I found myself reading Student's original paper on the t-test, and found on page 14 the following observations (quote transcription and emphasis both by me):
Another thing which interferes with the comparison is the comparatively large groups in which the observations occur. The heights are arranged in 1 inch groups, the standard deviation being only 2.54 inches: while the finger lengths were originally grouped in millimetres, but unfortunately I did not at the time see the importance of having a smaller unit, and condensed them into two millimetre groups, in terms of which the standard deviation is 2.74.
Several curious results follow from taking samples of 4 from material disposed in such wide groups. The following points may be noticed: (1) The means only occur as multiples of .25 (2) The standard deviations occur as the square roots of the following types of numbers n, n+.19, n+.25, n+.50, n+.69, 2n+.75. (3) A standard deviation belonging to one of these groups can only be associated with a mean of a particular kind; thus a standard deviation of √2 can only occur if the mean differs by a whole number from the group we take as origin, while √1.69 will only occur when the mean is at n±.25.
Are you aware of this? Do the observations in this quote generalize to useful additions to your method?