I am teaching some very basic statistics for grouped data.
In a different class a few years ago I made an 'assumption of uniformity' about the data:
within bins, data is uniformly distributed
E.g. if I had three data points in the $[0,1)$ bin, under this assumption, the data points would be $0.25$, $0.5$ and $0.75$.
I understand that this isn't a very good assumption but is certainly a working one. Previously, I used this assumption to give the mode as the midpoint of the modal class and I could also find the median in the following way:
Suppose we have 0-1 with two entries, 1-2 with three and 2-3 with one. There are six data points so the median will be at position 3.5 at the midpoint of the two middle data, points, so at 1.375 according to the assumption of uniformity.
Since that time I have learnt that we can use the ogive to give a more reasonable approximation for the median (without this assumption) and I also seen that there is a formula that gives an approximation to the mode without using this assumption.
Just recently I have seen a formula for the approximate location of the median that doesn't use this assumption either.
This is all great and I am quite happy to show my students all of this. However, it is under this assumption of uniformity that the formula for the mean of a grouped data set is correct! I find this quite an inconsistency and I am not comfortable using the assumption for the mean and not for the median and mode.
I normally like deriving all the formulae I use in an appendix: with time pressure this is not possible...
My questions therefore are:
- Is there a well-known formula for the mean which doesn't use this assumption of uniformity?
- If not, can anyone account for this apparent anomaly? Or is the usual formula a better estimator than I give it credit for and can be derived without using an assumption of uniformity?
I don't see how overestimations can necessarily be compensated with underestimations. Surely assuming that all the data is centred at the midpoint is an even stronger assumption.