In descriptive statistics, continuous variables are often presented using class intervals of uniform width, for instance:
Annual salary (€) Frequency [20000, 40000[ 10 [40000, 60000[ 25 [60000, 80000[ 5
and one is told to compute the mean or variance by assuming that all data in a class are at the center of the class. For instance, one would compute the mean annual salary from the table above as
$ \frac{10 \times 30000 + 25 \times 50000 + 5 \times 70000}{40} = 47500. $
It is easy to show that the uncertainty (maximal error) when computing the mean this way is half the width of a class interval (in this case, $10000$). In other words, the true mean is in the interval $[37500, 57500[$.
My question is: are there simple bounds for the uncertainty of the variance?