Created at 181123
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# [[Epistemic status]]
Last modified date: 181123
# Central limit theorem
The central limit theorem states that the distribution of sample means approximates a normal distribution as the sample size becomes large, regardless of the population's distribution.
Consider a dice roll. The outcomes can be 1, 2, 3, 4, 5, or 6, each with an equal probability of 1/6. If we roll the dice once, the distribution of outcomes is uniform. However, if we roll the dice multiple times and calculate the average outcome, the distribution of these averages will approximate a normal distribution, as per the CLT.
Imagine you have a fair six-sided die. Each side (1, 2, 3, 4, 5, 6) has an equal probability of 1/6.
1. Single Roll: If you roll the die once, the outcome is equally likely to be any of the six numbers. The distribution of outcomes is uniform.
2. Multiple Rolls: Now, let's say you roll the die 10 times and calculate the average outcome. You might get something like (3+5+2+6+1+4+3+2+6+4)/10 = 3.6.
3. Repeat the Process: Repeat this process of rolling 10 times and calculating the average many times, say 1000 times.
4. Distribution of Averages: Now, if you plot the distribution of these 1000 averages, it will not be uniform anymore. Instead, it will approximate a normal distribution, even though the original distribution (a single die roll) is not normal. This is the essence of the Central Limit Theorem.