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could you please state what the empirical rule is?

"the

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You might have answered your own question. How do those two definitions relate to one another?

hi,

there should be a misunderstanding here: the empirical rule you quote is not an empirical rule, it is a math fact given the formula of the normal distribution. I'd be very surprised if this small factoid explained the CLT, knowing how difficult the proof of that theorem is

regards

there should be a misunderstanding here: the empirical rule you quote is not an empirical rule, it is a math fact given the formula of the normal distribution. I'd be very surprised if this small factoid explained the CLT, knowing how difficult the proof of that theorem is

regards

btw, this was a question from an exam.

The CLT and the empirical rule are unrelated. Let me make a few points.

1. The Gaussian (Normal distribution) is a mathematical abstraction i.e., there is not any single set of data that will exactly follow a normal distribution. This is why there are statistical tests to determine if a set of data approximate a normal distribution e.g. Anderson-Darlington test, Kolmogorov-Smirnov test, Shapiro-Wilks, etc.

2. The primary CLT (note that there are corollaries to the primary Central limit Theorem) is associated with the sampling distribution of the Means - not an underlying distribution of a data set, such as IQ scores, which are well known to follow a normal distribution with a mean of 100 and standard deviation of 15. The CLT applies to any underlying distribution that has finite mean and finite variance - this would exclude theoretical distributions such as a Cauchy distribution or t-distributions with 1 or 2 degrees of freedom. The convergence rate of a sampling distribution of Means is contingent on the underlying distribution. For example, if the underlying non-normal distribution is continuous Uniform then the sampling distribution of means will converge more quickly (i.e., smaller sample size) then an sampling distribution of means that have an Exponential distribution.

3. The Empirical Rule is really associated with Gaussian type underlying distributions (not a sampling distributions of means) that can approximate the probabilities of the area under the curve e.g., 68.26% between -1 and +1 standard deviation above and below the mean.

1. The Gaussian (Normal distribution) is a mathematical abstraction i.e., there is not any single set of data that will exactly follow a normal distribution. This is why there are statistical tests to determine if a set of data approximate a normal distribution e.g. Anderson-Darlington test, Kolmogorov-Smirnov test, Shapiro-Wilks, etc.

2. The primary CLT (note that there are corollaries to the primary Central limit Theorem) is associated with the sampling distribution of the Means - not an underlying distribution of a data set, such as IQ scores, which are well known to follow a normal distribution with a mean of 100 and standard deviation of 15. The CLT applies to any underlying distribution that has finite mean and finite variance - this would exclude theoretical distributions such as a Cauchy distribution or t-distributions with 1 or 2 degrees of freedom. The convergence rate of a sampling distribution of Means is contingent on the underlying distribution. For example, if the underlying non-normal distribution is continuous Uniform then the sampling distribution of means will converge more quickly (i.e., smaller sample size) then an sampling distribution of means that have an Exponential distribution.

3. The Empirical Rule is really associated with Gaussian type underlying distributions (not a sampling distributions of means) that can approximate the probabilities of the area under the curve e.g., 68.26% between -1 and +1 standard deviation above and below the mean.

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