The Chieftain's Hatch: Range Maths

There is an important distinction to be made between data, analysis and information. In the military, it is particularly the case when referring to intelligence. Information coming into the intel office is data. Information being put out by the intel office is intelligence.

This sort of distinction can be applied to anything from operational estimates to mechanical design. In this case, a paper from the Aberdeen Proving Grounds' Ballistics Research Lab in December 1951 used mathematical principles in order to assess the situations and likelihoods involved in a tank meeting an AP shell. BLUF: If you don't like maths, just scroll down to the charts starting about half-way down.

The paper was called “The Range and Angular Distribution of AP Hits on Tanks.” Exciting, I know.

It said….

“A study is made of the distribution of ranges of engagement and angles of attack in anti-tank engagements in N.W. Europe during WW II. It is demonstrated that, in general, tanks fired when they saw each other. Thus the ranges of engagement depended upon the terrain of the region of engagement. A mathematical model of terrain is derived describing the maximum range of sighting and the angular distribution of aspects of the tank when sighted that is in good agreement with the data on the range of tank engagements and the angular distribution of aspects of fire falling on a tank. The range constant for this terrain model can be derived from a combat map. The expression for the range and angular frequency distribution of attack and for sighting is:

Where R= Range of engagement in yards, Rw/Bar = Mean range of engagement (660 yards), Ɵ = Azimuth angle of attack in radians, a = constant (1 for hull, ¾ for turret)

It is shown that about half of the time the turret was facing forward when a tank was knocked out and half of the turret azimuth angle had the frequency distribution (1/2π)(1+Cos Ɵ). “

For the three of you out there who are statisticians, I’m sure you will be salivating over the coming two articles. I’ll be going over the range information in this one, and turret facing in the next.

For the rest of us, who can barely keep up, perhaps the mathematical part of this will be skimmed over, but you may yet find interesting some of the data and methodology to give an impression of the level of detail gone into these sorts of things.

I go back to the paper from the Ballistic Research Labs.


The proper apportionment of armor over a tank depends on the distribution in range and angle of directions from which a tank is to be attacked. From terminal ballistics tests and calculations, the probability of a kill or any other degree of damage, such as knocking out the mobility, by a particular weapon can be found as a function of range and azimuth. The expected or average value of this probability over all ranges and angles of encounters is an index of the passive vulnerability of the tank to that weapon.

A study of optimum gun characteristics, namely accuracy and velocity, requires a knowledge of the ranges at which tanks are to engage. It is also important to know how gun characteristics will affect the range of engagement.

An evaluation of slewing rates and the time required for a tank to sight a target and bring the gun to bear on it requires information on the expected direction of targets and direction in which the turret is facing when it is needed for action.

The purpose of this paper is to present some recent findings on the subject of ranges and angles at which AP firing weapons are encountgered by tanks. The three fold objective is (1) to extend the data used in previous studies of this type, (2) to fit these data with simple empirical expressions, and (3) to discuss a mathematical model that accounts for these distributions.


The distribution of ranges of attack has been studied in England by Dr. K. Pennycuick. Dr Pennycuick considered the ranges of engagements of 202 US tanks that were knocked out by AP Projectiles. He then fitted the distribution with a fourth degree polynomial. Dr Pennycuick got his data from the records of the US First Army during the first half of the European campaign. Since then, data have become available from this source for later on in the war. Similar data have also become available froim the US Third Army and the Second British Army. The equation which is to be given in this report agrees closely with the polynomial that Dr Pennycuick chose.

The report, “Tank Armour Distribution Theory”, summarizes the work that was done during and after WWII on the angular distribution of hits about a tank. Dr Iliffe laid the ground work for later development when he defined the “directional probability variation” or “d.p.v”, as the chance that a gunner has of firing at a particular aspect of a tank. It is pointed out that, in general, the d.p.v. is not the angular distribution of attack, because it is only the chance a gunner has of firing. Thus, if a gunner realizes that the tank is invulnerable to him, he will not bother to fire. However, the German 75mm and 88mm guns are capable of penetrating allied tanks from almost all directions at the ranges that were encountered. Thus, for all intents and purposes, it may be assumed that the d.p.v. of German guns vs Allied armour was essentially equal to the angular distribution of attack. Dr Iliffe stated several possible functions for d.p.v.s but did not come to any definite conclusion. J. M. Whitaker based a d.p.v. on the time that a given sector of a tank was exposed to firte from a line of guns as the tank approached the line, bisecting it at right angles. The function describing his d.p.v. is defined by three equations in various ranges of the angle. When limited data became available, it was found that this distribution was in good agreement with the actual case. Whitaker’s d.p.v., as it is referred to, has been in use in England as the angular distribution for some time.

[Chieftain’s note: I actually have an issue with one of the assumptions here. There is plenty of anecdotal evidence to suggest that just because a gunner did not expect to be able to penetrate a tank, that did not stop him from shooting anyway. There is a psychological effect on the target, repeated hits (or hits with HE) could have a disabling effect, and, frankly, it just beat sitting around doing nothing while waiting to see if the other guy kills you. The notable exception might be in ambush, when the ambushing vehicle might chose to wait for a more advantageous angle before announcing its presence. That said, I do agree with the premise for the purposes of this paper that since they were analyzing the hits on Allied vehicles, and that the Germans would have a reasonable expectation of causing damage, the Germans, at least, would have likely fired at first reasonable opportunity. So, I don’t have issue with the maths, just that interesting assumption by the mathematicians which seems to ignore the human factor]

Very little data are given [Chieftain: That’s twice he’s said that ‘data are given.’ I’m sure it’s right, but you don’t hear that any more] on the actual angles from which the tanks are attacked. Data are given on angles but the angle is with respect to the surface. Two values of azimuth lead to the same angle of penetration, hence the data are ambiguous for this purpose. Thus methods had to be worked out to interpret the data in terms of the particular angular distribution being considered. Different authors have used different methods.

The hull and turret of the tank have two separate angular distributions. This is due to the fact that the turret is not always facing in the forward direction, but when it turns to engage a target to the side or rear, it exposes its rear to frontal attack. When a distribution in the direction from which the turret is facing is assumed, it is possible to find the angular distribution of attack on the turret in terms of the corresponding distribution for the hull.


The frequency distribution function, F(R, Ɵ) will be defined in such a way that the probability that an attack will come from ranges between R and R+dR and from the sector between Ɵ and Ɵ+d Ɵ is F(R, Ɵ)dRd Ɵ . [Chieftain: Obviously]. Since this is a distribution function, then:


It is reasonable to make the assumption that the angular and range distributions are independent. Under these conditions F(R, Ɵ) can be written as f(R)F(Ɵ). [Chieftain: Actually, I’m not sure it’s entirely independent. I can see an argument for holding fire on a broadside target at range if the required lead is uncertain, for example. But I suspect this may be rare enough to not affect the overall maths]

 Range Distribution: [Chieftain: OK, the good news is that much of the following is understandable by mere mortals, and the charts are easy to interpret]

Data on the ranges of encounter of Allied tank casualties in Europe during WWII are available from three sources. These sources are the intelligence reports of the US First and Third Armies, and the British Report, “Survey of Casualties Amongst Armoured Units”.

The first of these, namely the report of the US First Army consists of a tabulation of the following items where available: Unit, tank model, date damaged, approximate range, approximate angle, where hit, penetration, and “Hit by”. A total of 601 tank casualties are listed. According to the “First Army Report of Operation”, the First Army’s total tank losses in Europe during a period from 1 August 1944 through 22 February 1945 were 1,251. Of the 601 casualties on which data exist, 519 were in the period covered by this operations report. Hence data exist on about 519 out of 1,251 recorded casualties. This means that the sample represents about 40% of all the First Army’s casualties. Hence, this sample is considered representative, although no information is given in the basic report of the manner in which these data were gathered.

A characteristic common to all three sources of data is the lack of identification of the attacking weapon. Thus, “75mm”, “88mm”, “105mm” given in the data could apply to tanks and towed antitank guns alike. [Chieftain: OK, maybe not 105mm, but it could still be something like a StuH]. Thus, the family of armor piercing weapons must be studied as a while. The First Army casualty data give the estimated ranges of 314 engagements with AP firing weapons which resulted in US casualties. These ranges are shown in Figure 1. The grouping of the data around convenient ranges such as 600, 1000 and 1500 yards indicate the estimated nature of the data. Only twelve cases were estimated at ranges greater than 200 yards.

The Third Army data consist of forms filled out in the field by the unit’s commanding officers. The “Third Army After Action Report” gives the total losses for all the war except July as 1,257 light and medium tanks. Data exist on 244 casualties during this period. Hence the sample represents about 20% of the population. Data exist on only 18 casualties during the part of the war not covered by the report.

[Chieftain’s note: I have commented in the past as to just how survivable US tanks were. It is to be noted that in addition to First and Third Armies, there was also in 12th Army Group the Ninth Army and Fifteenth Army, though that latter undertook limited operation. Later they were joined by Sixth Army Group, which also brought along Seventh Army. Thus the 2,508 tanks reported as lost by First and Third combined in the time period combined are only a portion of US Army tank losses in France and Germany. Total Armored Force deaths in NorthWest Europe of which, thus, those 2,508 tanks also represent only a portion is reported by the Adjutant General’s office as 1,372 men. Total wounded, 4,406. It is to be noted that this tally does not count cavalry corps personnel who may have been in Stuarts and Chaffees, nor officers, as officers came from other branches. Regardless, some may find the numbers to be astonishingly low given the public reputation of US tank crew survivability.]

There are a total of 160 AP casualties for which the ranges are given. These ranges are shown in Figure 2. Here only one case was recorded at a range greater than 2,000 yards. Much more care seems to have been given to estimating these ranges than those for the First Army, thus producing a smoother distribution.

The data on British tank casualties are contained in the British report, “A Survey of Casualties Amongst Armoured Units”. The report consists of as much information as was available in the circumstances of the engagement on all the casualties suffered by 19 British Armoured Regiments, between the crossing of the Rhine and the end of hostilities. The data were gathered by an operational research team who saw most of the tanks and talked to most of the surviving crews. Of the 333 casualties reported, 86 cases were AP hits for which the estimated range is given. These data are shown in Figure 3.

 The cumulative distributions of the estimated engagement ranges in the three armies are compared in Figure 4.

Although the three armies fought over different terrain and under different conditions, the similarity between the three distributions is considered to justify the assumption that the samples of ranges of casualties from the three sources were drawn from roughly the same population. The overall distribution for the three armies is shown in Figure 5.

The data are readily described in terms of the cumulative distribution function G(R) where:


 G(R) is the fraction of casualties from ranges greater than R, and Rw/Bar is the average range of the distribution. Special graph paper has been made of the function e-x(X+1) so that a graph of this cumulative distribution will occur as a straight line as a function of range. The data of Figure 5 are plotted in Figure 6 on graph paper of this type.

A straight line that is a good fit to these data is that which corresponds to a G(R) with an average range of about 660 yards. The method of deriving the mean of G(R) is given in the section of this paper concerned with a model for range distribution. This average range agrees satisfactorily with the data averages shown in Table 1.

 Table 1.

 Mean range of engagement for tanks in NW Europe:

                                                                No. of Casualties              Mean Range - yards

US First Army                                                     314                                         760

US Third Army                                                   156                                         615

Second British Army                                         86                                           644


Total                                                                    556                                         701


Thus it was seen that the assumed range distribution is in good agreement with the data*. The percentage of attacks from the interval between two ranges can be found by taking the difference of values of this function at those two ranges. For example, from the graph 87% of all engagements were at ranges greater than 200 yards and 65% were at ranges greater than 400 yards. Hence 22% or about one fifth of all engagements were at ranges between 200 and 400 yards. Only 2% of all engagements were at ranges greater than 2,000 yards.

 *Dr F.E. Grubbs has pointed out to the author that this curve is known to statisticians as a Pearson Type III. This form has the desireable characteristics of being related, as will be shown, to certain measurements and assumptions about the terrain and although the selected frequency distribution does not fit the observed data exceptionally well on the statistical ground it appears to be logically preferable to others not so related to the terrain characteristics.

 The frequency function f(R) corresponding to the cumulative distrubition G(R) is:


F(R)dR is the probability that an encounter was between R and R+dR. Figure 7. It is to be noted that the range at which the most encounters took place was 330 yards, or one-half the average range.

[End Extract]

The report then goes into the statistics as to from which direction the rounds were fired. We’ll return in Part 2, I’m sure many of you will be eagerly awaiting it as much as the Inside the Hatch video on the Char B1, which seems to be delayed again.

A reminder on the above data: The ranges given are for attacks against tanks by guns. They do not specify whether the guns in question were mounted on tanks, tank destroyers, assault guns or were simply towed, so be cautious about drawing too much from it about the nature of tank vs tank combat. Instead, what the data could present to us (as well as tank designers of WW2, with the wonderful benefit of hindsight) is the matter of just how much armour would be required to provide a meaningful level of protection against whatever was being thrown at folks. If 330m of armor is the most common range of encounter, and 660m is the average range, what levels of armour would be required to reliably protect against a 75/48, 75/46, a 75/70 or an 88/56 or an 88/71 at those ranges? In other words, if you add additional weight of tracks, applique armor or sandbags etc, are you making a meaningful contribution to your own safety, or are you instead merely reducing your own reliability and mobility for no practical benefit? You can figure this out for “most likely”, “50-50” or pick your own value, such as 80% of encounters.

The other question is that of acquisition at range: At shorter ranges, a wide field of vision and a faster turret slew are more important than at longer ranges as the chances are that a large angle will have to be covered between ‘front’ and ‘target.’

Anyway, back next week in Part 2 of this report, which has a perhaps surprising conclusion about the direction the target tank is looking when it gets hit.

Bob, re-imaged perhaps, but no less dedicated, will take you to the forum discussion.

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