The distributions of the run‐lengths are readily available for independent variables but those of the run‐sums are not. It is therefore desirable to obtain their moments. In this paper the first four moments of the run‐sums are shown to be expressed in terms of the moments of the truncated variable and the probabilities corresponding to the truncation level for independent sequences of random variables. The explicit expressions for the moments of the truncated variable are given for the case of normal, lognormal and gamma distributions. It is shown that the expected values of the positive and negative run‐sums have the same magnitude when the sequence under consideration is truncated at the mean.