Estimating Mean under Interval Uncertainty and Variance Constraint

In many practical situations, we have a sample of objects of a given type. When we measure the values of a certain quantity for these objects, we get a sequence of values x1, ..., xn. When the sample is large enough, then the arithmetic mean E of the values xi is a good approximation for the average value of this quantity for all the objects from this class.

The values xi come from measurements, and measurements are never absolutely accurate. Often, the only information that we have about the measurement error is the upper bound Di on this error. In this case, once we have the measurement result Xi, the condition that |Xi-xi| <= Di implies that the actual (unknown) value xi belongs to the interval [Xi - Di, Xi + Di].