the mean of the data. If there are \(r_i\) observations of
the value \(n_i\) then the variance is computed by
\(\mathrm{E}[X^2]-\mathrm{E}[X]^2\), where
\(\mathrm{E}[X]\) is computed using $$\sum_i\frac{r_i\times
n_i}{\sum_i{r_i}}$$ , and
\(\mathrm{E}[X^2]\) is computed by $$\sum_i\frac{r_i\times
n_i^2}{\sum_i{r_i}}$$. We realise that the
computational formula,
\(\mathrm{E}[X^2]-\mathrm{E}[X]^2\), is usually not
regarded as computationally stable, but the magnitude of the numbers
involved is such that, that this is not likely to cause an issue.