# Talk:Closed Form Precision

For the most part the discussion of the Bessel, Gaussian and Rayleigh correction factors is off the deep end. All elementary text books would use (n-1) as the divisor when calculating the standard deviation, but few would name it. The Gaussian correction is small compared to the overall size of the confidence interval with small samples. For measures like group size it is also unnecessary. For small samples the distribution is significantly skewed so you should use tables based on Monte Carlo results which would have the correction factor "built in." By the time you get enough data for group size measurements to be really be normally distributed the Gaussian correction factor is then totally negligible. Herb (talk) 16:36, 24 May 2015 (EDT)

**Perhaps the correction factors could be buried deeper or on another page. But we can't just wave them away and we shouldn't pull mysterious correction factors out of thin air. This is, after all, the closed form analysis. Yes, you can get the same results via Monte Carlo and use that approach instead, but for those who want to see the pure math these terms are part of it. David (talk) 17:56, 24 May 2015 (EDT)**

The fact that most of the measures are really skewed distributions deserves more serious discussion. Herb (talk) 16:36, 24 May 2015 (EDT)

**Agreed. Higher moments are noted in a few of the spreadsheets, like Media:Sigma1RangeStatistics.xls, but a discourse would be a worthwhile addition. David (talk) 17:56, 24 May 2015 (EDT)**

I must admit a bit of curiosity here. I don't know if the Student's T test has the Gaussian correction built into the tables or not. For small samples (2-5 degrees of freedom) the confidence interval is so large that it wouldn't make a lot of difference, and when doing various error combining to get an "effective" number for the degrees of freedom then having factor built in might mess things up. I looked and can't find the answer readily available. I'll try to remember to ask the question on the math stats forum. Herb (talk) 16:36, 24 May 2015 (EDT)

## σ

The paragraph under heading "Estimating σ" misses the salient points entirely. First there are so many different uses of \(\sigma\) that it is confusing when hopping around on wiki. \(\sigma_{RSD}\) should be used consistently for the radial standard deviation of the population of the Rayleigh distribution. Second \(s_{RSD}\) can't be calculated directly from the experimental data like *s* can for the normal distribution. You have to calculate \(s_r\) from the data, then use a theoretical proportionality constant to get \(s_{RSD}\). Herb (talk) 16:36, 24 May 2015 (EDT)

**As noted at the top of the page this use of σ is***not*confusing because it turns out to be the same σ used in the parameterization of the normal distribution used as the model, which is the same as the standard deviation. David (talk) 17:56, 24 May 2015 (EDT)

The discussion about the *unbiased Rayleigh estimator* is just wrong. I'll admit another limit on my knowledge here. You could calculate \(s_{RSD}\) from either the experimental \(\bar{r}\) or \(s_r\). It seems some combination of the two would give the "best" result. I don't have a clue how to do that error combination. I do know that in general the measurement of \(\bar{r}\) would have a greater relative precision (as % error) than \(s_r\). In fact the ratio of the two estimates could be used as a test for outliers. The standard deviation is more sensitive to an outlier than the mean.

**I don't understand what's wrong: the math is right there in Closed Form Precision#Variance_Estimates and Closed Form Precision#Rayleigh_Estimates. Expand the equations and they are identical. David (talk) 17:56, 24 May 2015 (EDT)**

- The point that I am trying to make here, and above in my criticism of the paragraph under heading "Estimating σ", is that there are two different uses of \(\sigma\) connected to the Rayleigh distribution, and the two are not equal.

\(\sigma_r\) is the standard deviation of the mean radius measurement

\(\sigma_{RSD}\) is the shape factor used in the PDF

Also the wikipedia section http://en.wikipedia.org/wiki/Rayleigh_distribution#Parameter_estimation is a bit confusing, but it essentially refers to N dimensions, not N data points in the sample. In our case N=2.

Herb (talk) 22:08, 24 May 2015 (EDT)

- The point that I am trying to make here, and above in my criticism of the paragraph under heading "Estimating σ", is that there are two different uses of \(\sigma\) connected to the Rayleigh distribution, and the two are not equal.

- read this: http://ballistipedia.com/index.php?title=User_talk:Herb#Radial_Standard_Deviation_of_the_Rayleigh_Distribution

Herb (talk) 12:39, 25 May 2015 (EDT)

- read this: http://ballistipedia.com/index.php?title=User_talk:Herb#Radial_Standard_Deviation_of_the_Rayleigh_Distribution

**In order to avoid this confusion I make a point here of suggesting we avoid talking about "radial standard deviation" (i.e., the standard deviation of radii). If we do that then the only σ we're talking about is the one used in the Rayleigh parameterization which (via transformation to polar coordinates)***is*the same as the σ parameter of the normal distribution associated with this model. If you see any other σ with another meaning please point it out so we can either remove it or clarify if it can't be removed. David (talk) 12:31, 25 May 2015 (EDT)

- Making the simplification doesn't avoid the confusion it creates it. Wikipedia doesn't create unique σ's for every function and use by labeling them with subscripts since there are thousands of cases. We should since the σ in the Rayleigh distribution has other relationships to the distribution that σ in the normal distribution doesn't have.

- As I noted above read this: http://ballistipedia.com/index.php?title=User_talk:Herb#Radial_Standard_Deviation_of_the_Rayleigh_Distribution

Herb (talk) 13:06, 25 May 2015 (EDT)

- As I noted above read this: http://ballistipedia.com/index.php?title=User_talk:Herb#Radial_Standard_Deviation_of_the_Rayleigh_Distribution

**"Radial Standard Deviation" is the problem. If by RSD you mean "the standard deviation of radii" then it is***not*the Rayleigh scale parameter. But if you introduce the term then people*will*confuse it with the Rayleigh parameter ... when they're not confusing it with earlier definitions like \(\sqrt{\sigma_h^2 + \sigma_v^2}\). I haven't found a reason to even refer to the standard deviation of radii: it is not used in any calculation or estimation, and it is not a helpful measure given the others in use.**As for the unlabelled σ: When used as the Rayleigh parameter it is done in the context of the symmetric bivariate normal distribution, in which case every σ***is*identical. Elsewhere when we talk about non-symmetric distributions, or specific axes, we give it a subscript. Where else should it be labelled? David (talk) 14:03, 25 May 2015 (EDT)

## Mean Diameter

In the section on *Mean Radius* the use of the phrase *Mean Diameter* seems out of whack. Using a "diameter" as a radius is just confusing for no good reason. Seems like you should just use 2\(\bar{r}\). Herb (talk) 16:36, 24 May 2015 (EDT)

*Mean Diameter*has come up as an attractive standard, since it corresponds to the 96% CEP which is closer to the measures people are used to than 50% CEP or the mean radius, and rolls off the tongue more easily than "two mean radii" or any other variation I've heard of. David (talk) 17:56, 24 May 2015 (EDT)

## Sighter Shots

A number of things are stuffed on this wiki page because they fit some erroneous notion of "Closed form". For example the discussion of sighter shots. That should be a a separate wiki page. The whole thing about the errors for sighter shots also depends on the assumptions for the dispersion for which there are 4 general cases. Herb (talk) 16:36, 24 May 2015 (EDT)