# Circular Data Analysis: Calculating Sample Circular Dispersion

Using Fisher (1993) Statistical analysis of circular data (p30-35), I calculated the sample circular dispersion of some timestamp data. However, I got circular dispersion greater than $1,000$. Is this possible? Does a circular dispersion of, for example, $361$ equivalent to a circular dispersion of $1$? Or did I do some massive calculation error?

Just as a reference, I also followed this post. Thus, quoting Kees Mulder:

1. Denote the data by $\boldsymbol\theta = \{\theta_1, \dots, \theta_n\}.$ An estimate of the mean $\bar\theta$ can be calculated with

2. Calculate $\bar{R} = \frac{\sqrt{S^2 + C^2}}{n}$

3. Calculate the dispersion as suggested by whuber. I'm not sure why, but Wikipedia's definition seems to differ slightly from Fisher. I will use Fisher's:

• $\hat\delta = \frac{1 - \left[ (1/n) \sum_{i=1}^{n} \cos 2 (\theta_i - \hat\mu) \right]}{2\bar{R}^2}$
4. Then, choose some constant $c$ (1 is probably fine, but you may fine-tune). Then, the interval is given by

• $\left[\hat\mu - c \hat\delta, \hat\mu + c \hat\delta \right]$

Also, unfortunately, I cannot post my data.

Given the questions and comments below, I just want to clarify what I am looking for. I am primarily interested in whether the circular dispersion has an upper bound and how to interpret large circular dispersions. I am actually fairly confident that my calculations are correct, as long as the notes that I posted are correct.

• What is your mean resultant length and sample size, in this case? – Kees Mulder Nov 21 '16 at 19:15
• Could you add a fuller reference/citation than "Fisher (1995)", as you would in a paper or dissertation? – Silverfish Nov 21 '16 at 19:52
• To make this self-contained, please define circular dispersion and give a small dataset with puzzling result sufficient to replicate (or correct) what you find. – Nick Cox Nov 21 '16 at 19:54
• If you want to know for sure whether you made a calculation error, I can't see an alternative to you posting your data! Or are you just interested in whether the circular dispersion has an upper bound? – Silverfish Nov 21 '16 at 19:58
• You are making a calculation error. You don't have to reproduce your actual data: make a very small dataset with similar characteristics and see whether the problem recurs with it. Then you can post that dataset. – whuber Nov 22 '16 at 16:25