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I am not a statistician so I only know the bits of statistics I have
needed as I go along, but here are a couple of not normal
distributions to think about.

If you take measurements truncated to a certain number of digits and
then average them, this is the average of  independent identically
distributed uniform variables.

The resulting distribution is the Bates Distribution
https://en.wikipedia.org/wiki/Bates_distribution. Obviously the
central limit theorem tells you it tends towards normal in the limit
of averaging infinitely many, but for a small number you see the
kurtosis (it is a short tailed distribution)

Another two of interest to navigators might be the wrapped normal and
von Mises distributions that are used for angles. Obviously an angle
cant actually be normally distributed as it wraps around.  Maybe the
errors in a compass that can spin around wildly if you do some tight
turns before settling down, observed by a computer so you cant use
common sense to ignore extreme values.

Bill

On Tue, 5 Feb 2019 at 21:00, Frank Reed  wrote:
>
> Dave, you wrote:
> "One little gem which will delight you I’m sure is Anderson’s ‘Is the Gaussian Distribution Normal?’"
>
> The idea that common measurements ("normal" measurements) are not "normally 
distributed" (pulled from a Gaussian distribution) is well-known. It's 
introductory statistics. Folks rediscover this periodically and think they've 
discovered something quite profound. There are other distributions that one 
can use. But how can we generate simulated data that broadly resembles real 
navigational data with its relatively high prevalence of outliers?
>
> There is a relatively easy way to model the higher prevalence of outliers in 
sextant observations and other sorts of navigation measurements which allows 
us to use many of the mathematical properties of Gaussian distributions. We 
imagine our numbers as being pulled from two bins (both assumed to have mean 
value equal to zero). One bin, call it bin A, has a relatively "normal" 
standard deviation, call it s0 (for altitude sights in celestial navigation, 
s0 might be 0.5 minutes of arc). The other bin, bin B, has a higher standard 
deviation, call it s1. And s1 might be, in a typical real-world modelling 
case, three times larger than s0. Number are drawn from bin A some large 
fraction of the time, e.g. 80% of instances, and drawn from bin B the rest of 
the time. Of course the numbers aren't labeled with the bin that they came 
from so all you get in the end is a bunch of numbers with some statistical 
properties. Those numbers will have a net standard deviation somewhat greater 
than s0, but the key property is that they, collectively, will not correspond 
to a Gaussian distribution because there will be more "outliers". The "tails" 
of the distribution are thicker, more heavily-populated. A statistical 
measure of this is known by the rather ugly, jargon-y term "kurtosity".
>
> So if you're simulating observations, don't use a simpleGaussian. Use a pair 
of Gaussians, as above. This is a nice, easily-implemented technique for 
generating model data for navigation simulations, and it can help find cases 
where the "normal" math might lead you astray.
>
> Frank Reed
>
> 

   

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