Certainly. And for small perturbations it would be linear; just in 
different proportions depending on the perturbation vector.

How this proportion varies by perturbation vector direction I have no idea.

Note that since chi-squared variance is 2 * DOF, perturbation squared 
magnitude (actually a re-scaled chi-squared with mean 1) variance is thus 
2/DOF giving a standard deviation of about 0.12.

Best wishes,

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