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For example, I have a distribution with mean = 55.46; med = 54.5; mode = 45. The Shapiro-Wilk is non significant, the data are unimodal, and there is no significant skew or kurtosis.

At a glance, the mode looks different from the mean and median. Does the non significance of the Shapiro-Wilk etc. mean that they are the same? If so, why bother discussing equivalence of the mean, median and mode, since it's not telling us any new information. If not, how to you work out the cutoffs for these numbers? How close is close enough?

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    $\begingroup$ Symmetric distribution is not the same as mean = median = mode (as your title implies). For example, there can be symmetric bimodal distribution where mean =median but the two do not equal mode. $\endgroup$ Apr 13, 2016 at 14:51
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    $\begingroup$ There can be asymmetric distributions with mean, median and mode equal and they need not be pathological. For example, binomial distributions often qualify. More seriously, near equality of these measures doesn't rule out a small fraction of outliers which could dominate analysis. That is a bigger deal than the result of Shapiro-Wilk. What's the sample size here? $\endgroup$
    – Nick Cox
    Apr 13, 2016 at 15:01
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    $\begingroup$ And re what I mean by "empirical dataset": I mean that you are testing whether an empirically observed sample comes from a symmetric distribution. The contrast would be to determine whether a theoretical distribution is symmetric or not. For instance, a mixture of a $N(0,1)$ and a $N(5,1)$ distribution with equal weights will be symmetric (but certainly not unimodal), while a mixture with weights 0.49 and 0.51 will not by symmetric any more - but if all you have is a sample from such a mixture, you may not be able to detect the (slight) asymmetry. $\endgroup$ Apr 14, 2016 at 9:55
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    $\begingroup$ This question remains difficult to understand because we don't know what you've been taught and what the expectations are. As several comments already imply, it is hard to discuss these questions even at intemediate level without mentioning complications and exceptions that are possible, or even common, but often not emphasised in elementary or introductory courses. $\endgroup$
    – Nick Cox
    Apr 14, 2016 at 10:41
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    $\begingroup$ FWIW, my own view is that teaching Shapiro-Wilk at an introductory level often produces as many learning difficulties as it solves, but that's true of a lot of statistics learning. Learning statistics is highly spiral: as you revisit even basic topics you continually revise your view of what's crucial. Even what's unimodal is a judgement call as often a secondary peak on a histogram will be dismissed by experienced people as likely to be a sampling quirk. Not that this answers your question.... $\endgroup$
    – Nick Cox
    Apr 14, 2016 at 10:44

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