Talk:Illustration of the central limit theorem

Error on the page!
For what it's worth, it occurs to me (1) figures with the probability curves in red (with black or gray for the box) would "read" better, and (2) it could be nice to post some lines of Octave so that people can "try it at home". I'm sure there are other ways to improve this page. I'll get around to the stuff above eventually. Wile E. Heresiarch 02:41, 6 Apr 2004 (UTC)

Error on the page!
The standard deviation of the tested distribution is definitely not 1. As a result, the first sentence of the second paragraph for each chart has to be changed or removed. The graphs, however, appears to be correct.

Unfortunately, the exact value of the standard deviation is hard to guess from the graph, so I can't just correct it. (I tried to replicate the distribution, and the standard deviation is somewhere between 0.4 and 0.5.)

Could the author of the graphs correct the text, please?


 * Hmm, yes, you're right. I'll redraw the figures. I'd like to make the figures have red lines on black & white to make them easier to see anyway. I'll try to fix up the figures in the next few days. In the meantime I'll just cut out the mistaken statement about the standard deviation. Wile E. Heresiarch 19:09, 12 Oct 2004 (UTC)


 * I've redrawn the figures so that all four have mean 0 and std deviation 1. Also, image:central limit thm 1.png has a list of the Octave commands used to generate the figures. (Should also work in Matlab, except the plotting commands are probably different.) Hope this helps, Wile E. Heresiarch 15:29, 14 Oct 2004 (UTC)


 * Isn't supposed to be seen some formulas on that page? I cannot see any analytic expressions, just blank paragraphs...5;59PM 01 Jan 2006

Merge proposal
The article Concrete illustration of the central limit theorem appears to me to be a spinoff with insufficient substance for being an independent article. --Lambiam Talk 17:34, 12 April 2007 (UTC)
 * Maybe it's a POV fork. 193.95.165.190 12:46, 1 August 2007 (UTC)


 * I say, merge and change the name to Demonstrations of the central limit theorem or something like that. I rather like the discrete illustration given here as it is easier for beginners to see than the example with the continuous (or rather, piecewise continuous) distribution. VectorPosse 08:41, 27 August 2007 (UTC)

Desired Explanation
Can someone add an explanation why the "sum of independent variables" is a convolution? It seems to me that a convolution, rather, represents repetitive applications of a distribution as a dispersive event from a mean.
 * Same idea for me. The sum considered here is NOT a convolution. MaCRoEco 22:20, 16 June 2007 (UTC)


 * Actually it is. Given two independent real-valued random variables having continuous probability distributions with density functions f and g, the density of their sum is given by the function h defined by:
 * $$h(z) = \int_{-\infty}^\infty f(x)g(z-x)\,dx \,,$$
 * which is a convolution if there ever was one. In fact, our Convolution article describes a convolution as being "a kind of very general moving average", and thus as a generalization of what we have here. --Lambiam Talk  08:01, 17 June 2007 (UTC)
 * Thanks a lot for the explanations. MaCRoEco 17:09, 17 June 2007 (UTC)

Error on the page!
The Java applet on dice throwing does not demonstrate the central limit theorem. Instead, it demonstates a form of the law of large numbers (which, roughly stated, asserts that observed frequencies converge almost surely to probabilities as the sample size increases without bound). Note that the limiting distribution in each case of the applet is not a normal distribution; instead it is the distribution of total spot numbers for the particular case of dice tossing (which is only approximately normal when several dice are tossed). [by Marvin Ortel, on 2:14PM 11 November 2007, Honolulu, Hawaii]


 * I disagree. It's a valid demonstration of the CLT. The convergence to normality can be seen by going from 1 dice to 5, which gives a similar progression as in this article. Birge (talk) 20:49, 19 December 2007 (UTC)

Error on the page!
The final short section, "Probability mass function of the sum of 1,000 terms" has an error in either the text or the figure. The figure shows a histogram of a sample with a mean around 2000. However the expected sum of 1000 *standard* ([0,1]) uniform samples should be around 500. If the draws are coming from a [0,4] uniform distribution then the histogram should display larger variance. The histogram looks much more like the distribution of the sum of 4000 standard uniform samples. — Preceding unsigned comment added by 128.135.100.102 (talk) 18:33, 19 January 2012 (UTC)


 * It's a question of interpretation. My interpretation is that, given that the subsection is part of the "discrete distribution" section, both the "uniform distribution" and the "example on this page" are referring to the 3-point equal-probability discrete distribution on the values 1,2,3 that is used at the start of the section. If this is what is actually meant, then the text and/or figure caption just needs to be clarified. Melcombe (talk) 22:24, 19 January 2012 (UTC)

Is the illustration in the continuous case really optimal?
The illustration and discussion of the continuous case is not optimal in my opinion. The CLT alone does not guarantee that the sequence of densities converges pointwise to the normal density, but the illustration and discussion in this article could be easily misinterpreted in this way. Since this is a common misconception of the CLT, I would suggest to make this point clearer and at least note that what is shown in the illustration (pointwise convergence of the density) is an actually stronger type of convergence than convergence in distribution. The current illustration would be better suited for an article on local limit theorems. For this article, I think it would be better to plot histograms instead of densities plus the density of the standard normal distribution and describe in the text that the area of each bar of the histogram converges to the area in that interval under the normal density. This would be closer to the actual content of the CLT. --Zakoohl (talk) 12:27, 6 June 2021 (UTC)