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Comment by rcxdude | Hacker News Reader

Nah, there's nothing in a box plot which assumes a bell-shape. It does, however just visualise the parameters which reasonably well characterize a smooth single-mode distribution regardless of the underlying distribution. So it's a valid criticism of using box plots, especially when the alternatives can just as well visualise a bell-shaped distribution, as well as showing when it is not.

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The problem is that the four quantile groups contain equal numbers of items, but are not represented by equal areas, even if we replace the whiskers with a bar of the same width as the box.

The bottom whisker contains 25% of the data, yet is just a thin line, which can furthermore be arbitrarily short.

It really is a dumb visual presentation.

The only way to use it is to recover the five parameters from it, and then stop looking at it.

For that purpose, a QR code would be just as good, if not better. You'd need a device with a camera to get the parameters (but "everyone" has that now), and when you're looking at it with your bare eyes, it doesn't tell you any visual lie.

> The only way to use it is to recover the five parameters from it, and then stop looking at it.

...which is its intended use case since Tukey invented it as a way of visualising the "5 number summary". I think part of his criteria were that it should be easy to make by hand which is clearly no longer a consideration so there are plenty of reasons to just do something else most of the time these days.

> smooth single-mode distribution

That IS a bell curve. While it's true that the Guassian distribution is often called a bell curve or even "the" bell curve, a non-Guassian single mode distribution is still absolutely bell shaped in a general sense.

So, although you started your comment with "nah", you're actually in agreement with the content you replied to.

In the mathematical sense this is clearly not true - it’s easy to come up with a smooth single mode distribution that doesn’t look like a bell.

Is it? How?

You could include a lot of little bells far from the single mode, but that's reading a little too much into the literal meaning of "single mode" - a "bimodal" distribution isn't one where the two most common values are both modes. It's one where there are two distinct local maxima.

The tails to the left and right must asymptotically approach zero (or you don't have a smooth distribution, because you have discontinuities somewhere), and if there's just one local maximum, your curve will look like a bell.

The exponential distribution (modal value 0) is not bell shaped. If you don't like it's range of non-negative, then take some smooth mollification

And the smooth mollification will look like...?

in the simplest case... just mirror it (some call this a Laplace distribution). if you don't like how it's not differentiable at the mode there are further smoothings (see, e.g., the wikipedia article for this distribution) but this simple construction is continuous.

It looks like a spike, not a bell.

A spike is not smooth (typically meaning continuous in the variable and its first derivative), which was one of the conditions.

Then take a Cauchy or a t-distribution. Basically anything with a longer tail than exp(x^2). The Gaussian summary will be misleading because of the tails.

Yes or the chi-square distribution with k=1 or 2 or any other of the gamma distributions[1] with the right parameters will have a shape that is one-sided with the mode at the lower extreme and no "low tail" in the normal sense.

[1] https://en.wikipedia.org/wiki/Gamma_distribution or https://stats.libretexts.org/Bookshelves/Probability_Theory/...

The bell curve IS the smooth single mode with LOWEST ENTROPY.

Do you mean greatest entropy? Not if the support is, for example, the positive reals.

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