# Talk:Directional statistics

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## Circular standard deviation | Factor of 2

I noticed an oddity in the definition of the circular variance, which probably has historical reasons: While it is not generally true that the estimator of the circular standard deviation squared yields the circular standard deviation, I would have expected this equality to hold for very small values of the circular standard deviation, which recovers the linear case. However, there is a factor of 2 missing! You can test this numerically yourself with the following line of code

sigma=0.1

std_circ=sqrt(-2*log(abs(mean(exp(1i*sigma*randn(1e5,1))))))

var_circ=1-abs(mean(exp(1i*sigma*randn(1e5,1))))

I therefore added the following sentence to the article:

Note that for small $S(z)$, we have $S(z)^2=2\mathrm{Var}(z)$.

Does anybody know a reference for this?

Ben Benjamin.friedrich (talk) 20:39, 25 August 2014 (UTC)

## ?

The final step of the formula given for the example is unclear; apparently it is written for result in radians. Wouldn't it be better written this way:

Mean angle = arctangent (mean sine / mean cosine). If mean cosine < 0, add 180 degrees to the result. If mean sine <0 and mean cosine >= 0, add 360 degrees to the result.

Jim

## Modulus method

The modulus method only works in few specific cases (such as the example given).

Consider the same example but rotated further to the left so that the three angles are 330, 340 and 350 degrees. Taking the modulus 360 of the sum (1020) results in 2 remainder 280, which divided by 3 is clearly not 340.

As the remainder (in a modulus 360 operation) can only range from 0 to 360, the 'average' can only range from 0 to 360/n.

Unless I'm misunderstanding the procedure, I would suggest taking it out.

Fink3412 09:28, 20 August 2007 (UTC)