While reading David Chandler's The Campaigns of Napoleon, Chandler reports some findings determined by the Prussian military near the end of the 18th century. Apparently the Prussian military determined the accuracy of muskets at various distances using sheets (and careful counting). As I understand it, the target was a sheet of cloth roughly a man's height, though a battalion's width. The numbers reported:
- At 75 yards, 60% of the shots fired hit the sheet
- At 150 yards, 40% hit
- At 225 yards, 25% hit
We can actually come up with an equation describing the proportion of shots hit as a function of distance. The trick is not to consider the proportion p = hit/(hit + miss), but instead the logarithm of the proportion p/(1 − p) = hit/miss. Why? Well, this is what we do with the logistic regression. Taking the linear regression ln(p/(1 − p)) = a + b x (where x is distance in yards) gives us coefficients:
- a = 1.1378733 with a standard error of 0.0734506
- b = −0.0100272 with a standard error of 0.0004533
This regression has an adjusted R-squared value of 0.9959, an F-statistic: 489.2 on 1 and 1 DF, p-value: 0.02876...so it is statistically significant at the conventional alpha=0.05 levels.
Reworking this model, we can explicitly write the proportion of shots hitting their target as:
- [Full model] p(x) = 1/(1 + exp(−1.1378733 + 0.0100272x))
This is approximately linear until 200 yards away:
- [Linear approximation] plin(x) = 0.7954567260747398 − 0.0029452336313330117x
For example, the linear approximation would give plin(250) = 0.0708341 (i.e., 7% of shots hit) but the full model gives a more optimistic p(250) = 0.202793 (i.e., 20% of shots hit). For "small x", the two models agree to within a few percent. (The linear approximation will be more optimistic by at most 3.8168%.)
I'm not sure about the accuracy of the "large distance" predictions of the full model; for example, 10% of shots fired 332 yards away will hit, while 5% of shots fired 407 yards away will hit, and 1% of shots fired 571.75 yards away will hit. Perhaps this is true, and it's just the degree of informality when people say, "It is completely inaccurate after 250 yards."
One concluding remark: if you were to use this in a model, I would also add a term for morale. Presumably a soldier on the target range shoots more accurately than after receiving a devestating volley on the battle field; consequently, one would also have to add a term to reflect this.