# Central limit theorem (CLT) de Moivre discovered that when you repeat tossing a fair coins many times (could be different number of tosses each time) then the average of heads will be centered around the 1/2. Constructing a histogram of number of experiments w.r.t average heads will always have a bell shape i.e., a Gaussian curve a.k.a the normal distribution. The same happens when sampling any distribution - the generated histogram is always a bell centered around the distribution's expectation. for example: when tossing a fair coin $10,000$ times, there is a probability of 0.95 that the mean of number of heads will be in the interval $(0.5-0.01,0.5+0.01)$. If we run an experiment and we get a mean of 0.53 then we conclude that the coin is not fair. More generally, when sampling independently the same distribution multiple times and compute the mean - the CLT state that the set of means generates a normal distribution. note that this is true no matter what type of distribution used in the original sampling. Philosophically, the CLT is interesting as it forces a random-phenomena (tossing coins) to have a well defined structure that can be analyzed mathematically. It is a structure that is revealed when looking at a large number of chaotic event. The [[Null Hypothesis]] method is based on the CLT. ## The CLT Algorithm - output: estimation of the mean of large set of values without measuring all values. - Randomly sample small number values - compute their mean - repeat the process many time - output the mean of the means. This is a good approximation of the large set mean according to CLT. Reference:[[@What is Random? Chance and Order in Mathematics and Life]] Related: [[Covariance]], [[Law of large numbers]] ## Created 2022-03-01 13:23