2
$\begingroup$

This histogram is part of a task about descriptive statistics. I thought it would be easy, and it is, but i am not sure about this one. First I described this histogram as slightly positively skewed. The skewness coefficient supported this. But then those three peaks caught my eyes and i was confused, because i've learned that it wouldn't make any sense to talk about skewness (or kurtosis) if the distribution is a bimodal or multimodal one. So, my question is one of several. Mainly: Is this a multimodal distribution? Can you say this for sure? Or is this a matter of interpretation? Along these lines the secondary questions are: If it is a multimodal distribution and especially if it is a matter of interpretation, would it be reasonable to describe the skewness, the kurtosis and the seemingly multimodal aspect? Because: from the histogram and the coefficients it seems i could say something about all three aspects. Or is it truely a strict rule, that you can't really talk about skewness/kurtosis, if it is bimodal or multimodal? In this case, i would be back to my initial question: Is this a multimodal distribution and can you be sure?

Histogram

$\endgroup$
  • 3
    $\begingroup$ Welcome to our site, Arina! This is a very well-posed question. You would likely enjoy glen_b's discussion of ambiguity in histograms in his answer at stats.stackexchange.com/questions/51718. $\endgroup$ – whuber Jun 2 '15 at 22:45
  • 3
    $\begingroup$ Small comment. I take it that the vertical axis is frequency so that the values shown must be integers. Axis labels including 2.5, 7.5, etc. are then not a good choice here. Better would be 0 5 10 15. or 0 4 8 12 16. Less small comment. A normal is not a good reference distribution for a bounded variable. You can see that substantial fractions predicted below 0 and above 100 make that an absurd reference. Naturally, both these details could be default choices from your software. $\endgroup$ – Nick Cox Oct 6 '17 at 13:48
  • $\begingroup$ See also: How to test if my distribution is multimodal?. $\endgroup$ – Scortchi - Reinstate Monica Oct 6 '17 at 13:48
1
$\begingroup$

You can fit various types of distributions, multimodal and unimodal, and assess model fit using statistics like BIC. I would guess, given your histogram, that the different distributions will have similar fit, so it will be difficult to claim that the distribution is in fact multimodal. If you had more pronounced dual (or more) peaks, then I would guess that the data would better support bimodality (or multimodality) based on measures of model fit. But it's hard to say without actually fitting those distributions and looking at the model fit statistics.

I want to comment on kurtosis though. I have seen people say that low kurtosis indicates bimodality, while large kurtosis indicates unimodality. This is patently false. Take a bimodal distribution with very small kurtosis. Now mix it with a much wider distribution, with small mixing probability. The resulting distribution will have exactly the same bimodality, but huge kurtosis. Kurtosis measures nothing about the peak (flatness, sharpness, or modality). It measures the outlier (potential rare, extreme observation) characteristic of a distribution only. See https://en.wikipedia.org/wiki/Talk:Kurtosis#Why_kurtosis_should_not_be_interpreted_as_.22peakedness.22 for a clear explanation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.