While this loss of precision may be tolerable and viewed as a minor flaw of the naïve algorithm, further increasing the offset makes the error catastrophic. Again the estimated population variance of 30 is computed correctly by the two-pass algorithm, but the naïve algorithm now computes it as −170.66666666666666.
Next consider the sample (), which gives rise to the same estimated variance as the first sample.
The two-pass algorithm computes this variance estimate correctly, but the naïve algorithm returns 29.333333333333332 instead of 30.
can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
Thus this algorithm should not be used in practice.
If and again choosing a value inside the range of values will stabilize the formula against catastrophic cancellation as well as make it more robust against big sums.
computing the sample valiance son a sample of size mtn given the means and valiance son two subsamples of sizes m and n.
we present resume and rounding error analyze son several numerical schemes.
we would like to calculate simple statistics like the weighted mean or weighted variance of the sample without having to store all samples, and by processing them one-by-one.
The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments.
One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of, e.g., the data storage format or the original measurement hardware.
However, the results of both of these simple algorithms ("Naïve" and "Two-pass") can depend inordinately on the ordering of the data and can give poor results for very large data sets due to repeated roundoff error in the accumulation of the sums.