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Jun 11, 2020 at 14:32 history edited CommunityBot
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Feb 12, 2020 at 2:11 history edited Glen_b CC BY-SA 4.0
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Feb 12, 2020 at 2:11 history edited Glen_b CC BY-SA 4.0
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Feb 1, 2020 at 12:46 vote accept Goala
Feb 1, 2020 at 11:50 comment added Goala Now I can see this difference. In terms of moments - the first moment is the arithmetic mean and the sample AM is its BLUE estimator. But it doesn't mean this measure is the best descriptor of the "central tendency" (regardless of how imprecise this term is), so people invented also the other measures (Wikipedia lists a lot of them!), trying to help us in certain cases, like harmonic, geometric... In case of the normal d. both the AM understood as the central tendency measure and as the 1st moment "serve" equally well. In other cases, the AM, even being a CT measure, has better alternatives.
Feb 1, 2020 at 11:46 comment added Goala For example, when I take the arithmetic mean and standard deviation calculated on data coming from the log-normal distribution, the average - 3xSD may give me negative outcomes, I saw an example in the internet. While I saw it, it was very obvious. This would never happen with the geometric mean divided by 3 geometric standard deviations. So the latter measures are "better descriptors" of this king of right-skewed data, not the arithmetic mean. BUT, from "theory perspective", rather than the "applied perspective", the arithmetic mean has a very important position - the first raw moment.
Feb 1, 2020 at 11:43 comment added Goala Thank you for this awesome answer. I struggle for years trying to understand the relationships and differences between the "expected value" (something we expect, so that's why I said "typical": "looking by the distribution I expect this to be typical, "not surprising me", "something I find normal in this data set") and the "central tendency" (something "central to the distribution", so ..... again :) But the terms can be confusing. The arithmetic mean doesn't look very "representative" to the log-normal distribution, also because it's more about additive outcomes, not multiplicative.Thank you!
Feb 1, 2020 at 5:04 history edited Glen_b CC BY-SA 4.0
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Feb 1, 2020 at 4:53 history edited Glen_b CC BY-SA 4.0
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Feb 1, 2020 at 4:47 history edited Glen_b CC BY-SA 4.0
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Feb 1, 2020 at 4:41 history answered Glen_b CC BY-SA 4.0