# Randomness and the Bell Curve **URL:** \ **Category:** Factual Questions\ **Created:** [June 2, 2022, 1:57am UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542 "2022-06-02T01:57:19Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 2, 2022, 11:28pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/21 "2022-06-02T23:28:49Z")
One notable feature of the Poisson distribution, as distinct from the Gaussian, is that it has no left tail whatsoever, as the distribution goes to 0 (and stays there) at 0 or below. And it’s not necessarily true that the mean of a Poisson is equal to the variance. It can be, but it can also be much larger or smaller. A Poisson distribution with a mean much greater than the standard deviation ends up looking very much like a Gaussian, but one with a small mean (relative to the standard deviation) looks very different. ---
**Author:** ![Maserschmidt](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/maserschmidt/32/18829_2.png) [@Maserschmidt](https://boards.straightdope.com/u/Maserschmidt)\ **Post date:** [June 2, 2022, 11:32pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/22 "2022-06-02T23:32:52Z")
> [@Chronos](#): > > And it’s not necessarily true that the mean of a Poisson is equal to the variance. That is very different from what I learned, would appreciate a cite on that. One other nice feature of Poisson regressions is that they are additive, meaning if you fit multiple (independent) regressions you can use the added means to create a combined distribution. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [June 2, 2022, 11:40pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/23 "2022-06-02T23:40:41Z")
Perhaps thinking of an exponential or Erlang, rather than Poisson? The mean and variance of a Poisson distribution are definitely the same. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 3, 2022, 2:26am UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/24 "2022-06-03T02:26:11Z")
On the other hand, if the mean and variance of a probability distribution are the same, it is by no means necessarily a Poisson distribution 🙂 ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [June 3, 2022, 4:04am UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/25 "2022-06-03T04:04:12Z")
> [@Chronos](#): > > And it’s not necessarily true that the mean of a Poisson is equal to the variance. It can be, but it can also be much larger or smaller. A Poisson distribution with a mean much greater than the standard deviation ends up looking very much like a Gaussian, but one with a small mean (relative to the standard deviation) looks very different. The mean and variance of a Poisson distribution are always equal essentially by definition. That is the distribution is defined by its probability mass and the mean and variance are equal to the parameter usually called lambda. But they you switch and use mean and standard deviation. When the mean is much larger than that standard deviation (not variance), then lambda is large and the skewness is 1/sqrt(lambda) and the kurtic excess is 1/lambda. In this sense the distribution for large lambda is more Gaussian than for small lambda. > **[Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution)** > > Γ > ( > ⌊ > k > + > 1 > ⌋ > , > λ > ) > > > ⌊ > k > ⌋ > ! > > > > , > > > {\\displaystyle {\\frac {\\Gamma (\\lfloor k+1\\rfloor ,\\lambda )}{\\lfloor k\\rfloor !}},} > > or > > > > > e > > ... ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 3, 2022, 10:50am UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/26 "2022-06-03T10:50:50Z")
I switched to standard deviation because I assumed that “variance” was a mistake, because mean and variance can’t be the same, as they have different dimension. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 3, 2022, 10:54am UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/27 "2022-06-03T10:54:09Z")
If you take a (continuous) exponential distribution, its mean should equal its standard deviation. ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [June 3, 2022, 3:44pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/28 "2022-06-03T15:44:27Z")
> [@Chronos](#): > > because mean and variance can’t be the same, as they have different dimension. This is incorrect. (wow the thrill of being able to correct the great Chronos. I feel unworthy) I understand where your coming from, in the case of scaled parameters you would need your standard deviation to be on the same scale as your mean. So if you have a life time of 9 months your standard deviation might be 3 months and so your variance would have to be 9 square months. If you changed the units to years, they your lifetime would be 0.75 years, your standard deviation would be 0.25 years and your variance would be 0.0625 squared years. So the variance that was equal to the mean in one set of units couldn’t be equal in an alternate set. In the case of the poisson distribution you are dealing with count data so there isn’t any issue with needing units to match. The Poisson distribution only has one parameter: the mean. All the rest of the characteristics of the distribution fall from that. If your geiger counter is getting an average of 10 hits per second, then the variance of the number counts after one second will be 10. This comes to an average of 600 counts per minute which will have a variance also of 600. This might seem counter intuitive since if the mean increase by a factor of 60, shouldn’t the variance increase by a factor of 60^2. But the extra time gives more time for the noise to cancel out and so the relative signal to noise ratio will be smaller. after a minute than it would be after a second. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [June 3, 2022, 4:31pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/29 "2022-06-03T16:31:43Z")
> [@Chronos](#): > > I switched to standard deviation because I assumed that “variance” was a mistake, because mean and variance can’t be the same, as they have different dimension. Nevertheless they are. The Poisson distribution is a distribution of counts so the moments are pure numbers without dimension. ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [June 3, 2022, 9:11pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/30 "2022-06-03T21:11:17Z")
> [@Stranger\_On\_A\_Train](#): > > “Normal” in the context of statistical distributions means that the moment is perpendicular, not that is typical or expected. Ah! Thanks for the correction! I didn’t know this. So, wouldn’t that mean binomial and Cauchy distributions are also “normal”? ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [June 3, 2022, 11:39pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/31 "2022-06-03T23:39:43Z")
They are ‘normal’ in that context but the term has come to be used in statistics specifically to refer to the (Laplace-)Gauss distribution, presumably because it is the most frequently encountered in introductory statistics courses and widely used in frequentist statistics such as hypothesis testing, error analysis, and basic reliability modeling. Stranger ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [June 4, 2022, 11:42am UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/32 "2022-06-04T11:42:04Z")
> [@Stranger\_On\_A\_Train](#): > > […] the term has come to be used in statistics specifically to refer to the (Laplace-)Gauss distribution, presumably because it is the most frequently encountered […] That sounds like the term _is_ used because it is “normal” (“typical”) for so many phenomena. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 4, 2022, 12:31pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/33 "2022-06-04T12:31:17Z")
I imagine that, functionally, that term is used for that distribution because by this point, it’s the name for that distribution. The rest is etymology, more than mathematics, and I suspect that most folks who use the term don’t know the etymology. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [June 4, 2022, 3:50pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/34 "2022-06-04T15:50:08Z")
Most ‘normal’ things are named for Joe Normal, PhD who was prolific and had many contributions in many areas of math and physics. 😉 ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [June 4, 2022, 4:03pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/35 "2022-06-04T16:03:50Z")
Ah. Like Professor Staff, who teaches in an amazing range of fields. ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [June 4, 2022, 6:20pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/36 "2022-06-04T18:20:56Z")
> [@Napier](#): > > That sounds like the term _is_ used because it is “normal” (“typical”) for so many phenomena. It is etymologically coincidental that the standard Gaussian distribution is both mathematically normal (perpendicular) and is the ‘typical’ distribution that people re familiar with (in many cases of casual user of statistics, the Gaussian distribution may be the only distribution they are familiar with). It is like how when you use the term “trigonometry” people think in terms of sine, cosine, and tangent functions and their inverses because those are the most commonly used in a basic analytic geometry class or in studying Fourier analyssi but there are actually three other much less used trigonometric functions (secant, cosecant, and cotangent) that are also included in the domain of trigonometry. Stranger ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 4, 2022, 6:39pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/37 "2022-06-04T18:39:02Z")
I have heard of a definition of skewness which is basically the third standardized moment about the mean. E.g. for any distribution which is symmetric with respect to the mean, all the odd central moments trivially vanish, so the skewness of such a distribution will be zero. As for a “normally distributed” variable, I am pretty sure people are justifiably going to assume the distribution is Gaussian. > [@Stranger\_On\_A\_Train](#): > > the standard Gaussian distribution is both mathematically normal (perpendicular) Perpendicular to what? ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [June 4, 2022, 7:08pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/38 "2022-06-04T19:08:10Z")
I think perpendicular in the sense mean and variance are independent- you can change either without changing the other. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [June 4, 2022, 7:34pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/39 "2022-06-04T19:34:58Z")
But this is true of any symmetric distribution for which the variance exists so the Gaussian distribution is only **a** normal distribution rather than **the** normal distribution by this logic. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [June 4, 2022, 7:57pm UTC](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542/40 "2022-06-04T19:57:37Z")
True. Then I don’t know what @Stranger_On_A_Train meant by perpendicular. [Previous page](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542.md?page=1) [Next page](https://boards.straightdope.com/t/randomness-and-the-bell-curve/965542.md?page=3)