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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 ${\displaystyle S(z)}$, we have ${\displaystyle S(z)^{2}=2\mathrm {Var} (z)}$.

Does anybody know a reference for this?

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

Thank for adding this. It also got me confused.

The Mardia & Jupp book says:

"The motivation for this definition is that wrapping the normal distribution

N ( \mu ,\sigma^2) round the circle gives the wrapped normal distribution W N ( \mu , rho )

with \rho = exp{ -\sigma^2/2) as in (3.5.63), so that 1 - \nu =\exp(- \sigma^2 / 2 ).

For small \nu, (3.4.14) reduces to \sigma = \sqrt(2\nu).

" Cosine12 (talk) 18:54, 6 May 2024 (UTC)[reply]

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

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)[reply]

Agreed - I've removed it. Also, this article is about Directional statistics itself, not about various methods to calculate a circular average. Tomixdf 09:42, 20 August 2007 (UTC)[reply]

I've added the wrapped normal distribution as an example of the way a wrapped distribution can be made from the pdf of another distribution. I'm also redirecting the nonexistent wrapped normal distribution to this page. I hope this is considered useful! digfarenough (talk) 22:18, 11 November 2008 (UTC)[reply]

Good work, thanks. Tomixdf (talk) 07:07, 12 November 2008 (UTC)[reply]

I started a new Goodness of fit section - in case it should be put elsewhere. - Rod57 (talk) 14:31, 6 April 2015 (UTC)[reply]

The current title of this article is directional statistics, which is a synonym of circular statistics. A quick search on Google Scholar and Google Books will show that the latter term, "circular statistics," is more widely used than "directional statistics". Here are the results.

• "Circular statistics": 9,670 results on Google Scholar and 811 results on Google Books
• "Directional statistics": 4,060 on Google Scholar and 766 results on Google Books

Thus, I would like to propose renaming this article to Circular statistics. Any thoughts? danielkueh (talk) 02:48, 28 August 2015 (UTC)[reply]

• Oppose: circular is 2D, a special case of spherical statistics or 3D directions. fgnievinski (talk) 03:10, 28 August 2015 (UTC)[reply]

Point taken. danielkueh (talk) 03:15, 28 August 2015 (UTC)[reply]