Traditionally, practitioners start a statistical analysis of a given sample x1, … , xn by computing the sample mean E and the sample variance V. The sample values xi usually come from measurements. Measurements are never absolutely accurate and often, the only information that we have about the corresponding measurement errors are the upper bounds Δi on these errors. In such situations, after obtaining the measurement result , the only information that we have about the actual (unknown) value xi of the ith quantity is that xi belongs to the interval . Different values xi from the corresponding intervals lead, in general, to different values of the sample mean and sample variance. It is therefore desirable to find the range of possible values of these characteristics when xi ∈ xi.
Often, we know that the values xi cannot differ too much from each other, i.e., we know the upper bound V0 on the sample variance V : V ⩽ V0. It is therefore desirable to find the range of E under this constraint. This is the main problem that we solve in this paper.