pool of data. A sampling distribution is a collection of sample values from the population distribution obtained through repeated sampling, which can be used to draw inferences about the entire population.

## How do we find the sample mean?

The sample mean refers to the mean of the sample values collected from the data set, while the population mean is the mean of all values in the data set. If the population mean is known, then the sample mean will be the same as the population mean, provided the sample size is sufficiently large. The reason behind this is that, for large sample sizes, the variance of the sampling distribution of the mean is low, which makes the sample mean the best point estimate for the population mean.

For instance, if the population mean (μ) is 20, simply enter 20 for μ and voila! The central limit theorem calculator would show that the sample mean (x̅) is also 20!

## How do we find the sample standard deviation?

The standard deviation is a measure of the spread of the data from the mean value. Given the population standard deviation and the sample size, the sample standard deviation, s, can be calculated using the following central limit theorem formula:

$s = \frac{\sigma}{\sqrt{n}}$

Where σ is the population standard deviation and n is the sample size. Our standard deviation calculator might come in handy for preprocessing the data.

Defining the sample set, which is a subset of the population, is necessary to find the sample standard deviation because this depends on the size of the sample selected.

## Central limit theorem example

Let's say that we know the population standard deviation, σ, of the people's ages in a city is 35 years with a mean age of 60 years, and we're choosing 49 individuals through random sampling. In this central limit theorem calculator, do the following:

1. Type 60 as a population mean μ.
2. Input 35 for σ.
3. Input 49 for n.
4. Tada! The calculator shows the following results:
• The sample mean is the same as the population mean:
$\qquad \overline{x} = 60$
• The sample standard deviation (s) is 5 years, which is calculated as follows:
$\qquad s = 35 / √49 = 35 / 7 = 5$

Our central limit theorem calculator is omnidirectional, which means that you can also find the population standard deviation by keying in the sample standard deviation and sample size!

## Central limit theorem conditions

For the central limit theorem formulas to be applicable, certain conditions must hold true -

• The data must be sampled randomly.
• The sample values must be independent of each other. Usually, if random sampling is used, the data will automatically be devoid of any dependency bias.
• The sample size must be at least 30.
• The sample size must be at most 10% of the total population.

## How are samples selected in statistics?

There are several ways of sampling from a population, based on the requirement. Some of the well-known sampling techniques are:

• Simple Random Sampling;
• Clustered Sampling;
• Systematic Sampling;
• Stratified Sampling; and
• Convenience Sampling, etc.

Now that we know how to apply the central limit theorem formulas, we can also work with metrics that are specific to certain distributions. For instance, you could find the z-score or the empirical rule to evaluate certain characteristics of a normal distribution.

Even if you're looking for something very different, our wide range of might give you some useful tools!

## FAQ

### How do you determine the maximum sample size for a given population?

A good rule of thumb for the maximum sample size of a given population is 10% of the population size, or 1000, whichever is lower.

### What is the minimum sample size for applying the central limit theorem?

If the sample size is too small, the sample might not accurately reflect the overall data and may offer a skewed representation. Hence, a minimum sample size of 30 is recommended.

### What is the most accurate sampling method?

The best sampling method offers a reasonably correct inference about the population by using an unbiased subset of the population. More often than not, Simple Random Sampling (SRS) gives a sufficiently close representation of the overall population since the samples are chosen randomly and therefore avoid most types of biases.

Population mean (μ)
Population standard deviation (σ)
Sample size (n)
Sample mean (x̅)
Sample standard deviation (s)
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