Central Limit Theorem (The Central Limit Theorem) is the basis for a lot of statistics to prove its practical significance of this paper describes the central limit theorem.
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Proof of Central Limit Theorem
The practical significance of the central limit theorem
Proof of Central Limit Theorem
From a uniform distribution (the probability of taking any value from 0 to 1 in the left picture is equal), calculate the mean value of a set of 20 samples at random, and draw a mean histogram ; then take 10 sets of samples, 20 sets of samples, and 30 sets in turn Sample,...100 groups of samples, calculate 10 mean, 20 mean, 30 mean,...100 mean..., draw a histogram corresponding to the mean, we will Surprisingly found that the mean distribution tends to the normal distribution , and this is the central limit theorem ;
In the same way, randomly sampling from an exponential distribution and drawing a mean histogram, we will again be surprised to find that the mean distribution also tends to a normal distribution ;
Regardless of whether the distribution of the initial data is uniform or exponential, after random sampling and calculation of the mean, the mean obeys the normal distribution.
After further experimentation, it is found that no matter what the initial distribution of the data , random sampling many times, the sample mean obeys the normal distribution .
The practical significance of the central limit theorem
You don’t need to know what kind of distribution the original data obeys. You only need to know that according to the central limit theorem, the data mean obeys a normal distribution. You can use this property to make corresponding explorations. The following is a wonderful statement of the course:
For multiple sets of data, we don't need to know the specific distribution of each set of data.
According to the central limit theorem, we only need to know that the sample mean obeys a normal distribution .
Using the normal distribution of the mean , we can find the confidence interval, t test whether there is a difference in the mean of the two sets of samples, analysis of variance to compare whether there is a difference in the mean of the three or more sets of samples, and all other tests that can be done with the mean.
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