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In the geosciences, a recurring problem is one of estimating spatial means ofa physical field using weighted averages of point observations. An importantvariant is when individual observations are counted with some probability lessthan one. This can occur in different contexts: from missing data to estimatingthe statistics across subsamples. In such situations, the spatial mean is aratio of random variables, whose statistics involve approximate estimatorsderived through series expansion. The present paper considers truncatedestimators of variance of the spatial mean and their general structure in thepresence of missing data. To all orders, the variance estimator depends only onthe first and second moments of the underlying field, and convergence requiresthese moments to be finite. Furthermore, convergence occurs if either theprobability of counting individual observations is larger than 1/2 or thenumber of point observations is large. In case the point observations areweighted uniformly, the estimators are easily found using combinatorics andinvolve Stirling numbers of the second kind.

Convergent estimators of variance of a spatial mean in the presence of missing observations