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Can someone give me separate cases of empirical distributions where:

  1. mean does not accurately describe the central tendency.
  2. median does not accurately describe the central tendency.
  3. mode does not accurately describe the central tendency.

So for the first one, it occurs when you have a far-off outlier in the data set. What about the other 2?

Sources:

Central Tendency: https://en.wikipedia.org/wiki/Central_tendency

kjetil b halvorsen
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NoName
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    Without a clear/precise definition of what you intend by "central tendency" the question seems too vague to answer. It probably won't help to scour the textbook for whatever class you encountered the term in (or whatever other context), because they're also generally so vague about it as to convey nothing of any use. I'm yet to encounter a definition that was both broadly applicable\* (i.e. not strongly limited, such as say to symmetric unimodal distributions) and that I thought was worth taking seriously. – Glen_b Jan 20 '20 at 00:13
  • Also note that "outlier" is typically a term relating to an observation (an element of a sample), rather than to a distribution, which is an attribute of a population. – Glen_b Jan 20 '20 at 00:19
  • @Glen_b-ReinstateMonica Okay I updated the sources to include definition of central tendency. – NoName Jan 20 '20 at 00:30
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    That's a link to an article on central tendency. The nearest it comes to a *definition* is "a central or typical value for a probability distribution" which is exactly what I was getting at - so vague as to be *of no use at all* for answering your question. If you have no better definition, we're still in the same place. – Glen_b Jan 20 '20 at 01:23
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    Is this an exercise for a class? – Glen_b Jan 20 '20 at 01:23
  • @Glen_b-ReinstateMonica No, it is not homework. I want to know when to use each to determine the central tendency since "the most common measures of central tendency are the arithmetic mean, the median and the mode." – NoName Jan 20 '20 at 02:12
  • Without a clear definition of what it is we're trying to achieve (what this center *is* in general), how can we say one measure of center is better than another? And why one of those three rather than some other? – Glen_b Jan 20 '20 at 05:17
  • Relevant and closely related: [Why is median age a better statistic than mean age?](https://stats.stackexchange.com/questions/2547) – whuber Jan 20 '20 at 15:16
  • @whuber Yep, I already know cases where 1. is true. Just not sure about the other two. – NoName Jan 21 '20 at 15:17
  • When there are multiple modes, what would "central tendency" even mean? – whuber Jan 21 '20 at 16:49
  • @whuber I guess in mathematical terms: minimizing the summation of {the absolute distance of each value to x}, where x is the chosen central tendency. – NoName Jan 22 '20 at 04:10
  • That *defines* the median, so it looks like your question is tantamount to asking when the median and mean differ and when the median and a mode differ. Because the first has been answered (I referenced it earlier), it remains to construct a dataset with a mode far from the median. I suspect you could do that yourself easily. – whuber Jan 22 '20 at 17:11

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