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Here's something I've been wondering for a while. Normal distributions as most of you know look like this (standard normal from -4 to 4):

Standard normal distribution produced in R

But in textbooks and other serious sources, one often sees images of distributions presented as "normal" but that obviously aren't. These false normals are typically more rounded at the vertex, as if one was using a spherical approximation when trying to build a parabolic mirror. They look like this (from Investopedia):

enter image description here

or like this (on Wikipedia):

enter image description here

Here's an example (from here) of an especially bad non-normal curve used to introduce the normal distribution (most sources don't go this far):

enter image description here

What's up with these, why are these rounded-top distributions presented as normal in otherwise serious sources? Are they a legacy of times when it was hard to draw normal curves by hand? Do they merely allow more space for writing things under the curve?

EDIT: @Therkel produced these comparisons in R:

enter image description here enter image description here

As they point out, the Wikipedia image has a strange scaling.

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  • $\begingroup$ That first weird "one" looks to me like it just has a somewhat high variance relative to the length scale of the figure. The second "weird" one looks like the first one at the vertex to me, but the tails seem fatter than they should be (but then, that could be because it is a "schematic" and this makes the tails easier to see). That last one is horrible, though, it has inflection points that shouldn't be there. $\endgroup$ – Ian Feb 17 '17 at 18:21
  • $\begingroup$ It's hard to tell for sure because the variance and vertical scales vary between the pictures. It would be interesting to actually look for the best-fit Gaussian to each. $\endgroup$ – Nate Eldredge Feb 17 '17 at 18:49
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    $\begingroup$ @TheThunderChimp Great question! I have added R-generated versions to your two first pictures to easier see the difference. http://i.imgur.com/ZXmL4te.png http://i.imgur.com/AbSUs5k.png. Notice how the Wikipedia x-axis has a strange scaling. $\endgroup$ – Therkel Feb 18 '17 at 12:43
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    $\begingroup$ @Therkel Wow these are great, the differences are much less subtle than I thought, I'll add these to the question if you don't mind. $\endgroup$ – Pertinax Feb 18 '17 at 14:00
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    $\begingroup$ Very observant of you, but I submit that it is actually just a special case of the general situation that heavily-foregrounded phenomena often sacrifice some truth for the sake of impact, or convenience. As an extreme case, consider this classic joke, regarding commentary on the speech of a politician: A: “He murdered the truth!” B: “Impossible. He never got anywhere near it.” Also, this is why raindrops are usually depicted inaccurately, and, of course, the heart is usually depicted inaccurately. $\endgroup$ – Mike Jones Feb 22 '17 at 16:04
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The most likely reason is that the plots provided by the book author are reproduced by graphic designers using drawing programs. It is quite clear in your last example where one can recognize circular arcs and straight lines. Better approximations like your other examples are probably obtained with Bézier curves in programs like Photoshop or Illustrator..

enter image description here

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  • $\begingroup$ I added normal comparisons by @Therkel of the two other pseudo-normals. Do you think that Bézier curves would produce such a wide difference between actual normal curves and the graphic-designer ones? $\endgroup$ – Pertinax Feb 18 '17 at 14:12
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    $\begingroup$ @TheThunderChimp Bezier curves will get as close as one wants provided a sufficient number of control points. Never exactly on the bell curve, though. $\endgroup$ – A.G. Feb 18 '17 at 18:58

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