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I was taught, that for any distribution having the 1st raw moment defined, the "expected value" is equal to it and it's also the arithmetic mean.

"Expected value" sounds to me like "typical value".

When we calculate the arithmetic mean on normally distributed data, we feel tells us about the "typical observations". In this case, the expected value is also one of the central tendency measures - again, arithmetic mean, but also median and mode. And central tendency also sounds to me like "typical observations".

So, for the normal distribution (but maybe also for symmetric, low-skewed ones), the arithmetic mean is both the central tendency and expected value and speaks about typical observations.

OK, now let's talk about the right-skewed data, for simplicity let's take the log-normal distribution. Also there, the expected value is the first raw moment, which is is the arithmetic mean of the observations. But for such distributions, I noticed people over and over report the geometric mean, justifying it that such data comes from a multiplicative process rather than additive, so the geometric mean is better central tendency measure.

And this creates my question: if both "expected value" and "central tendency" suggest by its name it's about "typical observations", why so many people prefer the geometric mean (which is also a central tendency measure, one of many)? Do I understand it correctly, that "expected value" is not necessarily about "typical values" (while the "central tendency" - is), and only for the normal distribution both terms overlap, so we use the arithmetic mean as both "the best central tendency measure" and the expected value, but when it comes to skewed distributions, we may prefer different central tendency measures?

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The term central tendency appears to have originated in psychology. It is so vague as to be nearly meaningless. It conveys very little about what it intended except in such sufficiently nice cases where almost any measure of it would agree with almost any other. It's a term that I think serves to obfuscate more than it enlightens and I've yet to see a book that used it that I didn't think would have been better served by omitting the term (or any similarly vague substitute) altogether.

You mention three measures of location but there are a great many other possible measures.

By contrast, expected value has an explicit and unambiguous definition.

the "expected value" is equal to it and it's also the arithmetic mean.

Expected value is a population quantity, the first moment of the distribution, as you say. Arithmetic mean typically means a sample quantity (the first sample moment). Mean might be used in either case, however.

"Expected value" sounds to me like "typical value".

And central tendency also sounds to me like "typical observations".

It can be dangerous to rely over-much on what things sound like. It's best to rely on specific definitions. And there you strike a problem, because central tendency doesn't have one (or at best, if it does have a definition, it won't tend to be one that says enough for us to make anything of it).

(but maybe also for symmetric, low-skewed ones),

If it's symmetric, it cannot be skewed at all.

Certainly in the case of symmetric, unimodal distributions, what we mean by 'center' is usually unambiguous. It's outside that obvious case that we have to worry about what we really mean when we say 'center'.

And this creates my question: if both "expected value" and "central tendency" suggest by its name it's about "typical observations", why so many people prefer the geometric mean (which is also a central tendency measure, one of many)?

  1. As a sample quantity, the geometric mean tends to useful in the same sorts of situations that the lognormal distribution is often used as a model (even when the model itself isn't being used)

  2. If you're seeking a way to estimate one of the population parameters of the lognormal, the geometric mean contains "all the information" about it that's in the data. So if you are thinking of a lognormal model, you will do well to be thinking of the geometric mean (or perhaps its log).

Do I understand it correctly, that "expected value" is not necessarily about "typical values"

'Expected value' is explicit, 'typical value' is not. It's not clear whether or not expected value is a typical value or counts as a measure of 'central tendency' until we are clear about what, exactly we intend those otherwise vague terms to encompass.

only for the normal distribution both terms overlap,

Not only the normal. Consider the logistic distribution or the Laplace distribution for example (both symmetric unimodal) -- when we say a word like 'center' it's obvious what we mean.

when it comes to skewed distributions, we may prefer different central tendency measures?

Certainly. Or to be clearer, it forces us to think about what it is we're trying to find out (beyond vague words), and when we have been more specific about what we want to achieve, we can then think about good ways to achieve it.

Often we have a specific goal in mind, and the goal itself will help guide us.

If we don't have a specific goal beyond say "describe this distribution", then with a general (perhaps skewed, perhaps not unimodal, possibly heavy-tailed) distribution, a single quantity on its own -- or even a couple of them -- may not be especially informative.

(e.g. if I tell you the values of the population mean and mode, can you even tell if the median will be to the left of both, to the right of both, equal to one or the other, or in between the two? If not, how much are they really telling you about the distribution? Something, certainly, but not necessarily a lot.)

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  • $\begingroup$ 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! $\endgroup$ – Goala Feb 1 at 11:43
  • $\begingroup$ 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. $\endgroup$ – Goala Feb 1 at 11:46
  • $\begingroup$ 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. $\endgroup$ – Goala Feb 1 at 11:50

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