Scott's and Freedman–Diaconis rules of the thumb are based on the following formula:

In the article here it is stated that:

While these appear to be useful estimates for unimodal densities similar to a Gaussian distribution, they are known to be suboptimal for multimodal densities.

My question is, will Scott and Freedman–Diaconis rules of the thumb estimate the correct number of bins on distributions with more than one peak?

What are the disadvantages of these methods, and how to overcome them?

  • 2
    $\begingroup$ In my experience kernel density estimators are better at finding modes than are histograms. If your final output needs to use histograms, I suggest using a KDE first and then adjusting histogram bins to show modes you know are there. $\endgroup$
    – BruceET
    Sep 30, 2018 at 16:54

1 Answer 1


Comment continued. Here is a mixture of three normal samples (each of size 50) with means sufficiently far apart, relative to their standard deviations, to show separate modes. The default binning in R provides a histogram that does find the modes. The default KDE in R (with the default bandwidth) roughly matches the three modes (at 12, 18, and 25).

x = cbind(rnorm(50,12,2), rnorm(50,18,2), rnorm(50,25,2))
hist(x, prob=T, col="skyblue2"); rug(x)
lines(density(x), col="red", lwd=2)

enter image description here


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