Standard Deviation of Complex Numbers

[agmorgan]

Is it possible to calculate the Standard Deviation of Complex Numbers using Excel?


Are there built in functions to do this or would I have to write the whole thing using VBA?

[Foxer]

Have you looked up stdev?? Not sure if that suits your data.

[agmorgan]

STDEV is the standard deviation calculation although I haven't been able to get it to work with the COMPLEX worksheet function.

[RobDog888]

There was a link posted a few months ago on how inaccurate Excel is for complex mathematical calculations.

[dudzcom]

the standard deviation of a series of complex numbers is not mathematically defined. the usual equation for s.d. does not make any sense with complex numbers.

[aasdfasdfg]

I am reviving this thread since it is the first result in google search for the phrase "complex standard deviation".

Dudzcom (previous poster) is entirely incorrect and has probably misled many people unintentionally.

If you follow the link to http://en.wikipedia.org/wiki/Variance#Generalizations
you will find the definition of the variance of a distribution for complex numbers. It is a true generalization of variance since it always comes out to be a positive real number and reduces to the usual case when you are working with a real distribution.
Since the standard deviation is defined as the square root of the variance, this means that the standard deviation of a complex distribution does in fact make mathematical sense.
Just to be clear, we have the following method of calculating complex standard deviation:

s.d. = sqrt( E[ (X-u)*(X-u) ] )

in this equation, s.d. is standard deviation, sqrt is square root, E is the expectation value operator, u is the mean of the random variable X, and * denotes complex conjugate. Another way of writing this would be

s.d. = sqrt( E[ |X-u|^2 ] )

where the || denotes the norm of a complex number. When written in this form, it is apparent that we do in fact end up with a positive real number.


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