Khan Open Class-Statistics Study Notes: (6) central limit theorem

What is the central limit theorem?

Central Limit Theorem: sets the mean value to μ and the variance to σ ^ 2. (limited) samples with a sample size of n are extracted from any population. When n is sufficiently large, the sample distribution of the sample mean is approximately the normal distribution of the mean and the variance is σ ^ 2/n.

Note: The original distribution does not have to conform to the normal distribution. It can be any distribution, which can be discrete or continuous, that is, there is no requirement. N is the sample size, n samples are taken each time, and the mean of each sample is :. The distribution of the mean values of these samples is like a normal distribution. When n reaches a certain value, the frequency distribution of sample mean is well consistent with normal distribution.

In the same number of samples, for example, taking the mean value of 10000 samples, the larger the n value, the closer the skew and kurtosis are to zero, indicating that the larger the n value, the closer the normal distribution. It can be imagined that n = 1 is like the original distribution. When n → ∞, the mean is μ, and the normal distribution without variance is a straight line.

The normal distribution is applicable to the sample mean, regardless of the original distribution, that is, sampling distribution of the sample mean. When the sample size n is larger, it tends to be a normal distribution and converges quickly, that is, when n is 10 or smaller, it is very similar to the normal distribution, so it is widely used.

Note that it is a trend. For example, some distributed systems have a range, and the normal distribution is 0 only when x tends to be ± ∞.

Examples of central limit theorem

The central limit theorem has a wide range of applications. For example, we know the mean and standard deviation of a distribution (which can be non-normal distribution), for example, an example of outdoor drinking water, when we organize a team of 50 people, the average drinking water volume of these 50 people conforms to the normal distribution, and the probability of water consumption is 90%. This is an example of derivation and.

In actual application scenarios, the general sampling condition is known, but the general situation is unknown, and the general situation needs to be estimated, as shown in the following example of confidence interval: you sample 36 apples from you farm's harvest of over 200,000 apples. the mean weight of the sample is 112 grams (with a 40 gram sample standard deviation ). what is
Probability that the mean weight of all 200,000 apples is within 100 and 124 grams?

Here is a sample set. The 36 samples (or one sample set with n = 36) = 112, and the σ of the 36 samples is 40. Note that a sample set is provided here, not the sampling distribution of the sample mean of multiple sample sets. The total variance can be estimated from the sample variance.

Do you still remember the sample variance without deviation? Is divided by n-1, rather than n. The sample variance without deviation can be approximately the total variance σ, and then the variance of the sample mean can be obtained.

This question is equivalent

Then, check the Z table. The Z score is approximately 1.8. The probability of this interval is 92.8% in the Z table.

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