# Dispersion

The model used in the US manual was that shells from a single gun will tend to deviate over a rectangular or elliptical area whose longer axis is along the line between the gun and target. Imagine a rectangle covering the area the shots are landing (after throwing away any obvious outlyers). Now divide that rectangle into eight sections along the axis of fire. You'll find that roughly half the shells will land in the central two eights (25% long, 25% short). If you take the remaining six sections, roughly half the remainding shells will fall in the next sections out from the center. This results in this rectangle:

 2% 7% 16% 25% 25% 16% 7% 2%

The same effect happens in the other dimension, though the divisions are smaller since there is less dispersion in that direction. Putting it all together in both dimensions, you get (in percentages that a shell will hit each sector):

 0.04 0.14 0.32 0.5 0.5 0.32 0.14 0.04 0.14 0.49 1.12 1.75 1.75 1.12 0.49 0.14 0.32 1.12 2.56 4 4 2.56 1.12 0.32 0.5 1.75 4 6.25 6.25 4 1.75 0.5 0.5 1.75 4 6.25 6.25 4 1.75 0.5 0.32 1.12 2.56 4 4 2.56 1.12 0.32 0.14 0.49 1.12 1.75 1.75 1.12 0.49 0.14 0.04 0.14 0.32 0.5 0.5 0.32 0.14 0.04

Another way to look at this is as an expanding series of rectangular zones. 25% of the shells will land in the central four rectangles (the 2×2 area of "6.25's"); 68% in the first plus the next ring out (the 4×4 area in the center); 91% in the next ring out (6×6); and of course 100% in the whole 8×8 area:

The dimension of this rectangle is much longer in the direction of fire than in the lateral direction -- typically by a factor of four to five. Each 1/8 division along the long axis (the line of fire) is known as a range probable error; along the short axis is a deflection probable error. For the 155mm gun M1 at 18000 yards, the range probable error is 43 yards and the deflection probable error is 9 yards, so the whole rectangle is 344 × 72:

By using the range probable errors, you can compute the chance that a shell will directly hit a target of a particular size if the guns are aimed perfectly, for instance, in the above example a 86 yard × 18 yard target (to make the math easy: two range probable errors by two deflection probable errors) with its long axis aligned with the line of fire would be hit by 25% of the shells in the above example, since it's the same size as the black central 25% rectangle.

An anti-tank gun of size 4.3×0.9 yards, being 1/400th of that size, would be hit by about .0625% of the shells, so you'd have to fire about 1600 shells at it for a direct hit (though, since the burst radius is large, you'd have scared off the crew with much less ammunition).

Unfortunately, I only have a couple of range probable error figures that were used as examples in the FM6-40 manual. (If anyone out there has the WWII firing tables, I'd love to hear from you.)

 Ballistics Fuzes