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Is there a way to make clear that the kurtosis is 3 but the excess kurtosis (listed in table) is 0? Some readers may find this confusing, as it isn't explicitly labeled.
huh?-summer
what? PAR 00:42, 14 December 2006 (UTC)
I am not a matemathician or a statistician but in fact I came to this discussion page exactly to understand this. Kurtosis is indicated as 3 in many other sources, including http://www.wolframalpha.com/input/?i=normal+distribution , and the 0 value in this page is confusing for me -- Mantees de Tara ( talk) 20:37, 25 December 2009 (UTC)
I consider myself a pretty smart guy. I have a career in IT management, a degree, and 3 technical certifications. Granted, I am certainly not brilliant, nor am I an expert in statistics. However, I was interested in learning about the normal curve. I have a only a fair understanding of standard deviation (compared with the average person who has no idea what SD is) but wanted to really "get it" and wanted to know why the normal curve is so fundamental. Basically, I wanted to learn. So I googled "normal curve". As always, Wiki comes up first. But sadly, as (not always, but usually), the article is hardly co-herent. This article to me was written by the PhD for the PhD. it is not condusive to learning...it is condusive to impressing. It reminds me of a graduate student trying to impress a professor "look Dr. Stat, look at my super complex work". This article has defeated the purpose of wiki to me, which is to educate people. Now I will go back to Google and search for another article on the normal curve that was written for the average person who wants to learn, rather than the stat grad. Wiki is chronic for this. Either articles are meant as a politically biased rant (so much bias here), or written for a "niche" community (like this article). But so few of them are actually written to introduce, explain, and heighten learning. I read 2 paragraphs of this article, and that was more than enough. You might think I'm just too stupid to understand, and thats fine. But when I make contributions to articles that are about internet protocols and networking, I make sure that the layperson is kept in mind. This was not done here. What is so hard...seriously...about just introducing a topic and providing a nice explanation for people who do not have statistical degrees?—The preceding unsigned comment was added by 24.18.108.5 ( talk) 19:57, 1 May 2007 (UTC).
MisterSheik, do you have ANY evidence for your suspicion that anyone has ever gone out of their way to make things complicated? Can you point to ONE instance?
I've seen complaints like this on talk pages before. Often they say something to the general effect that:
Often they are right to say that. And in most cases I'd sympathize if they stopped there. But all too often they don't stop there and they go on to attribute evil motives to the authors of the article. They say:
Should I continue to sympathize when they say things like that? Can't they suggest improvements in the article, or even say there are vast amounts of material missing from the article that should be there in order to broaden the potential audience, without ALSO saying the reason those improvements haven't been made already is that those who have contributed to the article must have evil motives? Michael Hardy 01:51, 2 May 2007 (UTC)
Hi Michael. I think that the user's complaint was definitely worded rudely, and so I understand your indignation. It's not like he's paying for some service, but he's looking for information and then complaining that it isn't tailored for him. So, rudeness aside.
I'm going to go through some pages, and you can tell me what you think. (Apologies in advance to the contributors of this work.) Look at this version of mixture model: [1]. Two meanings? They're the same meaning.
But, what about this? [2] versus now pointwise mutual information.
There's a lot of this wordiness going on as well: [3], [4]], [5], [6], [7], [8] and [9].
And equations for their own sake: [10] and [11] (looks like useful information at first, but it's just an expansion of conditional entropy.)
Maybe all of the examples aren't perfect, but some are indefensible.
I like to see things explained succinctly, but making the material instructional instead of a making it function as a good reference is a bad idea, I think. And that's one of the things I told the person: find the wikibook.
But I still haven't answered your point about
Maybe it's not happening intentionally, or even consciously, but how do people produce some of the examples above without first snapping into some kind of mode where they are trying to speak "like a professor does"?
MisterSheik 03:33, 2 May 2007 (UTC)
Hi Michael, it's fine to say that these ideas are inefficiently expressed, but why are they inefficiently expressed? I think it's because writers are subconsciously aiming to make things difficult in order to achieve a certain tone: the one that they associate with "a professor". In other words, I think that people are imagining a target tone rather than directly trying to convey information succinctly. ps they are both examples of a "mixture model", which has one definition ;) MisterSheik 23:00, 3 May 2007 (UTC)
Whoa. I'm not "shooting my mouth off". I made it really clear that it was my impression that sometimes I think that authors make things difficult to understand. How is that "improper". I'm just sharing my opinions about the motivations of authors unknown. No one is attacking you. I don't have a "heavy burden of proof", because they're just my opinions and you're entitled to disagree. I showed you some examples of what convinced me and asked you what you thought. Ask yourself if you're getting a bit too worked up over nothing here?
(On the other hand, when you use rhetoric like "Don't you know that?", I can't see that you're kidding, and so it sounds like you are shooting your mouth off.)
Regarding this article, I think its fine. I guess the "overview" section could be renamed "importance" since it's not an overview at all. And, the material could be reorganized a little bit since occurrence and importance have similar information, but maybe not.
You make a really good point about people feigning sounding like a professor, and we have both seen that kind of thing. That's not what I meant though. I was trying to get at professionals or academics who know the material going out of their way to word things awkwardly. Let's take one example: "A typical examplar is the following:" Are we supposed to believe that someone actually uses that kind of language day-to-day? Someone is trying to impress the reader with his vocabulary, or achieve an air of formality, or what? Whatever it is, it's bad writing that, due to its unnaturalness, seems intentional (to me). I'm not saying someone is intentionally trying to trip up the reader. I'm saying that someone is trying to achieve something other than inform the reader in the most succinct way. I was trying to illustrate with my examples "undue care" for the presentation of information. MisterSheik 00:12, 4 May 2007 (UTC)
Basically a normal distribution is a curve which shows the distribution of data for something measurable. You will have a target mean (average) to aim for to ensure your distribution is maintained within the tolerance levels (LSL/USL) and have warning levels (LWL/UWL) which indicate when the process is going out of control and action needs to be taken to bring it back in control, limiting any rejects outside the LSL/USL (OK that bit is control charts rather than normal distribution but still related). Cp is a measure of the process variation about the mean (the higher the better towards 3), with cpk a measure of the process variation about a target mean. A Cp of 2 would be ok, but if the mean of the data is 20 when the target mean is 40 then thats not so good as it shows you have a controllable process but its all out of spec likely due to some incorrect setting. PPM part per million indicating how parts you are producing out of spec per million parts produced. —Preceding unsigned comment added by 77.102.17.0 ( talk) 01:00, 5 December 2009 (UTC)
{{
Technical}}
Lead to article is excellent, and the first few sections are readable, but topic is essential to a basic understanding of many fields of study and therefore a special effort should be made to improve the accessibility of the remaining sections. 69.140.159.215 ( talk) 13:00, 12 January 2008 (UTC)
Correct me if i am wrong but an easy way to make it easier for people highchool through Ph.D level would be to leave as is but work though easy examples in the beginning. —Preceding unsigned comment added by 69.145.154.29 ( talk) 23:29, 10 May 2008 (UTC)
I was never comfortable calling this distribution a "normal" distribution, too much baggage comes with the word "normal". However, what I think what might help more people get a handle on this probability distribution, is to try and describe how the word "normal" got associated with it. Fortran ( talk) 01:39, 6 April 2009 (UTC)
Along time since i learnt the history but think "normal" refers to the actual shape, as the bell shape for a normal collection of data will be a nice even bell shape curve, aka normal. Whereas if its skewed in some way due to some unknown variable then you are not achieving the target of a normal distribution curve? —Preceding unsigned comment added by 77.102.17.0 ( talk) 01:22, 5 December 2009 (UTC)
The Normal distribution is called the "Normal" distribution because several hundred years ago many people who were studying distributions noticed that in a large number of cases, the distributions looked similar. Thus if the distribution looked like most others, it was called "Normal." What Fortran is saying is that we now know the reason why many distributions all looked "Normal" (the Central Limit Theorem), and discussing how sampling and the CLT can lead to having a Normal distribution can be enlightening. —Preceding unsigned comment added by 141.211.66.134 ( talk) 16:33, 10 March 2010 (UTC)
Given the suggestion on the edit descriptions list that this article might be pushed towards GA status, it would be good if readers/editors would set down some areas for improvement. Any more suggestions as to what is needed? Melcombe ( talk) 09:10, 22 September 2009 (UTC)
(above comment split to allow addition of general discussion of changes needed for article) Melcombe ( talk) 08:58, 25 September 2009 (UTC)
Graphs should be improved too — the curves should be more thicker so that they are better visible; and also the labels violate MOS, as certain numbers are typeset in italics.
No section should consist of just a single test — they should be either expanded, or merged with some other sections.
The "too technical" tag shown on this talk page, but which might be missed; Melcombe 09:10, 22 September 2009 (UTC)
I removed the following paragraph from the lead, since the lead is already too long, and it'll be expanded even more to include references to multivariate normal and complex normal distributions, and Gaussian stochastic processes.
The normal distribution can be used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches. Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data are used.
The example can still be used somewhere later in the article, although we don’t have a conceivable “introduction” section before we go into hard math. … stpasha » 21:35, 23 September 2009 (UTC)
I think we should use only one notation for the exponential function. As you know, there is exp(x) and e^x. I skimmed through the article and found that exp(x) is more common. Tomeasy T C 06:51, 25 September 2009 (UTC)
The two images shown in the infobox are repeated further down. I think we should remove them. Any thoughts? Tomeasy T C 22:33, 25 September 2009 (UTC)
This subsection is largely a repetition of the section Definition. I would like to include the additional content of the subsection into the section, and remove the subsection. What do you think? Tomeasy T C 22:36, 25 September 2009 (UTC)
The article states the formula for a normal curve is as follows: But what is the origin/purpose of the constant: ? Might be useful to include that info. -- Steerpike ( talk) 20:32, 26 September 2009 (UTC)
I think it should. For example because it has less formulas in it :) … stpasha » 21:08, 27 September 2009 (UTC)
I propose to uniformly replace symbol φ (the pdf of the standard normal distribution) with ϕ (in LaTeX: \phi, in HTML: ϕ). The main reason for this change is to differentiate somehow standard normal distributions from characteristic functions, which are also denoted with φ. … stpasha » 21:35, 7 October 2009 (UTC)
This section in the article seems to be too biased towards the unbiasedness of estimation. At the same time it misses some important info about the t-statistic and construction of confidence intervals. Also the “maximum likelihood estimation” section is bloated — the detailed derivation is already present in the maximum likelihood article and probably doesn’t need to be repeated here. … stpasha » 02:07, 29 November 2009 (UTC)
The history section could be expanded. For once, it doesn't mention the important contributions of Maxwell, when he discovered that gas particles, being constantly subjected to bombardment from the other gas particles, will have their velocities distributed as 3d multivariate Gaussian rv's. I believe this discovery to be important because it demonstrated that normal distribution occurs not only as a mathematical approximation in games of chance or as a convenient tool in least squares analysis, but also exists in nature. … stpasha » 21:52, 5 October 2009 (UTC)
any variable that is the sum of a large number of independent factors
A sum of factors?? Am I getting this very wrong, or is this sentence indeed ill phrased. I would say a sum of summands or a product of factors, but still I would not really understand what this sentence tries to say. Please someone who understands the content, judge whether the wording sum of ... factors is correct. Tomeasy T C 22:25, 25 September 2009 (UTC)
OK, I see the logical flaw has been erased by somebody. Now that semantically the statement is correct, let's focus on the content. Any variable that is the sum of a large number of independent terms is distributed approximately normally. Really, is that so? I would guess most variables are not distributed at all, because they depend on independent but deterministic terms. I see that variable is linked to random variable. The qualifier random is key here to ensure the statement is not ridiculous. Therefore, the text must show this. Tomeasy T C 07:06, 2 October 2009 (UTC)
Yes, sum of factors. The word “factor” here should be understood as “An element or cause that contributes to a result (from Latin facere: one who acts)” (Collins). Of course it is so much unfortunate that this can be confused with the mathematical “factor” which is one of the terms in a product...
Another problem is the following: a typical layperson does not see the world in terms of random variables. For an everyman, the phenomenon is recognized as “random” if it recurs often and has pronouncedly different results each time: such as weather, or lottery, or coin tosses, etc. Other things such as heights or IQs aren’t really seen as “random” unless you force them to stop and think about it. For this reason, writing “any random variable which is the sum of independent terms” does not convey the important message: that this is not an abstract mathematical theorem but rather an approximation for great many random things encountered in the real life.
We can try the following: By the central limit theorem, any quantity which results from an influence of a large number (at least 10–15) of independent factors, will have approximately normal distribution. … stpasha » 21:02, 5 October 2009 (UTC)
Currently there is a separate article standard normal random variable (stub), whereas standard normal distribution redirects to the current article. I suggest that the first article be merged with the current, probably within the “Standardizing normal random variables” subsection. … stpasha » 10:12, 9 October 2009 (UTC)
Some action is needed for the redlinks shown under "tests of normality" ... either creating new articles, expanding existing ones that can be linked to, or providing direct citations; Melcombe 09:10, 22 September 2009 (UTC)
There seems a need to reduce the number of articles under "see also", preferably by saying something about them in the main text (if not already there}; Melcombe 09:10, 22 September 2009 (UTC)
More inline citations, restructuring of notes/references to more convenient form. I guess most detailed results will be findable in Johnson&Kotz so perhaps we could aim to provide page or section numbered pointers to this source. Melcombe 09:10, 22 September 2009 (UTC)
The use of the field "kurtosis" in the table seems not to be consistent across distributions. In some it seems to be the "normal" kurtosis and in some the excess kurtosis (-3). This is really problematic. I think it should either be named "excess kurtosis" in the table, or there should be two fields, one for each. Personally, I think one field should enough, and probably it should be the excess kurtosis, since this is usually more useful. However, it should be made clear, at least to people changing the page, that this is the excess kurtosis and not the other. If there is just one field, which is named "kurtosis" there will always be some who think, its the normal one and change it (see e.g. for the lognormal distribution, change from 21:13, 1 December 2009). Maybe it would be enough to change the template, so that it says "excess_kurtosis=..." instead of "kurtosis=...". Any other thoughts on this? Ezander ( talk) 15:45, 22 February 2010 (UTC)
There is a discussion on the WikiProject Statistics talk page about the financial variables section of this article. Regardless of the merits of the recent additions, and whether they are OR, the issues raised are more about difficulties with estimating the marginal distribution of a dependent, non-stationary sequence, and less about normality per se. This content is too detailed and not sufficiently relevant to be included here. Skbkekas ( talk) 16:26, 15 March 2010 (UTC)
“ |
THE NORMAL LAW OF ERROR STANDS OUT IN THE EXPERIENCE OF MANKIND AS ONE OF THE BROADEST GENERALIZATIONS OF NATURAL PHILOSOPHY ♦ IT SERVES AS THE GUIDING INSTRUMENT IN RESEARCHES IN THE PHYSICAL AND SOCIAL SCIENCES AND IN MEDICINE AGRICULTURE AND ENGINEERING ♦ IT IS AN INDISPENSABLE TOOL FOR THE ANALYSIS AND THE INTERPRETATION OF THE BASIC DATA OBTAINED BY OBSERVATION AND EXPERIMENT ♦ |
” |
// stpasha » 23:58, 21 March 2010 (UTC)
This article says:
In what sense can it be said that z-scores and percentiles "are derived from the normal distribution"? Michael Hardy ( talk) 16:16, 27 April 2010 (UTC)
The article presently has "Commonly the letter N is written in calligraphic font (typed as \mathcal{N} in LaTeX)." without a citation. All the sources I have use a non-script font and I have never seen it in a script font: it is certainly not common. WP:MSM says " it is good to use standard notation if you can" so why use something unnecessarily complicated, particulrly as there is no citation for this notation. Melcombe ( talk) 13:49, 18 May 2010 (UTC)
I would like to generate a set of numbers (x,y) with a known mean and CV; that is, I wish to generate a set of numbers that have a gaussian distribution, where I can set the mean and CV in advance. Thanks PS: maybe it doesn't go here, but a section on curve fitting software might help (please - no"r", if you know R, you already know a lot; stuff like IgorPro or Kaleidagraph etc, or excel thanks —Preceding unsigned comment added by 108.7.0.214 ( talk) 17:34, 22 June 2010 (UTC)
OK, let's start by assuming the covariance matrix is
so that ρ is the correlation. To be continued.... Michael Hardy ( talk) 18:23, 22 June 2010 (UTC)
By definition the entropy of the Normal distribution is not negative value. But what if σ → 0 in the finite formula of entropy? Thanks. Aleksey. —Preceding unsigned comment added by Kharevsky ( talk • contribs) 08:06, 5 July 2010 (UTC)
this article, under Definition, and the one on Gaussian function contain conflicting information on the meaning of constants a and c for the "bell curve."
— Preceding unsigned comment added by 68.37.143.246 ( talk) 01:52, 7 July 2010 (UTC)
Isn't this section a little much of a "how to" for Wikipedia? 018 ( talk) 17:22, 9 July 2010 (UTC)
Great, comprehensive page on the normal distribution, almost perfect. However, the detailed section on 'Gaussian' random number generators (which is also extremely informative) really does not belong in this top-level entry. —Preceding unsigned comment added by 129.125.178.72 ( talk) 16:04, 3 August 2010 (UTC)
I was missing a reference to the product of two gaussians. This could also go into the page for the gaussian function (there is a short mention of it, no mentioning of the resulting properties), but is is also relevant here. —Preceding unsigned comment added by 134.102.219.52 ( talk) 12:34, 7 September 2010 (UTC)
In the opening sentence the article states that the normal distribution is also known as a Gaussian distribution. I would argue however that the normal distribution is a special case of the Gaussian distribution, i.e. one that has an integral of 1, hence why it is called normal. The Gaussian distribution is in my opinion any general distribution described by the Gaussian function
If there aren't any objections I will edit the article to reflect this schroding79 ( talk) 00:08, 25 June 2008 (UTC)
Yes, the gaussian distribution is normal in shape. The standard normal distribution integrates to 1, whereas a frequency distribution which is normal or gaussian in shape does not necessarily integrate to 1. One aspect of interest to readers which is missing from the Wiki page about the Normal Distribution is the relationship between frequency distributions and probability distributions. Perhaps an introductory paragraph linking to Wiki pages about frequency distributions would be a good idea. It would help put this article into context. Lindy Louise ( talk) 09:58, 29 September 2010 (UTC)
I disagree and am curious to know why you think a frequency distribution "also always integrates to one". A frequency distribution does not always integrate to one. A probability distribution always integrates to one. This is why we normalise the normal distribution to get the standard normal distribution: the standard normal distribution integrates to one and therefore can be used as a probability distribution. This is basic stuff but is often omitted from the more esoteric textbooks. Lindy Louise ( talk) 13:18, 30 September 2010 (UTC)
I never said the standard normal distribution was the only probability distribution that integrated to one -- obviously any probability distribution function integrates to one. Neither did I say that gaussian and normal distributions are different. I agree with Melcombe. Lindy Louise ( talk) 17:16, 30 September 2010 (UTC)
Thanks O18 for your comment. I think I'm guilty of being too verbose, but I believe some readers confuse Normal Distribution with Standard Normal Distribution and I wanted to make the distinction. What I should have said is the Normal Distribution cannot be used directly as a Probability Distribution because the area under the Normal curve isn't equal to one. So we deliberately make the area under the Normal curve equal to one by doing some fancy maths: this normal distribution with an area of one is called the Standard Normal Distribution. It can then be used as a Probability Distribution simply because the area is equal to one. (In any probability system the sum of all the probabilities must equal one or, in other words, the area under a proability curve is equal to one.) Still verbose, sorry! Maybe I should have a go at updating the opening paragraph; I'll think about it. I was going to insert a link to Wiki pages about probability distributions and probability density functions but they're too difficult for non-mathematicians to understand, so I haven't. Lindy Louise ( talk) 21:29, 30 September 2010 (UTC)
If you integrate an absolute-frequency distribution you will not necessarily get unity for your answer. In fact I would think it a freak event if it were to happen! The only way you can be sure of obtaining unity by integration is if you use relative frequencies or probabilities. Hence the need for the Standard Normal Distribution, because we can be sure its integral is unity. The fact that the mean and variance of the Standard Normal Distribution are 0 and 1 is a consequence of the "normalisation" or "standardisation". The mean and variance of a Normal Distribution are not 0 and 1. That's one way of distinguishing between Normal and Standard Normal. Thanks for pointing me in the direction of the Gaussian function, but I am very familiar with the gaussian and normal functions (they're the same). Lindy Louise ( talk) 21:10, 10 December 2011 (UTC)
Calculating out by hand, the Fisher Information in the top right box seems incorrect and should instead be Khosra ( talk) 21:33, 9 September 2010 (UTC)
There used to be the time when the article started with “In probability theory, normal distribution is a continuous probability distribution which is often used to describe, at least approximately, any variable that tends to cluster around the mean”. Some people tend to revert the intro back to this sentence from time to time, which is why I think an explanation is due why such sentence is inappropriate in an encyclopedia.
First it must be stated that the distribution is not merely continuous, but absolutely continuous. Absolute continuity implies that the distribution possesses density, whereas simple continuity means very little. Second, about the “any variable that tends to cluster around the mean”. This is not an informative statement. Any unimodal distribution can be said to “cluster around the mean”, and some non-unimodal distributions too. This statement is so loose that it fails to describe anything. Finally, “is often used to describe, at least approximately” is a weasel-phrase. No serious researcher will use normal distribution to describe his data, unless he has good reasons to believe that the data IS actually normally distributed. There is a good quote from Fisher about this, see the Occurrence section. // stpasha » 09:23, 2 October 2010 (UTC)
BTW, Stpasha, you might want to check out the pages WP:TECHNICAL and Wikipedia:Lead section#Introductory text, which provide guidelines on how technical articles, and particularly the lead sections, should be written. Benwing ( talk) 23:01, 2 October 2010 (UTC)
Please see Anders Hald : A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713-1935. He has different views than Stigler.
1774 : asymptotic normality of the posterior distribution, derivation of the constant of the normal distribution (page 38). This is the first justification of the normal distribution and first appearance of the Bayesian Central Limit Theorem
1785 : further results (page 44)
Can we have a less controversial example in the lead? Current one (the heights of US adult males) isn't supported by a reference, and also contradicts a later claim in the article that the sizes of biological species are distributed approximately log-normally. Besides, “US adult males” is not a sufficiently homogeneous group: variability due to race / ethnicity make it a mixture of several log-normal distributions. // stpasha » 05:33, 5 March 2010 (UTC)
Long ago I suggested to remove this paragraph from the lead, which suggestion was refuted on the basis that it is “the only generally understandable information” there. Now that the lead has been improved in readability, maybe it’s ok to get this piece finally out? // stpasha » 02:19, 9 October 2010 (UTC)
I think maybe we should alter the definition to allow normal distributions with 0 variance. This is needed for consistency with the “ Multivariate normal distribution” article, where we say that a random vector X is distributed normally if and only if every linear combination of its components cX has univariate normal distribution. Since this linear combination can potentially have zero variance, such case must be allowed within the current article.
The cons for such inclusion are that we'll need to define pdf and cdf separately for the case σ² = 0. … stpasha » 20:43, 24 November 2009 (UTC)
I've been agnostic to the recent debate regarding whether to allow zero variance in the normal distribution. But I've rethought it, and concluded that almost certainly, we should not. The basic reason has to do with Wikipedia's "Verifiability not truth" maxim, which is a core principle ( WP:V, WP:VNT). Hence, we need to consult reliable sources, not use our mathematical intuition.
So far I've consulted two sources and they both agree that the variance must be specifically greater than, not greater than or equal to, zero. These include DeGroot and Schervish "Probability and Statistics" and Chris Bishop "Pattern Recognition and Machine Learning". I don't have any other books on hand, so I'd suggest other people check their own references. Note that on top of this, Bishop's description of the multivariate normal specifically says the covariance matrix must be positive definite, not non-negative definite. Benwing ( talk) 20:17, 10 October 2010 (UTC)
Right then now lets all stop complaining and help me write the Frechet distribution article which needs some work :) (its useful for Extreme Value theory). —Preceding unsigned comment added by 130.236.58.84 ( talk) 08:37, 11 October 2010 (UTC)
The annotation on my change got messed up accidentally. What I was trying to say was that "blah blah blah is called standard normal" sounds wrong vs. "blah blah blah is called the standard normal". But I don't know what's the "correct" convention (if there even is any at all). Benwing ( talk) 08:56, 10 October 2010 (UTC)
I am reading this in the text: "In addition, the probability of seeing a normally-distributed value that is far (i.e. more than a few standard deviations) from the mean drops off extremely rapidly. As a result, statistical inference using a normal distribution is not robust to the presence of outliers (data that is unexpectedly far from the mean, due to exceptional circumstances, observational error, etc.). When outliers are expected, data may be better described using a heavy-tailed distribution such as the Student’s t-distribution"
I am just wondering if this is accurate? My understanding of robust to outliers means that the model assigns very little (or zero) probability to values far away from the mean. So yes I think the heavy-tailed comment is correct but the first sentence should be "In addition, the probability of seeing a normally-distributed value that is far (i.e. more than a few standard deviations) from the mean drops off relatively slowly. As a result..."
For example from the article about the laplacian The pdf of the Laplace distribution is also reminiscent of the normal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean μ, the Laplace density is expressed in terms of the absolute difference from the mean. Consequently the Laplace distribution has fatter tails than the normal distribution.
So for example if we wish to use a more robust norm for the outliers we would use the L1 norm which leads to the Laplace (from a MLE point of view). But perhaps I am wrong. What does everyone else think? —Preceding unsigned comment added by 130.236.58.84 ( talk) 10:38, 10 October 2010 (UTC)
I removed the example of an electron in a 1s orbital being Gaussian. The distribution (for an electron in a 1/r Coulomb potential) is actually proportional to e-r. If I think of a similar example, I will add it in, because it was very striking! 128.111.10.89 ( talk) 13:25, 30 October 2010 (UTC)
The image in "Standard deviation and confidence intervals" part has wrong percentages. I suggest to change it with the image in Standard Score one or another, more accurate one. —Preceding unsigned comment added by 77.49.4.13 ( talk) 21:36, 10 December 2010 (UTC)
Here is the text after I changed it, I added the exact numbers in bold, which should explain why I choose to correct the article to it's current number rounding scheme:
If any one thinks this should be changes - please explain why - I'd be happy to know. Talgalili ( talk) 09:12, 11 December 2010 (UTC)
if X is normal, then what is the distribution of 1/X????? It is good to include this. even there is not solution to it. Jackzhp ( talk) 23:34, 28 December 2010 (UTC)
You transform it to u=1/x and you make calculations —Preceding unsigned comment added by 79.103.101.115 ( talk) 19:52, 9 January 2011 (UTC)
Someone has been messing around with this page, adding obscenities.
e.g . Under 'definition' there is 'The factor fuck you man in this expression ensures that the total area under the curve '
The page should be restored to its former condition.
But who knows what its properties are. You can plot it in R and it has a somewhat weird shape -- it has two modes on each side of the origin, is heavily skewed to the right (or to the left, on the negative side of the origin), and near the origin it drops suddenly and then has what looks like a completely flat section at 0 height near the origin. Benwing ( talk) 22:09, 9 April 2012 (UTC)
Like this:
Are all of the references to Mathworld/Wolfram really needed? See footnotes 12, 15, 16, 16, 26, and below the footnotes Weisstein, Eric W. "Normal distribution". MathWorld. Do all of these contribute something that is not already covered in the article? Mathstat ( talk) 23:49, 27 February 2011 (UTC)
The Gaussian function does also have numerous applications outside the field of statistics, for example regarding solution of diffusion equations and Hermite functions and regarding feature detection in computer vision. If this article would be merged under normal distributions, these connections would be lost. Hence, I think that it is more appropriate to keep the present article on the Gaussian with appropriate cross referencing and developing the article further. Tpl (talk) 11:53, 8 June 2011 (UTC)
The Kullback-Leibler divergence quoted in the article appears to be incorrect. In particular the log(sigma_1/sigma_2) term should not be in the brackets. It would be useful for someone to confirm this. The source quoted appears to be correct: http://www.allisons.org/ll/MML/KL/Normal/ Egkauston ( talk) 07:32, 29 November 2011 (UTC) Update: I checked again and I was wrong. The entry appears to be correct. Egkauston ( talk) 07:51, 29 November 2011 (UTC)
In the figures, it would be nice to show some Normal probability density function with mean and standard distribution values such as the maximum value would be higher than 1; for example, mu = .05, sigma = .003. The density can take values higher than 1; the constraint is for the cumulative density function (area under the PDF), which cannot exceds 1. It is a fairly common confussion between PDF and CDF, and I believe it is worth to remark. — Preceding unsigned comment added by 190.48.106.19 ( talk) 03:49, 25 July 2012 (UTC)
Hi, can someone check the main equation at the top of the page. I may be misunderstanding it, but its a probability distrubition so shoulden't a curve using it sum to 1? I put it into R and came out with 0.2. Checking on wolfram mathworld they use a slightly different equation. I may simply have misunderstood! Kev 109.12.210.202 ( talk) 09:07, 4 August 2012 (UTC)
The graph at the top of the page is good for the article. Thanks for having it there. It identifies the red curve as the standard normal distribution. Can the graph's author or a responsible party also identify and label the other curves, please? Thank you. 69.210.252.252 ( talk) 21:44, 15 August 2012 (UTC)
In the "Central Limit Theorem" section, the caption for the "De Moivre-Laplace" figure mentions "the function". It would be helpful if it were specified what function is meant. As it stands, the figure does not really aid understanding of the CLT. — Preceding unsigned comment added by 193.60.198.36 ( talk) 15:56, 26 November 2012 (UTC)
Hi there, just noticed there's an error in an equation in "Estimation of parameters" and it's not displaying properly. Not sure how to fix it or anything, but there it is. — Preceding unsigned comment added by 97.65.66.166 ( talk) 19:21, 25 March 2013 (UTC)
It should be noted that the Normal Distribution Function comes from the Stirling's approximation applied to the Binomial distribution (deMoivres-Laplace: http://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem.) In the binomial distribution, the probability of "each outcome" is known. That is Binomial distribution builds on that fact that "I can get k successes in n trials where each event has a probability p", and I plot the value of E(k) versus k. When I carry this to the Stirling approximation to form the Normal Distribution function, I assume each independent event has the same "p". What is this "p" that I refer to now in the context of a normal distribution function? In other words are the trials still "Bernoulli"? If yes what is the p used in the context of NDF.
If however one is simply assuming this is "distribution" function and the central limit theorem is just a coincidence, then note that most derivations of the central limit theorem also build from Binomial distribution. Can someone please clarify what is "Bernoulli" about the trails in that case? Is each E(x) associated with x still representing a success of a "Bernoulli outcome" at all??? The literature on this page, and the "central limit theorem" is not clear and is recursive..and always points back to Demoive-Laplace Theorem only.
An independent proof of the "Central limit Theorem", not relying on Binomial distribution would also help clarify this circular reference.
-Alok 11:31, 19 July 2013 (UTC) — Preceding unsigned comment added by Alokdube ( talk • contribs)
It should also be noted that wikipedia does not in anyway state that Normal Distribution function is sacrosanct but most text books and academicians tend to do so. However it would be really great if someone can show the assumptions made in the approach. -Alok 23:10, 23 July 2013 (UTC) — Preceding unsigned comment added by Alokdube ( talk • contribs)
The PDF can be re-arranged to the following form:
where Z is the Standard score (number of standard deviations from the mean). This makes it pretty obvious that the pdf is maximal when is small (narrow distribution) and when is small (towards the center of the distribution). I find this notation way simpler and more intuitive than the standard formula for the pdf. Should we include it in the main article (and where?) for the pedagogical purpose? — Preceding unsigned comment added by 129.215.5.255 ( talk) 10:46, 30 October 2013 (UTC)
The normal sum theorem for the sum of two normal variates is discussed in Lemons, Don (2002). An Introduction to Stochastic Processes in Physics. John Hopkins. p. 34. ISBN 0-8018-6867-X.. The proof of the theorem shows that the variance for the sum is the sum of the two variances. However, this doesn't prove the distribution for the sum is a normal distribution since more than one distribution can have the same variance. -- Jbergquist ( talk) 06:18, 30 November 2013 (UTC)
Would it be fair to say that few if any math majors turn to Wikipedia for help in their chosen field? If so, who exactly is this article written for? Unless they have post-secondary studies in Math, few people would have the knowledge or time to comprehend any of the terms used & these beginners, I would submit, are the vast majority of those who click on this article. We would just like, in layman's terms, an explanation of Normal Distributuion. Instead we've found a long, specialized article written for no one. — Preceding unsigned comment added by 96.55.2.6 ( talk) 22:40, 26 March 2013 (UTC)
You have links to the terms you don't understand. Also, N is an advanced subject in itself. i.e. it can not be simplified without being hollow and meaningless. Read about other types of distributions first if you want simpler examples of that type of math. The reason for the complexity, or rather lack of a comprehensive explanation for it, is that the distribution is not human constructed but an observed reality of life. It just happens to work for many common situations. -- 5.54.91.60 ( talk) 20:03, 21 June 2013 (UTC)
I'm confused. Shouldn't the ERF function be defined as ERF(a,b) = integral between a and b, instead of ERF(x) = integral between -x and +x? This would then allow for the proper definition of the CDF function as ERF(-infinity, x) instead of defining it as a single value function. Maybe the error introduced by using -x instead of -infinity is small.
130.76.64.109 ( talk) 16:15, 4 July 2013 (UTC)
I add to this, I'm a 4th year engineering student, and even then, this is going right over my head, it doesn't help that the way the formulas are shown, they cannot be selected, and as can be seen here,
Is the root function to the power of e, or is the whole term multiplied by e? — Preceding unsigned comment added by 114.76.42.246 ( talk) 23:29, 20 March 2014 (UTC)
Double factorials seem to be uncommon in mathematics, it may help with the exposition if the double factorials were replaced by their explicit formula MATThematical ( talk) 23:21, 9 May 2014 (UTC)
Normal curve never touches X axis. It was touching X axis in two figures which I have removed from the article. I would like to discuss on this point if someone has other opinion / reference. Thanks. -- Abhijeet Safai ( talk) 09:31, 29 May 2014 (UTC)
sqrt(-2*log(rand()))*cos(2*pi*rand()) — Preceding unsigned comment added by MClerc ( talk • contribs) 19:55, 6 August 2014 (UTC)
1. Cite on any regression can achieve normal residuals with proper modeling, please.
2. Some regressions explicitly assume other distributions, of course. Probit and logit come to mind.
3. I've seen weighting procedures to adjust for skewed residuals. But if the residuals have a kurtosis other than 3, how can normal kurtosis be achieved?
4. I'd like to keep this category under the Occurrence heading, but I'm honestly unclear about the proper treatment.
Everyone believes in the Gaussian law of errors, the experimenters because they think it is a mathematical theorem, the mathematicians because they think it is an empirical fact. Kennedy, quoting Poincare, but see this elaboration: http://boards.straightdope.com/sdmb/showpost.php?p=14046385&postcount=33. Measure for Measure ( talk) 20:56, 17 August 2014 (UTC)
Hello everybody,
I've just created an image that could replace another image in this article:
I think the new image is better, because
And it also has a CC0 license.
I could also re-make the other image in the same "style".
Best regards, -- MartinThoma ( talk) 19:38, 29 August 2014 (UTC)
According to the Characteristic function (probability theory) page, the CF of a distribution is the inverse Fourier transform of the PDF (and therefore the frequency-domain PDF is the Fourier transform of the time-domain CF ). We could just change instances of "Fourier transform" to "inverse Fourier transform", but the page goes on to say "...normal distribution on the frequency domain", so this we should also change to "...normal distribution on the time domain". I'm not missing something here, am I? Tsbertalan ( talk) 23:42, 4 December 2014 (UTC)
The Pascal CDF function, as shown does not translate the formula shown above it. As near as I can tell, it does not provide a correct result. I suggest that for this and other examples you use a more commonly used language: C or C++. — Preceding unsigned comment added by Statguy1 ( talk • contribs) 06:45, 16 February 2015 (UTC)
The Pascal code does not account for the double factorial in the denominator. This approximation of the CDF function is also given (with a reference) elsewhere in this WikiPedia article Normal_distribution#Numerical_approximations_for_the_normal_CDF — Preceding unsigned comment added by 138.73.5.2 ( talk) 15:02, 22 October 2015 (UTC)
The top line states "This article is about the univariate normal distribution", yet the description is in terms for 'random variables', (plural) i.e. the multivariate case. I'm not sure if the plural usage 'random variables' is a formal math usage I'm not familiar with, a british/american usage difference, or just poor usage. Also, the lead paragraph does not directly state what the Normal Distribution is, but infers the definition from the CLT. I suggest restating and splitting the 2nd lead paragraph as below, and submit it to discussion here first.
-Orig The normal distribution is remarkably useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.
-Rework The normal distribution is defined by the central limit theorem. Generalized, it states, under some conditions (which include finite variance), that the distribution of averages of a random variable independently drawn from independent distributions converge to the normal distribution, when the number of samples is sufficiently large.
Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed. LarryLACa ( talk) 03:47, 13 October 2015 (UTC)
One of the most famous and used distribution by scientist is bell type distributions. It is desired since it goes from maximum to minimum and vice versa. Distribution of phenomena and mathematical modeling can appear very easily.
Read more on reference: http://mathworld.wolfram.com/NormalDistribution.html
MansourJE ( talk) 21:47, 14 April 2016 (UTC)
The applications are briefly touched on, but the danger of misapplication is completely ignored.
MansourJE ( talk) 17:18, 14 April 2016 (UTC)
Hello, For the figure in the "Standard deviation and coverage" section, the vertical lines should be equally spaced. — Preceding unsigned comment added by 99.226.5.121 ( talk) 21:31, 8 February 2017 (UTC)
This article is very good, clear citation, the math is correct, has very reliable references. But it doesn't mention that Gaussian Distribution has a widely application in air pollution transportation model and diffusion model.
Jiamingshi (
talk)
05:35, 28 April 2017 (UTC)
Also the reciprocal of the standard deviation might be defined as the precision and the expression of the normal distribution becomes
According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, the fact that the PDF has unit height at zero,
Well:
and also for
So??? Madyno ( talk) 15:04, 11 May 2017 (UTC)
I think the formula for the Fourier transform is not in line with the definition in the lemma of Fourier transform. Madyno ( talk) 10:44, 22 October 2017 (UTC)
I am sure this article is very good, but I came to this page to find out why a Bell curve (or bell curve) is called as it is. Is it named after a shape or the person who first devised it or what? And maybe the article should say so? Kiltpin ( talk) 12:16, 27 October 2016 (UTC)
or
I like the first version more, as it looks more clearly arranged. -- Gunnar ( talk) 18:36, 4 November 2018 (UTC)
1. The entropy H, as a function of the density f, is called a functional.
2. The symbol d in an integral expression is a kind op operator, not a variable, and hence not set in italic. Madyno ( talk) 21:24, 14 June 2019 (UTC)
Okay, matter of taste. Madyno ( talk) 21:33, 14 June 2019 (UTC)
I miss the useful relation — Preceding unsigned comment added by 141.23.181.132 ( talk) 16:34, 18 August 2019 (UTC)
The section on the quantile function defines to be , but then 2 lines later uses to mean something related but different, without warning. Fathead99 ( talk) 15:17, 28 July 2017 (UTC)
Regarding diagrams of different PDFs: There should be a curve here with a maximum value above 1, just to illustrate that it is possible. — Preceding unsigned comment added by 2001:4643:E6E3:0:2C29:9E4F:EDF9:AD78 ( talk) 13:19, 12 September 2019 (UTC)
I'm missing the average deviation from the mean. Its simply sqrt(2/pi) * sigma ~ 0.797 sigma (see https://www.wolframalpha.com/input/?i=2+*+integral+from+0+to+infinity+of+x+*+exp%28-%28x%2Fsigma%29%5E2%2F2%29+%2F+%28sigma+*+sqrt%282+pi%29%29 ) This measure may not be a widely used quantity among mathematicians, but it's what less-mathematically-inclined people tend to report (e.g. bsc students, bankers etc)
May I add this to the table at top right? If so, how? Michi zh ( talk) 12:16, 7 January 2020 (UTC)
{{
Infobox probability distribution}}
doesn't have an entry for this, so there's currently no way to add it here, unless you also modify the infobox. That's probably not a great idea since this is such an unusual measure. I'm curious why you think less mathematically inclined people would prefer it (and what do you mean by "report"?). It's almost always more difficult to calculate then the std dev/variance. –
Deacon Vorbis (
carbon •
videos)
14:18, 7 January 2020 (UTC)
{{
Infobox probability distribution}}
I can place this suggestion. Cheers!
Michi zh (
talk)
20:10, 12 January 2020 (UTC)So the deed is done! I hope other people find it useful too! Feel free to delete this section if it's not necessary anymore. Best Michi zh ( talk) 22:37, 16 January 2020 (UTC)
The MAD is an acronym for several different measures (e.g. mean absolute deviation but also the median one). Please make sure to link the name to the specific definition used. Thanks. Tal Galili ( talk) 05:18, 17 January 2020 (UTC)
Could someone check the table of values? They seem to be wrong. — Preceding unsigned comment added by Piqm ( talk • contribs) 19:05, 2 November 2020 (UTC)
For some displays (like mine) the negative sign in the exponent doesn't show properly unless you've zoomed in. I'm not savvy enough to fix it — Preceding unsigned comment added by 2603:8080:1540:546:3175:C96:9C96:B48C ( talk) 20:58, 6 December 2020 (UTC)
This appears to be a bug with Chromium (FF and Safari show the minus properly). Logged a bug, let's see if it's fixable by Chromium: https://bugs.chromium.org/p/chromium/issues/detail?id=1159852 — Preceding unsigned comment added by 84.9.90.236 ( talk) 17:13, 17 December 2020 (UTC)
Is a sine wave of one period ( example) a type of bell curve? I didn't see it mentioned anywhere in the article. ➧ datumizer ☎ 13:16, 26 December 2020 (UTC)
I've only ever seen the term "bell curve" applied to an actual normal curve or to a curve that is close to normal in some sense. The restricted sine you mention would not qualify. FilipeS ( talk) 05:26, 30 December 2020 (UTC)
I feel the article is incomplete without some mention of the asymptotic behaviour of the tails of the curve. — Preceding unsigned comment added by 77.61.180.106 ( talk) 18:26, 11 January 2021 (UTC)
This article is terrible if you don't have advanced level maths already. There's not even an attempt to explain it in simple English (yes, I'm aware of Simple English Wiki; that version of this article is also not in Simple English). Who is the target audience of this article? People who already understand this sort of maths? I'd be surprised if even 5% of readers would learn anything from reading this article, I certainly haven't. — Preceding unsigned comment added by 217.155.20.204 ( talk) 14:18, 14 October 2020 (UTC)
Absolutely agree, Wikipedia is meant to be an accessible resource for finding out about something you don't know. Not a reminder for people with technical knowledge. This article is effectively useless, a school student stopping in here is going to take one look and navigate away. — Preceding unsigned comment added by 122.62.34.148 ( talk) 08:12, 11 March 2021 (UTC)
I quickly tried searching for tail bound but I saw no mention of tail bounds or concentration inequalities, which are very useful. The most useful being ones like
for one tail and twice for both tails.
See http://www.stat.yale.edu/~pollard/Books/Mini/Basic.pdf
https://www.math.wisc.edu/~roch/grad-prob/gradprob-notes7.pdf Wqwt ( talk) 05:17, 25 April 2021 (UTC)
Just like I commented on /info/en/?search=Multivariate_normal_distribution user Dvidby0 is citing his own paper with dubious contribution to the topic. It is more subtle here but I think it is worth to reconsider if citing yourself is moral ( are you promoting your 2020 paper?) and is it adding anything new to the topic? Maybe it is adding something but sure as hell it isn't adding clarity. People here want to learn something about normal distribution and you are putting your stuff in. Imagine everybody started adding things from their 1y old papers to get citations, wikipedia would look like complete garbage. — Preceding unsigned comment added by Vretka ( talk • contribs) 20:55, 12 March 2021 (UTC)
This article is or was the subject of a Wiki Education Foundation-supported course assignment. Further details are available on the course page. Peer reviewers: Jiamingshi.
Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT ( talk) 05:23, 17 January 2022 (UTC)
wiki is a general encylopedia for the avg person do you people really think the introduction is pitched to the avg person ? maybe ask your parents or grandparents jeez , grow up you math people and learn how to write English and don't you dare criticize me for being mean; i am really fed up with math people's total inability to write at the appropriate level really fed up — Preceding unsigned comment added by 2601:192:4700:1F70:BDC1:852D:B5D9:A6AC ( talk) 23:30, 30 December 2021 (UTC)
After the table with the moments of a Gaussian there is a sentence about the expected value of the "reciprocal". In fact the result is about for . In addition the result is a very weak upper bound: it is stated for a Gaussian with arbitrary mean but the result is independent of the mean. It seems unnecessary for such a result to appear in a section about the exact moments of the Gaussian. — Preceding unsigned comment added by Aterbiou ( talk • contribs) 09:34, 9 March 2022 (UTC)
I removed the following opening paragraph from the article. It seemed pointlessly vague and off-topic: articles on specific distributions aren't the place for handwaving explanations of basic statistical concepts. Leaving it here in case someone has a different opinion. 69.127.73.225 ( talk) 02:52, 4 April 2022 (UTC)
"A normal distribution is a probability distribution used to model phenomena that have a default behaviour and cumulative possible deviations from that behaviour. For instance, a proficient archer's arrows are expected to land around the bull's eye of the target; however, due to aggregating imperfections in the archer's technique, most arrows will miss the bull's eye by some distance. The average of this distance is known in archery as accuracy, while the amount of variation in the distances as precision. In the context of a normal distribution, accuracy and precision are referred to as the mean and the standard deviation, respectively. Thus, a narrow measure of an archer's proficiency can be expressed with two values: a mean and a standard deviation. In a normal distribution, these two values mean: there is a ~68% probability that an arrow will land within one standard deviation of the archer's average accuracy; a ~95% probability that an arrow will land within two standard deviations of the archer's average accuracy; ~99.7% within three; and so on, slowly increasing towards 100%."
Following the line "The CDF of the standard normal distribution can be expanded by Integration by parts into a series: " I think that in the equation for phi(x) the fraction "x^5/3.5" should read "x^5/15". 217.155.205.34 ( talk) 17:37, 12 May 2022 (UTC)
I tried implementing this and got tiny variances. Searching around, I found https://stats.stackexchange.com/questions/365192/bayesian-update-for-a-univariate-normal-distribution-with-unknown-mean-and-varia where someone else tried and got tiny variances. As far as I can tell, it works nicely if we don't divide by the total observations (or obs+pseudo) in the final distribution. I don't know how to prove such things. — Preceding unsigned comment added by 76.146.32.69 ( talk) 16:19, 25 July 2022 (UTC)
The pdf formula in the some sections is missing a minus sign. In the right column the pdf is correct. Ron.linssen ( talk) 08:33, 29 September 2022 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | Archive 4 |
Is there a way to make clear that the kurtosis is 3 but the excess kurtosis (listed in table) is 0? Some readers may find this confusing, as it isn't explicitly labeled.
huh?-summer
what? PAR 00:42, 14 December 2006 (UTC)
I am not a matemathician or a statistician but in fact I came to this discussion page exactly to understand this. Kurtosis is indicated as 3 in many other sources, including http://www.wolframalpha.com/input/?i=normal+distribution , and the 0 value in this page is confusing for me -- Mantees de Tara ( talk) 20:37, 25 December 2009 (UTC)
I consider myself a pretty smart guy. I have a career in IT management, a degree, and 3 technical certifications. Granted, I am certainly not brilliant, nor am I an expert in statistics. However, I was interested in learning about the normal curve. I have a only a fair understanding of standard deviation (compared with the average person who has no idea what SD is) but wanted to really "get it" and wanted to know why the normal curve is so fundamental. Basically, I wanted to learn. So I googled "normal curve". As always, Wiki comes up first. But sadly, as (not always, but usually), the article is hardly co-herent. This article to me was written by the PhD for the PhD. it is not condusive to learning...it is condusive to impressing. It reminds me of a graduate student trying to impress a professor "look Dr. Stat, look at my super complex work". This article has defeated the purpose of wiki to me, which is to educate people. Now I will go back to Google and search for another article on the normal curve that was written for the average person who wants to learn, rather than the stat grad. Wiki is chronic for this. Either articles are meant as a politically biased rant (so much bias here), or written for a "niche" community (like this article). But so few of them are actually written to introduce, explain, and heighten learning. I read 2 paragraphs of this article, and that was more than enough. You might think I'm just too stupid to understand, and thats fine. But when I make contributions to articles that are about internet protocols and networking, I make sure that the layperson is kept in mind. This was not done here. What is so hard...seriously...about just introducing a topic and providing a nice explanation for people who do not have statistical degrees?—The preceding unsigned comment was added by 24.18.108.5 ( talk) 19:57, 1 May 2007 (UTC).
MisterSheik, do you have ANY evidence for your suspicion that anyone has ever gone out of their way to make things complicated? Can you point to ONE instance?
I've seen complaints like this on talk pages before. Often they say something to the general effect that:
Often they are right to say that. And in most cases I'd sympathize if they stopped there. But all too often they don't stop there and they go on to attribute evil motives to the authors of the article. They say:
Should I continue to sympathize when they say things like that? Can't they suggest improvements in the article, or even say there are vast amounts of material missing from the article that should be there in order to broaden the potential audience, without ALSO saying the reason those improvements haven't been made already is that those who have contributed to the article must have evil motives? Michael Hardy 01:51, 2 May 2007 (UTC)
Hi Michael. I think that the user's complaint was definitely worded rudely, and so I understand your indignation. It's not like he's paying for some service, but he's looking for information and then complaining that it isn't tailored for him. So, rudeness aside.
I'm going to go through some pages, and you can tell me what you think. (Apologies in advance to the contributors of this work.) Look at this version of mixture model: [1]. Two meanings? They're the same meaning.
But, what about this? [2] versus now pointwise mutual information.
There's a lot of this wordiness going on as well: [3], [4]], [5], [6], [7], [8] and [9].
And equations for their own sake: [10] and [11] (looks like useful information at first, but it's just an expansion of conditional entropy.)
Maybe all of the examples aren't perfect, but some are indefensible.
I like to see things explained succinctly, but making the material instructional instead of a making it function as a good reference is a bad idea, I think. And that's one of the things I told the person: find the wikibook.
But I still haven't answered your point about
Maybe it's not happening intentionally, or even consciously, but how do people produce some of the examples above without first snapping into some kind of mode where they are trying to speak "like a professor does"?
MisterSheik 03:33, 2 May 2007 (UTC)
Hi Michael, it's fine to say that these ideas are inefficiently expressed, but why are they inefficiently expressed? I think it's because writers are subconsciously aiming to make things difficult in order to achieve a certain tone: the one that they associate with "a professor". In other words, I think that people are imagining a target tone rather than directly trying to convey information succinctly. ps they are both examples of a "mixture model", which has one definition ;) MisterSheik 23:00, 3 May 2007 (UTC)
Whoa. I'm not "shooting my mouth off". I made it really clear that it was my impression that sometimes I think that authors make things difficult to understand. How is that "improper". I'm just sharing my opinions about the motivations of authors unknown. No one is attacking you. I don't have a "heavy burden of proof", because they're just my opinions and you're entitled to disagree. I showed you some examples of what convinced me and asked you what you thought. Ask yourself if you're getting a bit too worked up over nothing here?
(On the other hand, when you use rhetoric like "Don't you know that?", I can't see that you're kidding, and so it sounds like you are shooting your mouth off.)
Regarding this article, I think its fine. I guess the "overview" section could be renamed "importance" since it's not an overview at all. And, the material could be reorganized a little bit since occurrence and importance have similar information, but maybe not.
You make a really good point about people feigning sounding like a professor, and we have both seen that kind of thing. That's not what I meant though. I was trying to get at professionals or academics who know the material going out of their way to word things awkwardly. Let's take one example: "A typical examplar is the following:" Are we supposed to believe that someone actually uses that kind of language day-to-day? Someone is trying to impress the reader with his vocabulary, or achieve an air of formality, or what? Whatever it is, it's bad writing that, due to its unnaturalness, seems intentional (to me). I'm not saying someone is intentionally trying to trip up the reader. I'm saying that someone is trying to achieve something other than inform the reader in the most succinct way. I was trying to illustrate with my examples "undue care" for the presentation of information. MisterSheik 00:12, 4 May 2007 (UTC)
Basically a normal distribution is a curve which shows the distribution of data for something measurable. You will have a target mean (average) to aim for to ensure your distribution is maintained within the tolerance levels (LSL/USL) and have warning levels (LWL/UWL) which indicate when the process is going out of control and action needs to be taken to bring it back in control, limiting any rejects outside the LSL/USL (OK that bit is control charts rather than normal distribution but still related). Cp is a measure of the process variation about the mean (the higher the better towards 3), with cpk a measure of the process variation about a target mean. A Cp of 2 would be ok, but if the mean of the data is 20 when the target mean is 40 then thats not so good as it shows you have a controllable process but its all out of spec likely due to some incorrect setting. PPM part per million indicating how parts you are producing out of spec per million parts produced. —Preceding unsigned comment added by 77.102.17.0 ( talk) 01:00, 5 December 2009 (UTC)
{{
Technical}}
Lead to article is excellent, and the first few sections are readable, but topic is essential to a basic understanding of many fields of study and therefore a special effort should be made to improve the accessibility of the remaining sections. 69.140.159.215 ( talk) 13:00, 12 January 2008 (UTC)
Correct me if i am wrong but an easy way to make it easier for people highchool through Ph.D level would be to leave as is but work though easy examples in the beginning. —Preceding unsigned comment added by 69.145.154.29 ( talk) 23:29, 10 May 2008 (UTC)
I was never comfortable calling this distribution a "normal" distribution, too much baggage comes with the word "normal". However, what I think what might help more people get a handle on this probability distribution, is to try and describe how the word "normal" got associated with it. Fortran ( talk) 01:39, 6 April 2009 (UTC)
Along time since i learnt the history but think "normal" refers to the actual shape, as the bell shape for a normal collection of data will be a nice even bell shape curve, aka normal. Whereas if its skewed in some way due to some unknown variable then you are not achieving the target of a normal distribution curve? —Preceding unsigned comment added by 77.102.17.0 ( talk) 01:22, 5 December 2009 (UTC)
The Normal distribution is called the "Normal" distribution because several hundred years ago many people who were studying distributions noticed that in a large number of cases, the distributions looked similar. Thus if the distribution looked like most others, it was called "Normal." What Fortran is saying is that we now know the reason why many distributions all looked "Normal" (the Central Limit Theorem), and discussing how sampling and the CLT can lead to having a Normal distribution can be enlightening. —Preceding unsigned comment added by 141.211.66.134 ( talk) 16:33, 10 March 2010 (UTC)
Given the suggestion on the edit descriptions list that this article might be pushed towards GA status, it would be good if readers/editors would set down some areas for improvement. Any more suggestions as to what is needed? Melcombe ( talk) 09:10, 22 September 2009 (UTC)
(above comment split to allow addition of general discussion of changes needed for article) Melcombe ( talk) 08:58, 25 September 2009 (UTC)
Graphs should be improved too — the curves should be more thicker so that they are better visible; and also the labels violate MOS, as certain numbers are typeset in italics.
No section should consist of just a single test — they should be either expanded, or merged with some other sections.
The "too technical" tag shown on this talk page, but which might be missed; Melcombe 09:10, 22 September 2009 (UTC)
I removed the following paragraph from the lead, since the lead is already too long, and it'll be expanded even more to include references to multivariate normal and complex normal distributions, and Gaussian stochastic processes.
The normal distribution can be used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches. Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data are used.
The example can still be used somewhere later in the article, although we don’t have a conceivable “introduction” section before we go into hard math. … stpasha » 21:35, 23 September 2009 (UTC)
I think we should use only one notation for the exponential function. As you know, there is exp(x) and e^x. I skimmed through the article and found that exp(x) is more common. Tomeasy T C 06:51, 25 September 2009 (UTC)
The two images shown in the infobox are repeated further down. I think we should remove them. Any thoughts? Tomeasy T C 22:33, 25 September 2009 (UTC)
This subsection is largely a repetition of the section Definition. I would like to include the additional content of the subsection into the section, and remove the subsection. What do you think? Tomeasy T C 22:36, 25 September 2009 (UTC)
The article states the formula for a normal curve is as follows: But what is the origin/purpose of the constant: ? Might be useful to include that info. -- Steerpike ( talk) 20:32, 26 September 2009 (UTC)
I think it should. For example because it has less formulas in it :) … stpasha » 21:08, 27 September 2009 (UTC)
I propose to uniformly replace symbol φ (the pdf of the standard normal distribution) with ϕ (in LaTeX: \phi, in HTML: ϕ). The main reason for this change is to differentiate somehow standard normal distributions from characteristic functions, which are also denoted with φ. … stpasha » 21:35, 7 October 2009 (UTC)
This section in the article seems to be too biased towards the unbiasedness of estimation. At the same time it misses some important info about the t-statistic and construction of confidence intervals. Also the “maximum likelihood estimation” section is bloated — the detailed derivation is already present in the maximum likelihood article and probably doesn’t need to be repeated here. … stpasha » 02:07, 29 November 2009 (UTC)
The history section could be expanded. For once, it doesn't mention the important contributions of Maxwell, when he discovered that gas particles, being constantly subjected to bombardment from the other gas particles, will have their velocities distributed as 3d multivariate Gaussian rv's. I believe this discovery to be important because it demonstrated that normal distribution occurs not only as a mathematical approximation in games of chance or as a convenient tool in least squares analysis, but also exists in nature. … stpasha » 21:52, 5 October 2009 (UTC)
any variable that is the sum of a large number of independent factors
A sum of factors?? Am I getting this very wrong, or is this sentence indeed ill phrased. I would say a sum of summands or a product of factors, but still I would not really understand what this sentence tries to say. Please someone who understands the content, judge whether the wording sum of ... factors is correct. Tomeasy T C 22:25, 25 September 2009 (UTC)
OK, I see the logical flaw has been erased by somebody. Now that semantically the statement is correct, let's focus on the content. Any variable that is the sum of a large number of independent terms is distributed approximately normally. Really, is that so? I would guess most variables are not distributed at all, because they depend on independent but deterministic terms. I see that variable is linked to random variable. The qualifier random is key here to ensure the statement is not ridiculous. Therefore, the text must show this. Tomeasy T C 07:06, 2 October 2009 (UTC)
Yes, sum of factors. The word “factor” here should be understood as “An element or cause that contributes to a result (from Latin facere: one who acts)” (Collins). Of course it is so much unfortunate that this can be confused with the mathematical “factor” which is one of the terms in a product...
Another problem is the following: a typical layperson does not see the world in terms of random variables. For an everyman, the phenomenon is recognized as “random” if it recurs often and has pronouncedly different results each time: such as weather, or lottery, or coin tosses, etc. Other things such as heights or IQs aren’t really seen as “random” unless you force them to stop and think about it. For this reason, writing “any random variable which is the sum of independent terms” does not convey the important message: that this is not an abstract mathematical theorem but rather an approximation for great many random things encountered in the real life.
We can try the following: By the central limit theorem, any quantity which results from an influence of a large number (at least 10–15) of independent factors, will have approximately normal distribution. … stpasha » 21:02, 5 October 2009 (UTC)
Currently there is a separate article standard normal random variable (stub), whereas standard normal distribution redirects to the current article. I suggest that the first article be merged with the current, probably within the “Standardizing normal random variables” subsection. … stpasha » 10:12, 9 October 2009 (UTC)
Some action is needed for the redlinks shown under "tests of normality" ... either creating new articles, expanding existing ones that can be linked to, or providing direct citations; Melcombe 09:10, 22 September 2009 (UTC)
There seems a need to reduce the number of articles under "see also", preferably by saying something about them in the main text (if not already there}; Melcombe 09:10, 22 September 2009 (UTC)
More inline citations, restructuring of notes/references to more convenient form. I guess most detailed results will be findable in Johnson&Kotz so perhaps we could aim to provide page or section numbered pointers to this source. Melcombe 09:10, 22 September 2009 (UTC)
The use of the field "kurtosis" in the table seems not to be consistent across distributions. In some it seems to be the "normal" kurtosis and in some the excess kurtosis (-3). This is really problematic. I think it should either be named "excess kurtosis" in the table, or there should be two fields, one for each. Personally, I think one field should enough, and probably it should be the excess kurtosis, since this is usually more useful. However, it should be made clear, at least to people changing the page, that this is the excess kurtosis and not the other. If there is just one field, which is named "kurtosis" there will always be some who think, its the normal one and change it (see e.g. for the lognormal distribution, change from 21:13, 1 December 2009). Maybe it would be enough to change the template, so that it says "excess_kurtosis=..." instead of "kurtosis=...". Any other thoughts on this? Ezander ( talk) 15:45, 22 February 2010 (UTC)
There is a discussion on the WikiProject Statistics talk page about the financial variables section of this article. Regardless of the merits of the recent additions, and whether they are OR, the issues raised are more about difficulties with estimating the marginal distribution of a dependent, non-stationary sequence, and less about normality per se. This content is too detailed and not sufficiently relevant to be included here. Skbkekas ( talk) 16:26, 15 March 2010 (UTC)
“ |
THE NORMAL LAW OF ERROR STANDS OUT IN THE EXPERIENCE OF MANKIND AS ONE OF THE BROADEST GENERALIZATIONS OF NATURAL PHILOSOPHY ♦ IT SERVES AS THE GUIDING INSTRUMENT IN RESEARCHES IN THE PHYSICAL AND SOCIAL SCIENCES AND IN MEDICINE AGRICULTURE AND ENGINEERING ♦ IT IS AN INDISPENSABLE TOOL FOR THE ANALYSIS AND THE INTERPRETATION OF THE BASIC DATA OBTAINED BY OBSERVATION AND EXPERIMENT ♦ |
” |
// stpasha » 23:58, 21 March 2010 (UTC)
This article says:
In what sense can it be said that z-scores and percentiles "are derived from the normal distribution"? Michael Hardy ( talk) 16:16, 27 April 2010 (UTC)
The article presently has "Commonly the letter N is written in calligraphic font (typed as \mathcal{N} in LaTeX)." without a citation. All the sources I have use a non-script font and I have never seen it in a script font: it is certainly not common. WP:MSM says " it is good to use standard notation if you can" so why use something unnecessarily complicated, particulrly as there is no citation for this notation. Melcombe ( talk) 13:49, 18 May 2010 (UTC)
I would like to generate a set of numbers (x,y) with a known mean and CV; that is, I wish to generate a set of numbers that have a gaussian distribution, where I can set the mean and CV in advance. Thanks PS: maybe it doesn't go here, but a section on curve fitting software might help (please - no"r", if you know R, you already know a lot; stuff like IgorPro or Kaleidagraph etc, or excel thanks —Preceding unsigned comment added by 108.7.0.214 ( talk) 17:34, 22 June 2010 (UTC)
OK, let's start by assuming the covariance matrix is
so that ρ is the correlation. To be continued.... Michael Hardy ( talk) 18:23, 22 June 2010 (UTC)
By definition the entropy of the Normal distribution is not negative value. But what if σ → 0 in the finite formula of entropy? Thanks. Aleksey. —Preceding unsigned comment added by Kharevsky ( talk • contribs) 08:06, 5 July 2010 (UTC)
this article, under Definition, and the one on Gaussian function contain conflicting information on the meaning of constants a and c for the "bell curve."
— Preceding unsigned comment added by 68.37.143.246 ( talk) 01:52, 7 July 2010 (UTC)
Isn't this section a little much of a "how to" for Wikipedia? 018 ( talk) 17:22, 9 July 2010 (UTC)
Great, comprehensive page on the normal distribution, almost perfect. However, the detailed section on 'Gaussian' random number generators (which is also extremely informative) really does not belong in this top-level entry. —Preceding unsigned comment added by 129.125.178.72 ( talk) 16:04, 3 August 2010 (UTC)
I was missing a reference to the product of two gaussians. This could also go into the page for the gaussian function (there is a short mention of it, no mentioning of the resulting properties), but is is also relevant here. —Preceding unsigned comment added by 134.102.219.52 ( talk) 12:34, 7 September 2010 (UTC)
In the opening sentence the article states that the normal distribution is also known as a Gaussian distribution. I would argue however that the normal distribution is a special case of the Gaussian distribution, i.e. one that has an integral of 1, hence why it is called normal. The Gaussian distribution is in my opinion any general distribution described by the Gaussian function
If there aren't any objections I will edit the article to reflect this schroding79 ( talk) 00:08, 25 June 2008 (UTC)
Yes, the gaussian distribution is normal in shape. The standard normal distribution integrates to 1, whereas a frequency distribution which is normal or gaussian in shape does not necessarily integrate to 1. One aspect of interest to readers which is missing from the Wiki page about the Normal Distribution is the relationship between frequency distributions and probability distributions. Perhaps an introductory paragraph linking to Wiki pages about frequency distributions would be a good idea. It would help put this article into context. Lindy Louise ( talk) 09:58, 29 September 2010 (UTC)
I disagree and am curious to know why you think a frequency distribution "also always integrates to one". A frequency distribution does not always integrate to one. A probability distribution always integrates to one. This is why we normalise the normal distribution to get the standard normal distribution: the standard normal distribution integrates to one and therefore can be used as a probability distribution. This is basic stuff but is often omitted from the more esoteric textbooks. Lindy Louise ( talk) 13:18, 30 September 2010 (UTC)
I never said the standard normal distribution was the only probability distribution that integrated to one -- obviously any probability distribution function integrates to one. Neither did I say that gaussian and normal distributions are different. I agree with Melcombe. Lindy Louise ( talk) 17:16, 30 September 2010 (UTC)
Thanks O18 for your comment. I think I'm guilty of being too verbose, but I believe some readers confuse Normal Distribution with Standard Normal Distribution and I wanted to make the distinction. What I should have said is the Normal Distribution cannot be used directly as a Probability Distribution because the area under the Normal curve isn't equal to one. So we deliberately make the area under the Normal curve equal to one by doing some fancy maths: this normal distribution with an area of one is called the Standard Normal Distribution. It can then be used as a Probability Distribution simply because the area is equal to one. (In any probability system the sum of all the probabilities must equal one or, in other words, the area under a proability curve is equal to one.) Still verbose, sorry! Maybe I should have a go at updating the opening paragraph; I'll think about it. I was going to insert a link to Wiki pages about probability distributions and probability density functions but they're too difficult for non-mathematicians to understand, so I haven't. Lindy Louise ( talk) 21:29, 30 September 2010 (UTC)
If you integrate an absolute-frequency distribution you will not necessarily get unity for your answer. In fact I would think it a freak event if it were to happen! The only way you can be sure of obtaining unity by integration is if you use relative frequencies or probabilities. Hence the need for the Standard Normal Distribution, because we can be sure its integral is unity. The fact that the mean and variance of the Standard Normal Distribution are 0 and 1 is a consequence of the "normalisation" or "standardisation". The mean and variance of a Normal Distribution are not 0 and 1. That's one way of distinguishing between Normal and Standard Normal. Thanks for pointing me in the direction of the Gaussian function, but I am very familiar with the gaussian and normal functions (they're the same). Lindy Louise ( talk) 21:10, 10 December 2011 (UTC)
Calculating out by hand, the Fisher Information in the top right box seems incorrect and should instead be Khosra ( talk) 21:33, 9 September 2010 (UTC)
There used to be the time when the article started with “In probability theory, normal distribution is a continuous probability distribution which is often used to describe, at least approximately, any variable that tends to cluster around the mean”. Some people tend to revert the intro back to this sentence from time to time, which is why I think an explanation is due why such sentence is inappropriate in an encyclopedia.
First it must be stated that the distribution is not merely continuous, but absolutely continuous. Absolute continuity implies that the distribution possesses density, whereas simple continuity means very little. Second, about the “any variable that tends to cluster around the mean”. This is not an informative statement. Any unimodal distribution can be said to “cluster around the mean”, and some non-unimodal distributions too. This statement is so loose that it fails to describe anything. Finally, “is often used to describe, at least approximately” is a weasel-phrase. No serious researcher will use normal distribution to describe his data, unless he has good reasons to believe that the data IS actually normally distributed. There is a good quote from Fisher about this, see the Occurrence section. // stpasha » 09:23, 2 October 2010 (UTC)
BTW, Stpasha, you might want to check out the pages WP:TECHNICAL and Wikipedia:Lead section#Introductory text, which provide guidelines on how technical articles, and particularly the lead sections, should be written. Benwing ( talk) 23:01, 2 October 2010 (UTC)
Please see Anders Hald : A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713-1935. He has different views than Stigler.
1774 : asymptotic normality of the posterior distribution, derivation of the constant of the normal distribution (page 38). This is the first justification of the normal distribution and first appearance of the Bayesian Central Limit Theorem
1785 : further results (page 44)
Can we have a less controversial example in the lead? Current one (the heights of US adult males) isn't supported by a reference, and also contradicts a later claim in the article that the sizes of biological species are distributed approximately log-normally. Besides, “US adult males” is not a sufficiently homogeneous group: variability due to race / ethnicity make it a mixture of several log-normal distributions. // stpasha » 05:33, 5 March 2010 (UTC)
Long ago I suggested to remove this paragraph from the lead, which suggestion was refuted on the basis that it is “the only generally understandable information” there. Now that the lead has been improved in readability, maybe it’s ok to get this piece finally out? // stpasha » 02:19, 9 October 2010 (UTC)
I think maybe we should alter the definition to allow normal distributions with 0 variance. This is needed for consistency with the “ Multivariate normal distribution” article, where we say that a random vector X is distributed normally if and only if every linear combination of its components cX has univariate normal distribution. Since this linear combination can potentially have zero variance, such case must be allowed within the current article.
The cons for such inclusion are that we'll need to define pdf and cdf separately for the case σ² = 0. … stpasha » 20:43, 24 November 2009 (UTC)
I've been agnostic to the recent debate regarding whether to allow zero variance in the normal distribution. But I've rethought it, and concluded that almost certainly, we should not. The basic reason has to do with Wikipedia's "Verifiability not truth" maxim, which is a core principle ( WP:V, WP:VNT). Hence, we need to consult reliable sources, not use our mathematical intuition.
So far I've consulted two sources and they both agree that the variance must be specifically greater than, not greater than or equal to, zero. These include DeGroot and Schervish "Probability and Statistics" and Chris Bishop "Pattern Recognition and Machine Learning". I don't have any other books on hand, so I'd suggest other people check their own references. Note that on top of this, Bishop's description of the multivariate normal specifically says the covariance matrix must be positive definite, not non-negative definite. Benwing ( talk) 20:17, 10 October 2010 (UTC)
Right then now lets all stop complaining and help me write the Frechet distribution article which needs some work :) (its useful for Extreme Value theory). —Preceding unsigned comment added by 130.236.58.84 ( talk) 08:37, 11 October 2010 (UTC)
The annotation on my change got messed up accidentally. What I was trying to say was that "blah blah blah is called standard normal" sounds wrong vs. "blah blah blah is called the standard normal". But I don't know what's the "correct" convention (if there even is any at all). Benwing ( talk) 08:56, 10 October 2010 (UTC)
I am reading this in the text: "In addition, the probability of seeing a normally-distributed value that is far (i.e. more than a few standard deviations) from the mean drops off extremely rapidly. As a result, statistical inference using a normal distribution is not robust to the presence of outliers (data that is unexpectedly far from the mean, due to exceptional circumstances, observational error, etc.). When outliers are expected, data may be better described using a heavy-tailed distribution such as the Student’s t-distribution"
I am just wondering if this is accurate? My understanding of robust to outliers means that the model assigns very little (or zero) probability to values far away from the mean. So yes I think the heavy-tailed comment is correct but the first sentence should be "In addition, the probability of seeing a normally-distributed value that is far (i.e. more than a few standard deviations) from the mean drops off relatively slowly. As a result..."
For example from the article about the laplacian The pdf of the Laplace distribution is also reminiscent of the normal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean μ, the Laplace density is expressed in terms of the absolute difference from the mean. Consequently the Laplace distribution has fatter tails than the normal distribution.
So for example if we wish to use a more robust norm for the outliers we would use the L1 norm which leads to the Laplace (from a MLE point of view). But perhaps I am wrong. What does everyone else think? —Preceding unsigned comment added by 130.236.58.84 ( talk) 10:38, 10 October 2010 (UTC)
I removed the example of an electron in a 1s orbital being Gaussian. The distribution (for an electron in a 1/r Coulomb potential) is actually proportional to e-r. If I think of a similar example, I will add it in, because it was very striking! 128.111.10.89 ( talk) 13:25, 30 October 2010 (UTC)
The image in "Standard deviation and confidence intervals" part has wrong percentages. I suggest to change it with the image in Standard Score one or another, more accurate one. —Preceding unsigned comment added by 77.49.4.13 ( talk) 21:36, 10 December 2010 (UTC)
Here is the text after I changed it, I added the exact numbers in bold, which should explain why I choose to correct the article to it's current number rounding scheme:
If any one thinks this should be changes - please explain why - I'd be happy to know. Talgalili ( talk) 09:12, 11 December 2010 (UTC)
if X is normal, then what is the distribution of 1/X????? It is good to include this. even there is not solution to it. Jackzhp ( talk) 23:34, 28 December 2010 (UTC)
You transform it to u=1/x and you make calculations —Preceding unsigned comment added by 79.103.101.115 ( talk) 19:52, 9 January 2011 (UTC)
Someone has been messing around with this page, adding obscenities.
e.g . Under 'definition' there is 'The factor fuck you man in this expression ensures that the total area under the curve '
The page should be restored to its former condition.
But who knows what its properties are. You can plot it in R and it has a somewhat weird shape -- it has two modes on each side of the origin, is heavily skewed to the right (or to the left, on the negative side of the origin), and near the origin it drops suddenly and then has what looks like a completely flat section at 0 height near the origin. Benwing ( talk) 22:09, 9 April 2012 (UTC)
Like this:
Are all of the references to Mathworld/Wolfram really needed? See footnotes 12, 15, 16, 16, 26, and below the footnotes Weisstein, Eric W. "Normal distribution". MathWorld. Do all of these contribute something that is not already covered in the article? Mathstat ( talk) 23:49, 27 February 2011 (UTC)
The Gaussian function does also have numerous applications outside the field of statistics, for example regarding solution of diffusion equations and Hermite functions and regarding feature detection in computer vision. If this article would be merged under normal distributions, these connections would be lost. Hence, I think that it is more appropriate to keep the present article on the Gaussian with appropriate cross referencing and developing the article further. Tpl (talk) 11:53, 8 June 2011 (UTC)
The Kullback-Leibler divergence quoted in the article appears to be incorrect. In particular the log(sigma_1/sigma_2) term should not be in the brackets. It would be useful for someone to confirm this. The source quoted appears to be correct: http://www.allisons.org/ll/MML/KL/Normal/ Egkauston ( talk) 07:32, 29 November 2011 (UTC) Update: I checked again and I was wrong. The entry appears to be correct. Egkauston ( talk) 07:51, 29 November 2011 (UTC)
In the figures, it would be nice to show some Normal probability density function with mean and standard distribution values such as the maximum value would be higher than 1; for example, mu = .05, sigma = .003. The density can take values higher than 1; the constraint is for the cumulative density function (area under the PDF), which cannot exceds 1. It is a fairly common confussion between PDF and CDF, and I believe it is worth to remark. — Preceding unsigned comment added by 190.48.106.19 ( talk) 03:49, 25 July 2012 (UTC)
Hi, can someone check the main equation at the top of the page. I may be misunderstanding it, but its a probability distrubition so shoulden't a curve using it sum to 1? I put it into R and came out with 0.2. Checking on wolfram mathworld they use a slightly different equation. I may simply have misunderstood! Kev 109.12.210.202 ( talk) 09:07, 4 August 2012 (UTC)
The graph at the top of the page is good for the article. Thanks for having it there. It identifies the red curve as the standard normal distribution. Can the graph's author or a responsible party also identify and label the other curves, please? Thank you. 69.210.252.252 ( talk) 21:44, 15 August 2012 (UTC)
In the "Central Limit Theorem" section, the caption for the "De Moivre-Laplace" figure mentions "the function". It would be helpful if it were specified what function is meant. As it stands, the figure does not really aid understanding of the CLT. — Preceding unsigned comment added by 193.60.198.36 ( talk) 15:56, 26 November 2012 (UTC)
Hi there, just noticed there's an error in an equation in "Estimation of parameters" and it's not displaying properly. Not sure how to fix it or anything, but there it is. — Preceding unsigned comment added by 97.65.66.166 ( talk) 19:21, 25 March 2013 (UTC)
It should be noted that the Normal Distribution Function comes from the Stirling's approximation applied to the Binomial distribution (deMoivres-Laplace: http://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem.) In the binomial distribution, the probability of "each outcome" is known. That is Binomial distribution builds on that fact that "I can get k successes in n trials where each event has a probability p", and I plot the value of E(k) versus k. When I carry this to the Stirling approximation to form the Normal Distribution function, I assume each independent event has the same "p". What is this "p" that I refer to now in the context of a normal distribution function? In other words are the trials still "Bernoulli"? If yes what is the p used in the context of NDF.
If however one is simply assuming this is "distribution" function and the central limit theorem is just a coincidence, then note that most derivations of the central limit theorem also build from Binomial distribution. Can someone please clarify what is "Bernoulli" about the trails in that case? Is each E(x) associated with x still representing a success of a "Bernoulli outcome" at all??? The literature on this page, and the "central limit theorem" is not clear and is recursive..and always points back to Demoive-Laplace Theorem only.
An independent proof of the "Central limit Theorem", not relying on Binomial distribution would also help clarify this circular reference.
-Alok 11:31, 19 July 2013 (UTC) — Preceding unsigned comment added by Alokdube ( talk • contribs)
It should also be noted that wikipedia does not in anyway state that Normal Distribution function is sacrosanct but most text books and academicians tend to do so. However it would be really great if someone can show the assumptions made in the approach. -Alok 23:10, 23 July 2013 (UTC) — Preceding unsigned comment added by Alokdube ( talk • contribs)
The PDF can be re-arranged to the following form:
where Z is the Standard score (number of standard deviations from the mean). This makes it pretty obvious that the pdf is maximal when is small (narrow distribution) and when is small (towards the center of the distribution). I find this notation way simpler and more intuitive than the standard formula for the pdf. Should we include it in the main article (and where?) for the pedagogical purpose? — Preceding unsigned comment added by 129.215.5.255 ( talk) 10:46, 30 October 2013 (UTC)
The normal sum theorem for the sum of two normal variates is discussed in Lemons, Don (2002). An Introduction to Stochastic Processes in Physics. John Hopkins. p. 34. ISBN 0-8018-6867-X.. The proof of the theorem shows that the variance for the sum is the sum of the two variances. However, this doesn't prove the distribution for the sum is a normal distribution since more than one distribution can have the same variance. -- Jbergquist ( talk) 06:18, 30 November 2013 (UTC)
Would it be fair to say that few if any math majors turn to Wikipedia for help in their chosen field? If so, who exactly is this article written for? Unless they have post-secondary studies in Math, few people would have the knowledge or time to comprehend any of the terms used & these beginners, I would submit, are the vast majority of those who click on this article. We would just like, in layman's terms, an explanation of Normal Distributuion. Instead we've found a long, specialized article written for no one. — Preceding unsigned comment added by 96.55.2.6 ( talk) 22:40, 26 March 2013 (UTC)
You have links to the terms you don't understand. Also, N is an advanced subject in itself. i.e. it can not be simplified without being hollow and meaningless. Read about other types of distributions first if you want simpler examples of that type of math. The reason for the complexity, or rather lack of a comprehensive explanation for it, is that the distribution is not human constructed but an observed reality of life. It just happens to work for many common situations. -- 5.54.91.60 ( talk) 20:03, 21 June 2013 (UTC)
I'm confused. Shouldn't the ERF function be defined as ERF(a,b) = integral between a and b, instead of ERF(x) = integral between -x and +x? This would then allow for the proper definition of the CDF function as ERF(-infinity, x) instead of defining it as a single value function. Maybe the error introduced by using -x instead of -infinity is small.
130.76.64.109 ( talk) 16:15, 4 July 2013 (UTC)
I add to this, I'm a 4th year engineering student, and even then, this is going right over my head, it doesn't help that the way the formulas are shown, they cannot be selected, and as can be seen here,
Is the root function to the power of e, or is the whole term multiplied by e? — Preceding unsigned comment added by 114.76.42.246 ( talk) 23:29, 20 March 2014 (UTC)
Double factorials seem to be uncommon in mathematics, it may help with the exposition if the double factorials were replaced by their explicit formula MATThematical ( talk) 23:21, 9 May 2014 (UTC)
Normal curve never touches X axis. It was touching X axis in two figures which I have removed from the article. I would like to discuss on this point if someone has other opinion / reference. Thanks. -- Abhijeet Safai ( talk) 09:31, 29 May 2014 (UTC)
sqrt(-2*log(rand()))*cos(2*pi*rand()) — Preceding unsigned comment added by MClerc ( talk • contribs) 19:55, 6 August 2014 (UTC)
1. Cite on any regression can achieve normal residuals with proper modeling, please.
2. Some regressions explicitly assume other distributions, of course. Probit and logit come to mind.
3. I've seen weighting procedures to adjust for skewed residuals. But if the residuals have a kurtosis other than 3, how can normal kurtosis be achieved?
4. I'd like to keep this category under the Occurrence heading, but I'm honestly unclear about the proper treatment.
Everyone believes in the Gaussian law of errors, the experimenters because they think it is a mathematical theorem, the mathematicians because they think it is an empirical fact. Kennedy, quoting Poincare, but see this elaboration: http://boards.straightdope.com/sdmb/showpost.php?p=14046385&postcount=33. Measure for Measure ( talk) 20:56, 17 August 2014 (UTC)
Hello everybody,
I've just created an image that could replace another image in this article:
I think the new image is better, because
And it also has a CC0 license.
I could also re-make the other image in the same "style".
Best regards, -- MartinThoma ( talk) 19:38, 29 August 2014 (UTC)
According to the Characteristic function (probability theory) page, the CF of a distribution is the inverse Fourier transform of the PDF (and therefore the frequency-domain PDF is the Fourier transform of the time-domain CF ). We could just change instances of "Fourier transform" to "inverse Fourier transform", but the page goes on to say "...normal distribution on the frequency domain", so this we should also change to "...normal distribution on the time domain". I'm not missing something here, am I? Tsbertalan ( talk) 23:42, 4 December 2014 (UTC)
The Pascal CDF function, as shown does not translate the formula shown above it. As near as I can tell, it does not provide a correct result. I suggest that for this and other examples you use a more commonly used language: C or C++. — Preceding unsigned comment added by Statguy1 ( talk • contribs) 06:45, 16 February 2015 (UTC)
The Pascal code does not account for the double factorial in the denominator. This approximation of the CDF function is also given (with a reference) elsewhere in this WikiPedia article Normal_distribution#Numerical_approximations_for_the_normal_CDF — Preceding unsigned comment added by 138.73.5.2 ( talk) 15:02, 22 October 2015 (UTC)
The top line states "This article is about the univariate normal distribution", yet the description is in terms for 'random variables', (plural) i.e. the multivariate case. I'm not sure if the plural usage 'random variables' is a formal math usage I'm not familiar with, a british/american usage difference, or just poor usage. Also, the lead paragraph does not directly state what the Normal Distribution is, but infers the definition from the CLT. I suggest restating and splitting the 2nd lead paragraph as below, and submit it to discussion here first.
-Orig The normal distribution is remarkably useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.
-Rework The normal distribution is defined by the central limit theorem. Generalized, it states, under some conditions (which include finite variance), that the distribution of averages of a random variable independently drawn from independent distributions converge to the normal distribution, when the number of samples is sufficiently large.
Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed. LarryLACa ( talk) 03:47, 13 October 2015 (UTC)
One of the most famous and used distribution by scientist is bell type distributions. It is desired since it goes from maximum to minimum and vice versa. Distribution of phenomena and mathematical modeling can appear very easily.
Read more on reference: http://mathworld.wolfram.com/NormalDistribution.html
MansourJE ( talk) 21:47, 14 April 2016 (UTC)
The applications are briefly touched on, but the danger of misapplication is completely ignored.
MansourJE ( talk) 17:18, 14 April 2016 (UTC)
Hello, For the figure in the "Standard deviation and coverage" section, the vertical lines should be equally spaced. — Preceding unsigned comment added by 99.226.5.121 ( talk) 21:31, 8 February 2017 (UTC)
This article is very good, clear citation, the math is correct, has very reliable references. But it doesn't mention that Gaussian Distribution has a widely application in air pollution transportation model and diffusion model.
Jiamingshi (
talk)
05:35, 28 April 2017 (UTC)
Also the reciprocal of the standard deviation might be defined as the precision and the expression of the normal distribution becomes
According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, the fact that the PDF has unit height at zero,
Well:
and also for
So??? Madyno ( talk) 15:04, 11 May 2017 (UTC)
I think the formula for the Fourier transform is not in line with the definition in the lemma of Fourier transform. Madyno ( talk) 10:44, 22 October 2017 (UTC)
I am sure this article is very good, but I came to this page to find out why a Bell curve (or bell curve) is called as it is. Is it named after a shape or the person who first devised it or what? And maybe the article should say so? Kiltpin ( talk) 12:16, 27 October 2016 (UTC)
or
I like the first version more, as it looks more clearly arranged. -- Gunnar ( talk) 18:36, 4 November 2018 (UTC)
1. The entropy H, as a function of the density f, is called a functional.
2. The symbol d in an integral expression is a kind op operator, not a variable, and hence not set in italic. Madyno ( talk) 21:24, 14 June 2019 (UTC)
Okay, matter of taste. Madyno ( talk) 21:33, 14 June 2019 (UTC)
I miss the useful relation — Preceding unsigned comment added by 141.23.181.132 ( talk) 16:34, 18 August 2019 (UTC)
The section on the quantile function defines to be , but then 2 lines later uses to mean something related but different, without warning. Fathead99 ( talk) 15:17, 28 July 2017 (UTC)
Regarding diagrams of different PDFs: There should be a curve here with a maximum value above 1, just to illustrate that it is possible. — Preceding unsigned comment added by 2001:4643:E6E3:0:2C29:9E4F:EDF9:AD78 ( talk) 13:19, 12 September 2019 (UTC)
I'm missing the average deviation from the mean. Its simply sqrt(2/pi) * sigma ~ 0.797 sigma (see https://www.wolframalpha.com/input/?i=2+*+integral+from+0+to+infinity+of+x+*+exp%28-%28x%2Fsigma%29%5E2%2F2%29+%2F+%28sigma+*+sqrt%282+pi%29%29 ) This measure may not be a widely used quantity among mathematicians, but it's what less-mathematically-inclined people tend to report (e.g. bsc students, bankers etc)
May I add this to the table at top right? If so, how? Michi zh ( talk) 12:16, 7 January 2020 (UTC)
{{
Infobox probability distribution}}
doesn't have an entry for this, so there's currently no way to add it here, unless you also modify the infobox. That's probably not a great idea since this is such an unusual measure. I'm curious why you think less mathematically inclined people would prefer it (and what do you mean by "report"?). It's almost always more difficult to calculate then the std dev/variance. –
Deacon Vorbis (
carbon •
videos)
14:18, 7 January 2020 (UTC)
{{
Infobox probability distribution}}
I can place this suggestion. Cheers!
Michi zh (
talk)
20:10, 12 January 2020 (UTC)So the deed is done! I hope other people find it useful too! Feel free to delete this section if it's not necessary anymore. Best Michi zh ( talk) 22:37, 16 January 2020 (UTC)
The MAD is an acronym for several different measures (e.g. mean absolute deviation but also the median one). Please make sure to link the name to the specific definition used. Thanks. Tal Galili ( talk) 05:18, 17 January 2020 (UTC)
Could someone check the table of values? They seem to be wrong. — Preceding unsigned comment added by Piqm ( talk • contribs) 19:05, 2 November 2020 (UTC)
For some displays (like mine) the negative sign in the exponent doesn't show properly unless you've zoomed in. I'm not savvy enough to fix it — Preceding unsigned comment added by 2603:8080:1540:546:3175:C96:9C96:B48C ( talk) 20:58, 6 December 2020 (UTC)
This appears to be a bug with Chromium (FF and Safari show the minus properly). Logged a bug, let's see if it's fixable by Chromium: https://bugs.chromium.org/p/chromium/issues/detail?id=1159852 — Preceding unsigned comment added by 84.9.90.236 ( talk) 17:13, 17 December 2020 (UTC)
Is a sine wave of one period ( example) a type of bell curve? I didn't see it mentioned anywhere in the article. ➧ datumizer ☎ 13:16, 26 December 2020 (UTC)
I've only ever seen the term "bell curve" applied to an actual normal curve or to a curve that is close to normal in some sense. The restricted sine you mention would not qualify. FilipeS ( talk) 05:26, 30 December 2020 (UTC)
I feel the article is incomplete without some mention of the asymptotic behaviour of the tails of the curve. — Preceding unsigned comment added by 77.61.180.106 ( talk) 18:26, 11 January 2021 (UTC)
This article is terrible if you don't have advanced level maths already. There's not even an attempt to explain it in simple English (yes, I'm aware of Simple English Wiki; that version of this article is also not in Simple English). Who is the target audience of this article? People who already understand this sort of maths? I'd be surprised if even 5% of readers would learn anything from reading this article, I certainly haven't. — Preceding unsigned comment added by 217.155.20.204 ( talk) 14:18, 14 October 2020 (UTC)
Absolutely agree, Wikipedia is meant to be an accessible resource for finding out about something you don't know. Not a reminder for people with technical knowledge. This article is effectively useless, a school student stopping in here is going to take one look and navigate away. — Preceding unsigned comment added by 122.62.34.148 ( talk) 08:12, 11 March 2021 (UTC)
I quickly tried searching for tail bound but I saw no mention of tail bounds or concentration inequalities, which are very useful. The most useful being ones like
for one tail and twice for both tails.
See http://www.stat.yale.edu/~pollard/Books/Mini/Basic.pdf
https://www.math.wisc.edu/~roch/grad-prob/gradprob-notes7.pdf Wqwt ( talk) 05:17, 25 April 2021 (UTC)
Just like I commented on /info/en/?search=Multivariate_normal_distribution user Dvidby0 is citing his own paper with dubious contribution to the topic. It is more subtle here but I think it is worth to reconsider if citing yourself is moral ( are you promoting your 2020 paper?) and is it adding anything new to the topic? Maybe it is adding something but sure as hell it isn't adding clarity. People here want to learn something about normal distribution and you are putting your stuff in. Imagine everybody started adding things from their 1y old papers to get citations, wikipedia would look like complete garbage. — Preceding unsigned comment added by Vretka ( talk • contribs) 20:55, 12 March 2021 (UTC)
This article is or was the subject of a Wiki Education Foundation-supported course assignment. Further details are available on the course page. Peer reviewers: Jiamingshi.
Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT ( talk) 05:23, 17 January 2022 (UTC)
wiki is a general encylopedia for the avg person do you people really think the introduction is pitched to the avg person ? maybe ask your parents or grandparents jeez , grow up you math people and learn how to write English and don't you dare criticize me for being mean; i am really fed up with math people's total inability to write at the appropriate level really fed up — Preceding unsigned comment added by 2601:192:4700:1F70:BDC1:852D:B5D9:A6AC ( talk) 23:30, 30 December 2021 (UTC)
After the table with the moments of a Gaussian there is a sentence about the expected value of the "reciprocal". In fact the result is about for . In addition the result is a very weak upper bound: it is stated for a Gaussian with arbitrary mean but the result is independent of the mean. It seems unnecessary for such a result to appear in a section about the exact moments of the Gaussian. — Preceding unsigned comment added by Aterbiou ( talk • contribs) 09:34, 9 March 2022 (UTC)
I removed the following opening paragraph from the article. It seemed pointlessly vague and off-topic: articles on specific distributions aren't the place for handwaving explanations of basic statistical concepts. Leaving it here in case someone has a different opinion. 69.127.73.225 ( talk) 02:52, 4 April 2022 (UTC)
"A normal distribution is a probability distribution used to model phenomena that have a default behaviour and cumulative possible deviations from that behaviour. For instance, a proficient archer's arrows are expected to land around the bull's eye of the target; however, due to aggregating imperfections in the archer's technique, most arrows will miss the bull's eye by some distance. The average of this distance is known in archery as accuracy, while the amount of variation in the distances as precision. In the context of a normal distribution, accuracy and precision are referred to as the mean and the standard deviation, respectively. Thus, a narrow measure of an archer's proficiency can be expressed with two values: a mean and a standard deviation. In a normal distribution, these two values mean: there is a ~68% probability that an arrow will land within one standard deviation of the archer's average accuracy; a ~95% probability that an arrow will land within two standard deviations of the archer's average accuracy; ~99.7% within three; and so on, slowly increasing towards 100%."
Following the line "The CDF of the standard normal distribution can be expanded by Integration by parts into a series: " I think that in the equation for phi(x) the fraction "x^5/3.5" should read "x^5/15". 217.155.205.34 ( talk) 17:37, 12 May 2022 (UTC)
I tried implementing this and got tiny variances. Searching around, I found https://stats.stackexchange.com/questions/365192/bayesian-update-for-a-univariate-normal-distribution-with-unknown-mean-and-varia where someone else tried and got tiny variances. As far as I can tell, it works nicely if we don't divide by the total observations (or obs+pseudo) in the final distribution. I don't know how to prove such things. — Preceding unsigned comment added by 76.146.32.69 ( talk) 16:19, 25 July 2022 (UTC)
The pdf formula in the some sections is missing a minus sign. In the right column the pdf is correct. Ron.linssen ( talk) 08:33, 29 September 2022 (UTC)