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Is there a deep difference between a Normal and a Gaussian distribution, I've seen many papers using them without distinction, and I usually also refer to them as the same thing.

However, my PI recently told me that a normal is the specific case of the Gaussian with mean=0 and std=1, which I also heard some time ago in another outlet, what is the consensus on this?

According to Wikipedia, what they call the normal, is the standard normal distribution, while the Normal is a synonym for the Gaussian, but then again, I'm not sure about Wikipedia either.

Thanks

GeoMatt22
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Leon palafox
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    Wikipedia is right, in this case. It usually is for topics like this. I would be more leery of it on controversial topics. – Peter Flom Apr 12 '13 at 17:33
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    There is a consensus. Your PI is confusing "Normal" with "Standard normal." The former refers to any version of the latter obtained via a change of location or scale. – whuber Apr 12 '13 at 17:43
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    Go with Wikipedia & Peter & whuber - & hire a different private investigator. – Scortchi - Reinstate Monica Apr 12 '13 at 18:23
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    Here's one moderately authoritative reference: http://mathworld.wolfram.com/GaussianFunction.html. – whuber Apr 12 '13 at 20:53
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    Peter Flom is right - as is Wikipedia, and whuber, and Scortchi. You can find any number of more authoritative works that support it - hundreds, perhaps thousands of standard texts for example and numerous papers. – Glen_b Apr 12 '13 at 23:02
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    As first commenter there, @PeterFlom should probably convert his comment into an answer so this Q. has a good answer. – Glen_b Dec 19 '14 at 04:42
  • I have heard the convention of using "Gaussian" only for the standard normal before (in the context of theoretical CS). This is not standard usage, and indeed is the opposite of what you've apparently heard, so I'd recommend against it. :) – Danica Dec 30 '16 at 01:05
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    Shortly: "Standard Normal Distribution" is a particular case of Gaussian(Normal) distribution where Mean_value=0 and Standard_Deviation=1. – Danylo Zherebetskyy Jul 12 '18 at 00:10
  • Normal Distribution = Gaussian Distribution. Standard normal Distribution is special case of Normal/Gaussian distribution where mean is 0 and standard deviation is 1. All Standard Normal Distributions are Normal/Gaussian Distributions. But not all Normal/Gaussian Distributions are Standard Normal. – Deepak Oct 09 '21 at 10:08

3 Answers3

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Wikipedia is right. The Gaussian is the same as the normal. Wikipedia can usually be trusted on this sort of question.

Peter Flom
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In http://mathworld.wolfram.com/NormalDistribution.html, there is a mention of a standard Normal distribution which looks like the one you were mentioning as mean = 0 and std = 1. But the Normal distribution is the same as Gaussian which can be converted to a standard normal distribution by representing using the variable z = (x-mean)/std.

Subramanian Tirunellayi Ramach
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If you just talk about probability distribution, Gaussian and Normal distributions are identical as Wikipedia mentioned. But a Gaussian function is not necessarily a Normal distribution when its integrate is not to 1.

Jerry
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    "But a Gaussian function is not necessarily a Normal distribution when its integrate is not to 1." This is not correct. All absolutely continuous probability distributions integrate to 1. This is part of how probabilities are conventionally defined (cf Kolmogorov axioms). – Sycorax May 02 '17 at 22:07
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    @Sycorax I think this may be in reference to the more generic "[Gaussian function](https://en.wikipedia.org/wiki/Gaussian_function)", which in some contexts need not be normalized (i.e. via a [Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral) factor). However I agree that the OP asked about a Gaussian *distribution*, so this answer is perhaps more of a comment. – GeoMatt22 May 03 '17 at 01:13
  • "All absolutely continuous probability distributions integrate to 1" That is what I meant actually, should have said when the Gaussian function's integrate is not to 1 instead. – Jerry May 03 '17 at 16:49