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What do you mean?

When calculating the mean of a list of numbers, the obvious approach is to sum them and divide by how many there are.

Suppose I give you a list of two numbers:

• 0
• 359

What is their mean? The obvious answer is 179.5.

If I told you that the numbers were compass bearings in degrees, what would your answer be then? Does 179.5 seem correct?

In the case of compass bearings, 0 is the same direction as 360. When talking about angles in the DSP world, we often talk about angles between -π and +π (in radians).

This conundrum is related to Steve Smith's Nuisance 7.

Circular Reasoning

This problem is well-studied [1]  and there is a clear solution to the problem [2]: use of vectorial (or phasor) addition for finding the mean. Instead of writing:

where x₁ and x₂ are directions between 0 and 360 degrees, we write

μ = arg ( exp[jπ x₁ /180] + exp[jπ x₂ /180 ] ).

where exp( ) is the exponential function, j is the square root of -1 and arg( ) is the argument (complex angle or phase) of the result.

In effect, this is a phasor average rather than a linear average.

References

[1] Kanti V. Mardia and Peter E. Jupp, "Directional Statistics," Wiley, 1999, ISBN-10: 0471953334.

[2] Lovell, Brian C. and Kootsookos, Peter J. and Williamson, R. C. (1991) The Circular Nature Of Discrete-Time Frequency Estimates. In IEEE International Conference on ASSP, May, 1991, pages 3369-3372, Toronto.

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