Most gunnery textbooks lead one to believe that the range distribution in a group of shots fired under conditions as nearly alike as possible, is always a symmetrical distribution typified by a pure probability curve like Figure 1. With this the MPI is obviously the best part of the group to put on the target.1
But in modern long-range gunnery it is becoming increasingly evident that this classic theory is not entirely applicable because in general the actual range distribution of such shots is frequently an unsymmetrical distribution typified by a curve like Figure 2.2 With distributions of this sort, the MPI is obviously not the best part of the group to put on the target.
Shots tumble “short” rather than “over,” and it is easier to imagine something going wrong with the flight of a projectile and causing a certain decrease in range, than it is to imagine an analogous effect causing an equal increase in range. The part of the dispersion caused by flight characteristics is therefore not safely assumed to be of symmetrical distribution. It may be expected generally to show a preponderance of “shorts.”
In firings at very small ranges where flight characteristics contribute only a relatively minor part of the dispersion (which total dispersion is then caused principally by the symmetrical velocity variations) it is not surprising to find that dispersions are very nearly symmetrically distributed. But to get any general rule, particularly any rule applicable to present-day conditions, results of firings at extreme ranges must also be taken into account.
For example, par. 1615 of Exterior Ballistics, 1926, gives in support of Figure 1, data now nearly fifty years old, showing the distribution in deflection of shots fired from a 4.7-inch gun at a range of 3,000 yards. As one might expect, these data from firings at such a small range show practically a symmetrical distribution of shots. Furthermore, the distribution of such shots could be expected to be more nearly symmetrical in deflection than in range.
Possibly this last throws some light on why those who continue to use these particular firing data in support of Figure I prefer to deal with deflection results rather than with range results. For they must have had range results also, and it would seem more natural to have used them had they supported Figure 1 as well as deflection results did. Furthermore, why should any of these data, which are probably of greater age than the reader, be now used if examples from modern firings will serve the same purpose?
So in this, and in other stock examples in the textbooks, there is little evidence to impress one that Figure 1 holds very closely for modern long-range firing, but much to make one suspect that it does not. Thus in modern long-range gunnery, where flight characteristics contribute a fairly large and an unsymmetrical part of the total dispersions, it is not surprising frequently to find range dispersions fairly unsymmetrical and represented by the family of curves which Figure 2 typifies. Under modern conditions Figure 1 seems, therefore, only the limiting case to which the more general family of curves typified by Figure 2 finally reduces as the range shortens so as to make range distribution more and more symmetrical.
The present-day statistician, when studying variations, approaches his subject open- mindedly with the idea of finding what sort of distribution exists. He accepts a symmetrical distribution only after proving he actually has one.3 So in this idea of unsymmetrical range distribution there is no novel thought which need surprise one. In fact it is the other way round—the surprising thing being rather that writers have clung so long to the assumption of necessarily symmetrical range distributions while apparently disregarding much of the ballistic data obtained under the very changed conditions of present-day gunnery.
Unsymmetrical variations are thoroughly treated in standard textbooks on statistics.4 There are three sorts of “centers” which a group of this kind may be considered as having.5 One is the simple “mean,” or the MPI commonly used in dealing with positions of the shots of a salvo. Another is the "median,” which has equal numbers of points each side of it, e.g., in a twelve-gun salvo the median is halfway between the shots which are sixth and seventh in order of range. In a nine-gun salvo the median is at the impact of the shot which is fifth in range. The other sort of “center” is the “mode,” or the point on each side of which the density of points systematically decreases.
In illustration of these terms we could, in speaking of ages in a large community, refer to the “mean” age of individuals, or refer to the “median” age (which exactly half the time is exceeded) or refer to the “modal” age (which is the age of the most numerous age group). And in general these three figures would not be identical, for they correspond exactly with one another only in the case of symmetrical distributions which do not in general exist. In Figure 2 the ordinate through the MPI passes through the center of gravity of the curve’s area, the ordinate through the median bisects the curve’s area, while the ordinate through the mode is the curve’s maximum ordinate.
In general, the mode, and not the MPI, is obviously the best part of the group to put on the target. But with the small number of shots in a salvo, one has not the same escape from the capricious workings of chance that he would have in dealing with very extensive data (e.g., age groups in a large community). A mode from two shots falling together at one end of a salvo would hardly indicate the ideal part of the group to spot on the target, because that particular agreement of ranges came probably from workings of chance not very likely to be approximated in the next salvo, nor even in the average salvo. The real average mode or the true mode which we wish to spot on the target is, therefore, not at all well directly indicated by the intervals between impacts in any one salvo. But we can know the true mode’s position indirectly from a steady approximate relation which holds between its average position and the not very variable relative positions of the MPI and of the median. At any rate, we can locate it sufficiently accurately to serve our present purpose.
The average mode lies on the same side of the MPI that the average median does, but is about three times as far away from the MPI.6 Thus, although the median is not the ideal part of the salvo to put on the target, it is at any rate better for this purpose than the MPI. Since no single salvo is likely to directly indicate the true mode by the intervals in the small number of impacts in that one salvo, about the best one can do in spotting is to use the median.
Can our spotters, in the few seconds they take to announce spots, be really judging the position of the MPI by some instinctive sort of true averaging process? Probably not. But since they are called upon to spot the MPI, they cannot be expected to go very far in voluntarily admitting such inability. If there are “doubtful” shots at the ends of the pattern there is insufficient time to apply any good criterion for retaining or for rejecting such shots. It is very hard to say what goes on in one’s mind in a process which has to be so nearly instantaneous.
But it seems no great reflection on spotters to suspect that what they really do is about as accurately described by saying that they spot the “50-50” point in a salvo as it would be by saying that they spot its MPI— or in other words, say that something which is practically median spotting is of necessity probably already with us, although not officially admitted.
Enough has been said to show, however, that far from having to feel apologetic about the use of the median, even in theory also this practice is to be preferred.7 The median coincides with the MPI when, with symmetrical distributions, the MPI is the proper thing to use. And at all other times the median lies even closer to the true mode than the MPI does.8 One practical advantage of using the median is that it saves the spotter worry about rejecting or retaining doubtful shots at the ends of pattern.
We could therefore, rewrite in more honest and in simpler terms much that existing textbooks say about spotting salvos by their MPI’s, and so legitimatize a kind of spotting which should not only be better, but which probably anyway is in actual use.
An extension of these ideas to one other matter appears logical. Since the median seems the better part of the salvo to put on the target, the “range” defined by it (instead of by the MPI) should be better to use in analysis of proving-ground and of target- practice results.9 Not only would the theoretical basis then be quite as sound or sounder than it is now, but possibly much numerical work could be saved.
1 “The maximum probability of hitting, i.e., the greatest percentage of hits, will occur when the MPI is at the center of the danger space of the target.” Par. 1616, Exterior Ballistics— Herrmann, 1926.
2 Compare, “An Introduction to the Theory of Statistics”—Yule, 1919, figs. 21 and 22.
3Memoriale de l'Artillerie Francaise, 1926, p. 336, shows for instance, the unsymmetrical variations of the barometer.
4 Yule, chapters on Frequency-Distribution and on Averages, particularly pp. 108, 116 and 120.
5 “See Century Dictionary (Supplement) for definitions of “mean,” “median” and "mode” as used in statistical work.
6 Yule, page 121, and Figure 2 opposite.
7 Compare Improvement Firing in field artillery work.
7 Even if one claims Figure I can be verified by experiment (Memorials de l’Artillerie Francaise, 1926, p. 456) this can be no basis for objecting to using the median in spotting.
8Revue d’Artillerie, 15 May, 1926, “Introduction to a rational theory of the errors of observation.” In this, General Estienne, by different reasoning, reaches a similar conclusion.