# Directional Statistics

Interesting mathematical problems arise when trying
to perform statistical analyses on data that are
essentially angular in nature.
For example, there is the (perhaps apocryphal)
story of sleep researchers who surveyed a
representative sample of people to determine
what time they usually go to bed, recording
the answers received using a 24 hour clock
convention. After performing a simple statistical
analysis, the researchers were shocked to find that
the average bedtime was 14.5 (2:30 in the afternoon)
with a standard deviation of 11 hours! What went wrong?
Obviously they forgot to (or didn't know how to)
take into account the fact that bedtimes just
before midnight and those just after are
in fact close to each other. That is, they were
treating the data as if they were real numbers
(using regular arithmetic) rather than hours
(using mod 24 arithmetic).
Similar problems arise in computer vision and
robotics applications when there is a need to
average multiple estimates of an angle.
Clearly, it is incorrect to just average the
angles together, i.e.

because of the problems with wraparound at 2 Pi.
However, you could get a reasonable answer by
treating each angle as a unit vector, adding the
vectors together, then taking the angle associated
with the vector resultant using arctangent, i.e.

Trying to mathematically formalize the above "trick",
and to extend it to multiple dimensions leads one
directly into the study of directional statistics,
which concerns questions about how to generalize
ordinary statistical analyses to handle point
observations on circles, spheres and hyperspheres.

## Some good references

K.V.Mardia,* Statistics of Directional Data,*
Academic Press, New York, 1972.

G.S.Watson,* Statistics on Spheres,*
John Wiley and Sons, New York, 1983.

P.E.Jupp and K.V.Mardia, "A Unified View of the Theory
of Directional Statistics : 1975--1988,"
* International Statistical Review,* Vol.57, pp.261-294.

R.V.Lenth, "Robust Measures of Location for Directional Data,"
* Technometrics,* Vol.23, 1981, pp.77-81.