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# how to calculate circular standard deviation ?

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how to calculate circular standard deviation ?

Guest Mar 22, 2017
#1
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$$\sigma =\dfrac{1}{N-1}\left(\displaystyle\sum^{N-1}_{i=0}x^{2}_{i}-\dfrac{1}{N}\left(\displaystyle\sum^{N-1}_{i=0}x_i\right)^2\right)$$

MaxWong  Mar 22, 2017
#4
+1093
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Hi Max,

There’s an omission in your equation.  The Sigma needs squared.

$$\large \sigma^2 =\normalsize \dfrac{1}{N-1}\left(\displaystyle\sum^{N-1}_{i=0}x^{2}_{i}-\dfrac{1}{N}\left(\displaystyle\sum^{N-1}_{i=0}x_i\right)^2\right)$$

Or use the square root of the equation:

$$\large \sigma =\normalsize \sqrt{\dfrac{1}{N-1}\left(\displaystyle\sum^{N-1}_{i=0}x^{2}_{i}-\dfrac{1}{N}\left(\displaystyle\sum^{N-1}_{i=0}x_i\right)^2\right)}$$

This is an efficient computational form of the equivalent:

$$\sigma = \sqrt { \frac{1}{N} \sum \limits_{i=1} ^ {N}(\chi - \mu)^2 }$$

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The formula for circular standard deviation is in post #3

GingerAle  Mar 23, 2017
#2
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Hello Max: In my 650-page Math Dictionary, there is no reference to "circular standard deviation", nor is there such a thing in Mathworld as well! The formula you have is frightening, just by looking at it!!

Guest Mar 22, 2017
#5
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Hello, Mr. (Blarney) Banker,

The reason your “650-page Math Dictionary” does not reference “Circular Standard Deviation” is because you have the large print volume. Unlike the standard print, the large print volume is abridged –they had to leave out some things.  You should get the unabridged versions that include all math concepts, including solutions for “P versus NP,” the “Hodge conjecture,” and the “Riemann hypothesis.” You can find a copy at your local bookstore.  Oh, you can purchase a magnifying glass there, too.

There were several references in Mathworld, you just didn’t know how to find them. To use a computer (and the internet) properly, you actually have to be smarter than the machine you use.  I understand it’s not easy, so I recommend the “Dummy” book series. “The Internet for Dummies” and “PCs for Dummies”   Sadly, these probably won’t help.

GingerAle  Mar 23, 2017
#6
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~The smartest cookie in the world

MaxWong  Mar 23, 2017
#3
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Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Rn), axes (lines through the origin in Rn) or rotations in Rn. More generally, directional statistics deals with observations on compact Riemannian manifolds. (https://en.wikipedia.org/wiki/Directional_statistics)

Here are the equations for Circular Standard Deviation (from the same source noted above):

$$S(z)={\sqrt {\ln(1/R^{2})}}={\sqrt {-2\ln(R)}}\small \hspace{25pt} \text {Equation for Sample Circular Standard Deviation } \\ {\overline {S}}(z)={\sqrt {\ln(1/{\overline {R}}^{2})}}={\sqrt {-2\ln({\overline {R}})}} \hspace{24pt} \small \text {Equation for Population Circular Standard Deviation } \\$$

GingerAle  Mar 23, 2017