The \(r^{th}\) population moment is defined as \(m_r = (1/N) \sum_{k \in U} (y_k - \bar{y}_U)^r\) where U is the set of population units, N is the population size, and \(\bar{y}_U\) is the population mean. When the input is for the whole population, wtd.moments evaluates this directly for \(r=2, 3, 4\). When the input is for a sample, the \(r^{th}\) moment is estimated as \(\hat{m}_r = (K/\hat{N}) \sum_{k \in s} ( w_k (y_k - \hat{\bar{y}}_U)^r ), r=2, 3, 4\) where \(s\) is the set of sample units, \(w_k\) is the weight for sample unit \(k\), \(\hat{N} = \sum_s w_k\), and \(\hat{\bar{y}}_U = \sum_{k \in s} w_k y_k / \hat{N}\). When \(r=2\), \(K=n/(n-1)\) so that the estimator equals the unbiased variance estimator if the sample is a simple random sample; if \(r=3,4\), then \(K=1\). The function also computes or estimates the population skewness, defined as \(m_3/m_2^{3/2}\) and the population kurtosis, \(m_4/m_2^2\).
The weights should be scaled for estimating population totals. The sample can be obtained from any complex design.