[Numpy-tickets] [NumPy] #559: var() on a size=1 array should raise exception.

NumPy numpy-tickets@scipy....
Mon Jul 30 09:25:01 CDT 2007

#559: var() on a size=1 array should raise exception.
 Reporter:  gpk     |       Owner:  somebody
     Type:  defect  |      Status:  new     
 Priority:  normal  |   Milestone:          
Component:  Other   |     Version:  none    
 Severity:  normal  |    Keywords:          
 Properly, a single value does not have a variance.
 Whoa!, you might say:  "I can compute the
 sum of the squares of the distance from the mean,
 even when there is only one value.  It's zero."
 That's true, but misleading.

 Since you call it "var", you are implying that
 it is a variance, and variance is undefined unless
 you have two or more values.

 More importantly, even the biassed estimator of
 variance ( sum((x-xbar)^2)/N ) does not have a
 well-defined value when N==1.   All estimates of
 the variance are equally good or equally bad when
 you have only one value.   Follow the logic to
 compute the minimum variance estimator of variance
 for a single sample, and you'll see all kinds of
 absurdities, including characteristic functions
 for the probability distribution that do not have a limiting value.

 From a practical point of view, anyone who takes the variance
 of a single value is probably in deep trouble.    Variances
 tend to get used for F-tests, which are undefined for zero
 degrees of freedom.   The get used to build confidence intervals
 for t-tests, but the t-distribution is undefined for zero degrees
 of freedom.    Some people will construct confidence intervals
 via   [mean-3*sqrt(var), mean+3*sqrt(var)] or some similar
 approximation:  this will fail badly and lead to tears.

 And, the same logic applies to the std(), of course.

 >>> import numpy
 >>> x = numpy.array([1.0])
 >>> x
 array([ 1.])
 >>> x.var()

Ticket URL: <http://projects.scipy.org/scipy/numpy/ticket/559>
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