How does the Central Limit Theorem explain the empirical rule? student question
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"The empirical rule is the statistical rule stating that for a normal distribution, almost all data will fall within three standard deviations of the mean. the empirical rule shows that 68% will fall within the first standard deviation, 95% within the first two standard deviations, and 99.7% will fall within the first three standard deviations of the mean." - Investopedia
"the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined (finite) expected value and finite variance, will be approximately normally distributed, regardless of the underlying distribution."- wiki
"the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined (finite) expected value and finite variance, will be approximately normally distributed, regardless of the underlying distribution."- wiki
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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.
Well, is it not as easy as saying this: Given a sufficiently large number of iterates of independent random variables... they will be normally distributed (with characteristics of the normal distribution) and thus the empirical rule applies? Now I'm curious as to the actual answer. It almost seems like this question is asking which came first, the CLT or the empirical rule? Maybe I don't understand the question, but it's vague nonetheless.
The CLT doesn't explain the empirical rule. They are separate theorems/facts only related because they both have something to do with the normal distribution. But the CLT has nothing to do with the explanation for why the empirical rule is true and the empirical rule has nothing to do with why the CLT is true.
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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