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I am a little confused about what assumptions are needed to meet the Empirical Rule in statistics. Does the population need to follow the normal distribution or the sample need to follow the normal distribution?

As far as I know, the ER is only applied to a normal distribution and if the sample follows a normal distribution then the population must follow a normal distribution. So assuming a sample follows a normal distribution also guarantees that the population it came from also follows a normal distribution. So, the assumption, the sample follows a normal distribution, is enough to say the population follows a normal distribution. If we say the population follows a normal distribution then the sample also follows a normal distribution. So the ER can be also applied with the assumption stating "the sample follows a normal distribution."

I am just trying to show my understanding about my question. If I am not correct, please correct me.

Nick Cox
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StoryMay
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  • The "Empirical Rule" is in effect one, two or three standard deviations from the mean, with the numbers based on the assumption of the two tails of a normal distribution. But this started from Fisher saying that in practice he found two standard deviations a practical number on which to base investigating further. – Henry Oct 28 '20 at 12:27

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The empirical rule applies when the observations appear to be drawn from a population that is normal. That population could be standard normal or $N(-57632, 838074)$. It’s just the normality that matters.

Dave
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  • Thank you for the answer. If the assumption considers the population to be normal, is there anyway to check if a population follows a normal distribution. Or do we just assume that the population follows a normal distribution? Can I also apply the ER to the Central Limit Theorem? – StoryMay Oct 28 '20 at 09:55
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    Graphical examination is the way to go. Three popular graphs for this are histograms, normal quantile-quantile plots, and kernel density estimates (more or less a smooth histogram). Some references will push formal hypothesis testing such as Kolmogorov-Smirnov or Shapiro-Wilk. The general sentiment on here is that those are not particularly helpful. With many observations, you will be able to reject normality, even if the deviation from normality is quite minor. (The empirical rule is approximate, anyway.) – Dave Oct 28 '20 at 10:00
  • I do not follow what you mean about applying ER to CLT. Please post that as a separate question. This question of mine might interest you, too: https://stats.stackexchange.com/questions/473455/debunking-wrong-clt-statement. – Dave Oct 28 '20 at 10:02
  • Yes, I should have put my comment in another question. Can you please stretch the last part, "With many observations, you will be able to reject normality, even if the deviation from normality is quite minor. (The empirical rule is approximate, anyway.)", in a bit more plainer way to understand? – StoryMay Oct 28 '20 at 10:22
  • (+1). Normal quantile plots I particularly support. Harold Jeffreys made the intriguing suggestion that high-quality datasets that essentially are measurement errors around a constant being estimated are typically closer to Student's $t$ with 7 df. Normal quantile plots are the method most likely to tell the difference, but you need a decent-sized sample in any case. – Nick Cox Oct 28 '20 at 10:54
  • @NickCox This is the first I've heard about $t_7$. Could you please post a reference? I'd be curious to read his rationale. – Dave Oct 28 '20 at 10:55
  • His rationale was _empirical_! It's summarized in his _Theory of Probability_, any edition. The original work was around 1938. I never found the original datasets, unsurprisingly. – Nick Cox Oct 28 '20 at 10:58
  • I think https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1938.0008 is the reference. It is rather dense and detailed but Hampel, Ronchetti, Rousseeuw and Stahel give a summary in their robustness book. – Nick Cox Oct 28 '20 at 15:28
  • On checking both references I see that I over-simplified the rather nuanced discussion. – Nick Cox Oct 29 '20 at 09:00
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I think the whole point is that it is an empirical rule, which often applies in practice (except that it is an approximation, and except when it doesn't).

The rationale for the empirical rule is just that many distributions resemble the normal or Gaussian in this respect. There is no theorem behind this, and it's easy to find exceptions.

The expression says as much about English usage as it does about statistics. The word rule is overloaded, and there are several meanings. Here are some:

  1. Identifying a procedure that must be followed, as say rules in elementary algebra for handling expressions given elements, operators, parentheses and the like, or rules in chess on how pieces may move, take each other, and so on. A rule of this kind requires understanding, but leaves no scope for discussion, except that if you want chess or an algebra with different rules, it's a different game. Sometimes a rule of this kind could be called a law.

  2. A law or regulation in some sense implying penalties if broken and apprehended, say to do with money handling or forbidden actions (being offside, swearing at an umpire or referee) in some sport.

  3. A rule (in British English, and perhaps in other dialects too, often "a rule of thumb") which is an approximation to what happens in nature or society. These run the spectrum from serious to facetious, as in a conifer growing 1 foot yearly, or 10 cm of fresh snow being the equivalent of 1 cm of rain, or 20% of the people doing 80% of the work.

The rule here is closest to sense #3.

Nick Cox
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