* delta_n2 * (n**2 - 3*n + 3) + 6 * delta_n2 * M2 - 4 * delta_n * M3 M3 = M3 + term1 * delta_n * (n - 2) - 3 * delta_n * M2 M2 = M2 + term1 # Note, you may also calculate variance using M2, and skewness using M3 # Caution: If all the inputs are the same, M2 will be 0, resulting in a division by 0. kurtosis = (n * M4) / (M2**2) - 3 return kurtosis Pébaÿ further extends these results to arbitrary-order central moments, for the incremental and the pairwise cases, and subsequently Pébaÿ et al. for weighted and compound moments. One can also find there similar formulas for covariance. Choi and Sweetman offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. 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. A relative histogram of a random variable can be constructed in the conventional way: the range of potential values is divided into bins and the number of occurrences within each bin are counted and plotted such that the area of each rectangle equals the portion of the sample values within that bin: where and represent the frequency and the relative frequency at bin and is the total area of the histogram. After this normalization, the raw moments and central moments of can be calculated from the relative histogram: where the superscript indicates the moments are calculated from the histogram. For constant bin width these two expressions can be simplified using : The second approach from Choi and Sweetman is an analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times. If sets of statistical moments are known: for ,