## Central Limit Theorem Definition

The central limit theorem states that the random samples of a population random variable with any distribution will approach towards being a normal probability distribution as the size of the sample increases and it assumes that as the size of the sample in the population exceeds 30, the mean of the sample which the average of all the observations for the sample will b close to equal to the average for the population.

### Central Limit Theorem Formula

We have already discussed that when the sample size exceeds 30, the distribution takes the shape of a normal distribution. For determining the normal distribution of a variable, it is important to know its mean and its variance. A normal distribution can be stated as

**X ~ N (µ, α)**

Where

- N= no of observations
- µ= mean of the observations
- α= standard deviation

In most of the cases, the observations do not reveal much in its raw form. So it is vital to standardize the observations to be able to compare that. It is done with the help of the z-score. It is required to calculate the Z-score for an observation. The formula to calculate the z-score is

**Z=(X- µ) / α /√n**

Where

- Z= Z-score of the observations
- µ= mean of the observations
- α= standard deviation
- n= sample size

### Explanation

The central limit theorem states that the random samples of a random population variable with any distribution will approach towards being a normal probability distribution as the size of the sample increases. The central limit theorem assumes that as the size of the sample in the population exceeds 30, the mean of the sample, which the average of all the observations for the sample, will be close to equal to the average for the population. Also, the standard deviation of the sample when the size of the sample exceeds 30 will be equal to the standard deviation of the population. As the sample is randomly chosen from the whole population and the size of the sample is more than 30, then it helps in hypothesis testing and constructing the confidence interval for the hypothesis testing.

### Examples of Central Limit Theorem Formula (with Excel Template)

#### Example #1

**Let’s understand the concept of a normal distribution with the help of an example. The average return from a mutual fund is 12%, and the standard deviation from the mean return for the mutual fund investment is 18%. If we assume that the distribution of the return is normally distributed, then let us interpret the distribution for the return in the investment of the mutual fund.**

Given,

- The mean return for the investment will be 12%
- The standard deviation will be 18%

So, to find out the return for a 95% confidence interval, we can find it out by solving the equation as

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**Upper Range = 12 + 1.96(18) = 47%****Lower Range = 12 – 1.96(18) = -23%**

The result signifies that 95% of the time, the return from the mutual fund will be in the range of 47% to -23%. In this example, the sample size, which is the return of a random sample of more than 30 observations of return, will provide us the result for the population return of the mutual fund as the sample distribution will be normally distributed.

#### Example #2

**Continuing with the same example, let us determine what will be the result for a 90% confidence interval**

Given,

- The mean return for the investment will be 12%
- The standard deviation will be 18%

So, to find out the return for a 90% confidence interval, we can find it out by solving the equation as

**Upper Range = 12 + 1.65(18) = 42%****Lower Range = 12 – 1.65(18) = -18%**

The result signifies that 90% of the time, the return from the mutual fund will be in the range of 42% to -18%.

#### Example #3

**Continuing with the same example, let us determine what will be the result for a 99% confidence interval**

Given,

- The mean return for the investment will be 12%
- The standard deviation will be 18%

So, to find out the return for a 90% confidence interval, we can find it out by solving the equation as

**Upper Range = 12 + 2.58(18) = 58%****Lower Range = 12 – 2.58(18) = -34%**

The result signifies that 99% of the time, return from the mutual fund will be in the range of 58% to -34%.

### Relevance and Use

The central limit theorem is extremely beneficial as it allows the researcher to predict the mean and the standard deviation of the whole population with the help of the sample. As the sample is randomly chosen from the entire population and the size of the sample is more than 30, then any random sample size taken from the population will approach towards being distributed normally, which will help in hypothesis testing and constructing the confidence interval for the hypothesis testing. Based on the central limit theorem, the researcher is able to choose any random sample from the whole population, and when the size of the sample is more than 30, then it can predict the population with the help of the sample as the sample will follow a normal distribution and also as the mean and the standard deviation of the sample will be same as the mean and the standard deviation of the population.

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