If a discrete probability distribution in a model being tested for
goodness-of-fit is not close to uniform, then forming the Pearson chi-square
statistic can involve division by nearly zero. This often leads to serious
trouble in practice -- even in the absence of round-off errors -- as the
present article illustrates via numerous examples.

The classic chi-squared statistic for testing goodness-of-fit has long been a
cornerstone of modern statistical practice. The statistic consists of a sum in
which each summand involves multiplying by the inverse of (i.e., dividing by)
the probability associated with the corresponding bin in the distribution being
tested for goodness-of-fit. This inversion typically precipitates rebinning to
uniformize the probabilities associated with the bins, in order to make the
test reasonably powerful. With the now widespread availability of computers,
there is no longer any need for this.

We describe an algorithm that, given any full-rank matrix A having fewer rows
than columns, can rapidly compute the orthogonal projection of any vector onto
the null space of A, as well as the orthogonal projection onto the row space of
A, provided that both A and its adjoint can be applied rapidly to arbitrary
vectors. As an intermediate step, the algorithm solves the overdetermined
linear least-squares regression involving the adjoint of A (and so can be used
for this, too).