Bruce Weaver wrote:
> In
> classes I took, a textbook author or lecturer would proceed
> roughly as follows:
>
> 1) Present the conceptual formula for some statistic (e.g.,
> the conceptual formula for the variance).
>
> 2) Point out that if you used the conceptual formula to
> calculate the statistic in question, you would often end up
> with an inaccurate answer due to rounding error.
>
> 3) Give you a computational formula for the same statistic
> that allowed you to avoid the rounding error, but provided
> little or no conceputal insight into the statistic.
The thing usually presented as a "computational formula"
for variance does NOT avoid rounding error, but on the contrary is
known to be a far worse offender in this regard than the conceptually
simple debiased-mean-square-residual formula. In particular, in the
nightmare scenario, it can yield a negative value for variance, causing
the program to crash when standard deviation is computed.
It is used (when it is) because it reduces (slightly) the number of
operations and (significantly) the space complexity and number of memory
calls.
I seem to recall from my long-ago exposure to numerical analysis that
there are formulae that work significantly better than the "standard"
ones in terms of rounding error, but I've never seen them in a stats
text. Anbody
know a good reference?
-Robert Dawson
.
.
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