Rather than considering \(x_k\) to be the kth data value, take advantage of the grouping to perhaps save a bit on arithmetic. Indeed, let’s assume that data is grouped into m categories \(x_1, x_2, ..., x_m\) with corresponding frequencies \(f_1, f_2, ..., f_m\text{.}\) Then, for example, when computing the mean rather than adding \(x_1\) with itself \(f_1\) times just compute \(x_1 \times f_1\) for the first category and continuing through the remaining categories. This gives the following grouped data formula for the mean
\begin{equation*}
\mu = \frac{x_1 f_1 + ... + x_m f_m}{f_1 + ... + f_m} = \frac{\sum_{k=1}^m x_k f_k}{\sum_{k=1}^m f_k}.
\end{equation*}
and the following grouped data formula for the variance (along with one equivalent form)
\begin{equation*}
\sigma^2 = \frac{\sum_{k=1}^m ( x_k-\mu )^2 f_k}{\sum_{k=1}^m f_k} = \frac{\sum_{k=1}^m x_k^2 f_k}{\sum_{k=1}^m f_k} - \mu^2
\end{equation*}