Strohm, Luke S. "An Introduction to the Sources of Delivery Error for Direct-Fire Ballistic Projectiles". Army Research Laboratory. July 2013. (Link.)
2.2 Normal (Gaussian) Distributions
For direct-fire ballistic projectiles, it is common to assume that error sources and the shot distributions they produce can be characterized by normal (Gaussian) distributions. Normal distributions are defined by a mean (μ) and standard deviation (σ, SD), which produce a bell curve that is unique to the distribution. The mean is the average of the distribution, while the SD quantifies the spread or precision of the distribution. For a one-dimensional normal distribution, approximately 68% of the distribution is within one SD of the mean (+/–) and 95% within two SDs (figure 2).
... In two dimensions, target impact distributions follow a bivariate normal distribution, meaning that the impact locations vary normally in two directions—in this case the horizontal and vertical directions of the target plane.
(Note that the bivariate normal model is the same as I've used in various archery simulations on this site.) Consider also the empirical test of shotgun shell spread presented here: Lowry, Ed. "Properties of Shotshell Patterns". American Rifleman. 1990. (Link.)
Now let's reflect on the rule as written in Chainmail for "Fire Optional"scatter:
Fire Optional: Roll two different colored dice. One color is for an over-shoot and the other is for an under-shoot. To decide which number of use you take the higher of the two. Miss is in inches, shown by dice spots. If they tie then the rock lands at the specified range. This method is simple but effective.Taking the higher of the two dice biases the scatter towards the high end of the range. This is shown as "Chainmail Fire Option A" below. Note that the resulting probability distribution is distinctly anti-Guassian; it is impossible for a shot to land exactly 1" away from the target; and generally speaking, it's simply total lunacy, some kind of Lovecraftian non-Euclidean physics:
But if we change one critical word to make the rule instead "take the lower of the two" dice, then this mechanic, while still very simple, does in fact generate a quasi-bell-shaped distribution as suggested by the American Rifleman and Army Research Laboratory publications above (shown as "Chainmail Fire Option B" below). It seems patently obvious that this is the better rule: