# Circular mean for directional data

Blog Post created by Dan_Patterson on Jan 10, 2016

Original:    Aug 2013

Modified:  Sept 2015  to include Numpy version

Reposted:  Jan 2016

General references

-   SciPy implementation  scipy/morestats.py at master · scipy/scipy ·GitHublines 2601 - 2642  using complex numbers

When working with angles or time (minutes, days, months or years) the mean (average) of a list of values can't be determined in the same fashion as we normally do.  This example show one how to calculate the values of angles (azimuths specifically).  The function can be used in any script once the circular_mean.py script is imported.

I will get around to other descriptive measures for circular data but they are covered in the references

```"""
CircularMean
Author: Dan.Patterson@carleton.ca
Date:     Aug 2013
Modified: Sept 2015
Purpose:
To calculate the mean (average) for points along a circle using the
angular measure to these points from the center of the circle
Notes:
If the angles are given in degrees then there is no need to convert
If the units are time, such as hours (24 hour clock) or months (12) then
these need to be converted by
angle in degrees = time quantity / number of equal intervals
ie 0.5 = (12 hours / 24 hours per day)
or 180 degrees
required modules
numpy, math
"""
import math
import numpy as np
def circ_mean(angles):
"""angles in degrees"
"""
cosList = []
cosSum = 0.0
sinSum = 0.0
for i in angles:
cosSum += theCos
sinSum += theSin
N = len(angles)
C = cosSum/N
S = sinSum/N
theMean = math.atan2(S,C)
if theMean < 0.0:
return theMean
def circ_mean_np(angles,azimuth=True):
""" numpy version of above"""
if azimuth:
ang_deg = np.mod(ang_deg,360.)
if __name__ == "__main__":

sampleData = [ [350,10],
[350, 5],
[355, 10],
[90,270],
[180,0],
[180,360],
[0,360],
[90,180,270]
]
print("\nCircular Mean Demo...python version\n")
frmt ="{:15s}{:15s}{:15s}"
for angles in sampleData:
theMean = circ_mean(angles)
args = (angles, theMean, math.degrees(theMean))
print("{!s:>15}{:>12.4}{:>15.4}".format(*args))
print("\nCircular Mean Demo...numpy version\n")
frmt ="{:15s}{:15s}{:15s}"
#
for angles in sampleData:
result = circ_mean_np(angles)
print("{!s:>15}{:>12.4f}{:>15.1f}".format(angles, result[0],result[1]))
```

Results from both

```angles         Mean radians   Mean degrees
[350, 10]       6.283          360.0
[350, 5]        6.24          357.5
[355, 10]     0.04363            2.5
[90, 270]       3.142          180.0
[180, 0]       1.571           90.0
[180, 360]       4.712          270.0
[0, 360]       6.283          360.0
[90, 180, 270]       3.142          180.0

```