# What is the Central Limit Theorem (TCL)?

The Central Limit Theorena (TCL) is a theory that is explained in statistics and that states that if we have a large sample drawn from the population, the distribution of the sample means will follow a normal distribution.

If the sample size increases, the sample mean will be closer and closer to the population mean, so the TCL will be a good indicator to determine the distribution of the sample mean of a population with a known variance.

## Properties of the Central Limit Theorem

The most outstanding properties are the following:

• Population mean and sample mean are equal. The mean of the distribution of all sample means is equal to the mean of the entire population.
• The variance of the distribution of the sample means is calculated: σ² / n (population variance between sample size).
• If the sample size is large (n> 30), the distribution of the sample means will follow a normal distribution. This applies regardless of the form of the distribution with which we are working.

Finally, indicating that the distribution of the sample means resembles a normal one is very useful and beneficial. In this way, we can apply hypothesis tests and confidence intervals when we need it (statistical inference). Let's just say things don't get that complicated.

The TCL allows us, in this way, to make inference about the population mean through the sample mean and to apply valid statistical methods of great utility. This allows people who have to perform an analysis, contrast or inference, to collect data from the entire population in a simpler way: through its sample mean.

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