Abstract
In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance \(v=\frac{1}{2}\cdot\sum^{n}_{i=1}(x_{i}-E)^{2}\,(where E=\frac{1}{n}\sum^{n}_{i=1}x_{i})\) when we only know the intervals \([{\tilde x}_{i}-\Delta_{i},{\tilde x}_{i}+\Delta_{i}]\) of possible values of the xi. In general, this problem is NP-hard; a polynomialtime algorithm is known for the case when the measurements are sufficiently accurate, i.e., when \(|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{\Delta_{i}}{n}+\frac{\Delta_{j}}{n}\) for all \(i\neq j.\) In this paper, we show that we can efficiently compute V under a weaker (and more general) condition \(|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{|\Delta_{i}-\Delta_{j}|}{n}\).
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Dantsin, E., Kreinovich, V., Wolpert, A. et al. Population Variance under Interval Uncertainty: A New Algorithm. Reliable Comput 12, 273–280 (2006). https://doi.org/10.1007/s11155-006-9001-x
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DOI: https://doi.org/10.1007/s11155-006-9001-x