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I have a huge list of times and would like to calculate an average or typical time. If I would just use the median (or other "normal" types of calculating an average), for example 23:59 and 00:01 would yield 12:00 when it should 00:00. Is there a better method?

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    $\begingroup$ What would you like the average of the sample 00:00, 06:00, 12:00, 18:00 to be? $\endgroup$
    – Did
    Jan 3, 2012 at 12:08
  • $\begingroup$ @DidierPiau: It would be ok if it would yield anything or undefined. However my data samples are rather huge so that would be rather unlikely. $\endgroup$
    – Nihu
    Jan 3, 2012 at 12:20
  • $\begingroup$ My point is that you will stumble on the same problem with huge data samples, namely, these samples will be roughly evenly distributed on the unit circle. Then it is difficult to summarize the sample by anything else than a point near the center of the circle, which is meaningless because this will not be ON the circle... (Re-reading your post, I realize that even a median, whose existence you take for granted, seems difficult to define.) $\endgroup$
    – Did
    Jan 3, 2012 at 12:26

2 Answers 2

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I see two approaches for this. If there's some time of day when nothing happens, e.g. 4 a.m., you can let the wraparound occur at that time; for instance times from 1 a.m. to 4 a.m. would become times from 25:00 to 28:00.

If there's no such natural cutoff, you could use directional statistics; from the Wikipedia article:

Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week, month, year, etc.), [...]

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  • $\begingroup$ Unfortunately there is no such natural cutoff. However I'm interested in directional statistics. Do you know any starting ressources? $\endgroup$
    – Nihu
    Jan 3, 2012 at 12:27
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    $\begingroup$ Maybe the WP page linked to... $\endgroup$
    – Did
    Jan 3, 2012 at 12:28
  • $\begingroup$ Take a look at the answer I posted. I hope www.youtube.com/watch?v=cYVmcaRAbJg will be suitable as a starting resource. $\endgroup$
    – Druid
    Aug 19, 2021 at 9:43
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The variables you mention are points belonging to the manifold, which is the circle. Therefore, they cannot be treated as if they belonged to Euclidean space.

I recommend the material that I have prepared on this subject and today I am sharing it on YouTube: Circular means - Introduction to directional statistics.

There are two main types of circular mean: extrinsic and intrinsic.

Extrinsic mean is simply the mean calculated as the centroid of the points in the plane projected onto the circle. $$ \bar{\vec{x}}=\frac{1}{N}\sum_{j=1}^N \vec{x}_j=\frac{1}{N}\sum_{j=1}^N [x_j,y_j]=\frac{1}{N}\sum_{j=1}^N [\cos{\phi_j},\sin{\phi_j}] $$ $$ \hat{\bar{x}}=\frac{\bar{\vec{x}}}{|\bar{x}|} $$ $$ \DeclareMathOperator{\atantwo}{atan2} \bar{\phi}_{ex}=\atantwo(\hat{\bar{x}}) $$ It is NOT a mean calculated using the natural metric along the circle itself.

Intrinsic mean, on the other hand, does have this property. This mean can be obtained by minimizing the Fréchet function. $$ \DeclareMathOperator*{\argmin}{argmin} \bar{\phi}_{in}=\argmin_{\phi_0\in C} \sum_{j=1}^N (\phi_j-\phi_0)^2 $$

For discrete data, you can also analytically determine the $N$ points suspected of being the mean and then compare them using the Fréchet function.

$$ \bar{\phi}_k=\arg \sqrt[N]{\prod_{j=1}^N e^{i\phi_j} }=\bar{\phi}_0+k\frac{2\pi}{N} $$ Where the N-th root is a N-valued function with outputs indexed with $ k\in\{1,\dots,N\} $. They are distributed evenly on the circle. And $ \bar{\phi}_0 $ is a usual mean calculated in an arbitrary range of angle values of length of $2\pi$. If somebody dislikes the complex numbers $$ \bar{\phi}_k=\frac{1}{N} \left(\sum_{j=1}^N \phi_j+k2\pi\right) $$ The result is, of course, the same.

Then you have to compare the points suspected of being the mean using the Fréchet function. $$ \DeclareMathOperator*{\argmin}{argmin} \bar{\phi}_{in}=\argmin_{k\in\{1,\dots,N\}} \sum_{j=1}^N (\phi_j-\bar{\phi}_k)^2 $$ Where the search for minimum runs over $N$ discreet indices.

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