Central Limit Theorem

The central limit theorem (CLT) states that if we take a large enough number of samples from a population (or a large enough sample size from a population with a random outcome), the distribution of the sample means (the average values from each sample) will approximate a Normal Distribution, regardless of the original population distribution.

The phenomenon can be observed with the Galton Board.

The idea is the following:

• Start with a random variable $X$ (like a die)
• Add $N$ samples of this variable ($X_1+X_2+\cdots +X_N$)
• The distribution of this sum looks more like a gaussian distribution as $N\to \infty$.

Or more formally:

$$\underset{N\to \infty }{\lim }P(a<\frac{(X_1+\cdots +x_N)-N\cdot \mu }{\sigma \cdot \sqrt{N}}<b)={\int }_a^b\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}(\frac{x}{\sigma }{)}^2}dx$$

which can be read like this: “If we take $N$ random variables, we compute how many std devs away form the mean is their sum, then the probability of this value being between two values $a$ and $b$ is equal to the area under the normal distribution curve between those same points”

Three assumptions have to be true in order for this theorem to be applicable:

1. All the $X_i$ must be independent from each other;
2. Each $X_i$ is drawn from the same distribution;

These two assumptions, most of the times related as i.i.d (independent and identically distributed), can be relaxed. In the Galton Board these are not respected, but a normal distribution comes out anyway.

3. The variance of the $X_i$ must be finite.

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tags: statistics resources:

• But what is the Central Limit Theorem? - YouTube