# How a statistical formula won the war



## syscom3 (Jul 22, 2006)

To help solve the intelligence riddle of the production rates of the German war machine, one has to adopt the view of "Germans being Germans" and that they are quite logical and predictable.

Guardian Unlimited | The Guardian | Gavyn Davies does the maths

Gavyn Davies does the maths 

How a statistical formula won the war 

Thursday July 20, 2006
The Guardian 


Here is a story about mathematical deduction that I love, mainly because it is said to be true, and because it had an impact (albeit small) on the outcome of the second world war. It is the story of how a simple statistical formula successfully estimated the number of tanks the enemy was producing, at a time when this could not be directly observed by the allied spy network.
By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat, but they were worried about the capabilities of the new marks IV and V. More troubling, they had really very little idea of how many tanks the enemy was capable of producing in a year. Without this information, they were unsure whether any invasion of the continent on the western front could succeed.

One solution was to ask intelligence to guess the number by secretly observing the output of German factories, or by trying to count tanks on the battlefield. Both the British and the Americans tried this, but they found that the estimates returned by intelligence were contradictory and unreliable. Therefore they asked statistical intelligence to see whether the accuracy of the estimates could be improved.
The statisticians had one key piece of information, which was the serial numbers on captured mark V tanks. The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced. And this deduction turned out to be right. It was enough to enable them to make an estimate of the total number of tanks that had been produced up to any given moment.

The basic idea was that the highest serial number among the captured tanks could be used to calculate the overall total. The German tanks were numbered as follows: 1, 2, 3 ... N, where N was the desired total number of tanks produced. Imagine that they had captured five tanks, with serial numbers 20, 31, 43, 78 and 92. They now had a sample of five, with a maximum serial number of 92. Call the sample size S and the maximum serial number M. After some experimentation with other series, the statisticians reckoned that a good estimator of the number of tanks would probably be provided by the simple equation (M-1)(S+1)/S. In the example given, this translates to (92-1)(5+1)/5, which is equal to 109.2. Therefore the estimate of tanks produced at that time would be 109

By using this formula, statisticians reportedly estimated that the Germans produced 246 tanks per month between June 1940 and September 1942. At that time, standard intelligence estimates had believed the number was far, far higher, at around 1,400. After the war, the allies captured German production records, showing that the true number of tanks produced in those three years was 245 per month, almost exactly what the statisticians had calculated, and less than one fifth of what standard intelligence had thought likely.

Emboldened, the allies attacked the western front in 1944 and overcame the Panzers on their way to Berlin. And so it was that statisticians won the war - in their own estimation, at any rate


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## Soren (Jul 22, 2006)

"_By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat_"  The guy who wrote this obviously doesn't know much about the history of WWII armoured warfare.


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## syscom3 (Jul 22, 2006)

Soren said:


> "_By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat_"  The guy who wrote this obviously doesn't know much about the history of WWII armoured warfare.



I saw that too.


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## delcyros (Jul 22, 2006)

Very cool info, Syscom!8)


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## the lancaster kicks ass (Jul 22, 2006)

i was more bugged by the fact he said 



> US and even British tanks



firstly america was only just in the war by 1942 and hadn't engaged jerry in tank warfare and at the time the British Matilda series for example could take on german tanks and were superior to american designs (unless the sherman was out then... but we improved it anyway)......


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## syscom3 (Jul 22, 2006)

the lancaster kicks *** said:


> i was more bugged by the fact he said
> 
> 
> 
> firstly america was only just in the war by 1942 and hadn't engaged jerry in tank warfare and at the time the British Matilda series for example could take on german tanks and were superior to american designs (unless the sherman was out then... but we improved it anyway)......



Dont be bugged by the fact he was wrong on that point.

Be thankfull some allied statistics experts figured out the German tank production rates.


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## delcyros (Jul 22, 2006)

It would be interesting to know whom did the staff believed, the statistical intellegence or the standart intellegence during wartime. It would also be interesting to know which conclusions did they draw from the material...


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## syscom3 (Jul 22, 2006)

This question is for you WW2 German tank experts out there.....

Did the S/N of an individual tank contain the date of manufacture (or acceptance)?


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## schwarzpanzer (Jul 22, 2006)

Formula that won the war = E=MC2 

I wouldn't be surprised if some Panzers had no markings at all Syscom.



> By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat, but they were worried about the capabilities of the new marks IV and V.



It could be said that, 'till the advent of the PzIV F2 (Special) that the Stuart, Grant and Sherman were the best tanks of '41-'42, but it up for debate...



> It would be interesting to know whom did the staff believed, the statistical intellegence or the standart intellegence during wartime.



The Allies were often kept ignorant as much as possible, unsurprising really; if they knew about the superiority of German equipment - that could be bad.

It's like with tactics too - look at what Rommel did to morale when his capabilities were known to the soldiers.


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## Soren (Jul 23, 2006)

syscom3 said:


> I saw that too.



And thats not all he got wrong, he apparently also thinks that the PzKpfw.V came before the PzKpfw.VI : 
_"By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat, but they were worried about the capabilities of the *new marks IV and V*."_


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## syscom3 (Jul 23, 2006)

schwarzpanzer said:


> ......
> I wouldn't be surprised if some Panzers had no markings at all Syscom.



That would not be normal for the German "mindset". 



> .......The Allies were often kept ignorant as much as possible, unsurprising really; if they knew about the superiority of German equipment - that could be bad...



This was high level industrial production analysis. The statisticians probably werent even privy to tactical information.



> ....It's like with tactics too - look at what Rommel did to morale when his capabilities were known to the soldiers.



a better question should be .."how accurate was the German analysis of US and british production"


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## DerAdlerIstGelandet (Jul 23, 2006)

Very interesting article but I do have to have a laugh at something.



> By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat,





Tell that to Rommel in France and the Poles and the British in France!


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## HealzDevo (Feb 28, 2007)

I think it was more weight of numbers rather than a superiority of British and American tanks that led to victory. There were very few British and American tanks that could go toe to toe with a German Panther tank. Only one US tank could defeat a Tiger in a tank to tank slug-fest. So no, UK and US WW2 tanks were not superior to German WW2 tanks. They were inferior and only won by numbers.


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## timshatz (Feb 28, 2007)

Actually, the guy got part of it right. "The Germans being Germans" happened at least one other time of great significance. Enigma wheels had a given number of digits on them (I believe there were 26 but am not sure). But the actual digit at each spot could be anything. The Poles (I am pretty sure it was them, going on memory) when they were making their "Bombe" (which was a copy of Enigma made from scratch) did the same thing the stat types did and figured the digits were sequential (Germans being an orderly people). One through 26, all the way around the dial. Turns out, they were right. It was a stroke of genius and it worked. 

As for the statistical analysis, it seems to work. Questions I would have about it would be:

1. Which serial number did they use? Is there a master number or is there one for the Transmission, Gun, Engine, Frame, ect? Which of the parts is most difficult to make or is the entire tank the real tricky part. 

2. As somebody else alluded to in this discussion, what did the serial numbers mean? Probably not tough to find with all the captured soliders but it would be something you would need to know. 

The Germans had figured out that the Allies were keeping track of their serials in at least one case. U-boats did not have sequential hull numbers after a certain point but skipped some digits to keep the Allies thinking there were more subs out there than there really were. 

The Germans did do a statistical analysis of the US production (not sure if they did it for the other allies but I think they did). Were on the money to slightly high for 1942 but missed 1943/1944 badly. Way underestimated on both years. 

Lastly, the Brits were doing plenty of statistical analysis throughout the war. One only needs to look at the bombing survey in which it was found that one in five bombers got anywhere close to their targets to see that. Brought about a huge change in Bomber Command (starting with the replacement of Portal with Harris).

Good post Sys, good topic.


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## Cojimar 1945 (Mar 2, 2007)

The German tiger and panther tanks seem rather disappointing. It seems really wasteful to build such large vehicles but put small calibre guns on them. The armor protection also seemed low given the size of the vehicles.


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## Chief (Mar 2, 2007)

The allies at one point modified some M4's with 105mm guns. How effective were they?


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## Glider (Mar 3, 2007)

Cojimar 1945 said:


> The German tiger and panther tanks seem rather disappointing. It seems really wasteful to build such large vehicles but put small calibre guns on them. The armor protection also seemed low given the size of the vehicles.



Are you for real or is this a wind up?


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## DerAdlerIstGelandet (Mar 3, 2007)

Cojimar 1945 said:


> The German tiger and panther tanks seem rather disappointing. It seems really wasteful to build such large vehicles but put small calibre guns on them. The armor protection also seemed low given the size of the vehicles.



Okay but nothing except for the T-34 and the Firefly could really penetrate them unless they got in real close and the Tiger and Panthers "small" (as you call them) guns could take out the Sherman and most allied vehicals before they were in distance to fire on the Tiger or Panther.


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## Gnomey (Mar 3, 2007)

If he classes a 75mm/88mm as a small gun, I wonder what a big gun is 150mm plus? Even nowadays the MBT's have only 120-125mm guns so he has to be smoking something...


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## HealzDevo (Mar 4, 2007)

I think I can remember something about the plotting by statisticians of bullet holes on an outline of the aircraft and where there were few to no bullet-holes, was where the bullets had hit the Allied planes that didn't make it back. Therefore more armor was put into those places. It seemed to have worked as a strategy...


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## timshatz (Mar 5, 2007)

HealzDevo said:


> I think I can remember something about the plotting by statisticians of bullet holes on an outline of the aircraft and where there were few to no bullet-holes, was where the bullets had hit the Allied planes that didn't make it back. Therefore more armor was put into those places. It seemed to have worked as a strategy...



Sounds like a good idea, in leu of anything else.


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## HealzDevo (Mar 11, 2007)

Found it the M27-46 Pershing Tank was the US tank that could conduct a slug-fest with a German Tiger and have a hope of winning and it was only getting introduced in 1944-1945!


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## bigZ (Mar 11, 2007)

Peter Riedel's book, "German Air Attache" makes intresting reading also describes how he accurately and legally compilled US aircraft production during his time in Washington. With this he was able to predict the production figures from 41 onwards.


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## HealzDevo (Mar 12, 2007)

So did this information get to Germany from Peter Riedel considering the name seems German?


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## timshatz (Mar 12, 2007)

bigZ said:


> Peter Riedel's book, "German Air Attache" makes intresting reading also describes how he accurately and legally compilled US aircraft production during his time in Washington. With this he was able to predict the production figures from 41 onwards.



If you do your homework and check your sources, this is something that can be done now. Same as back then. Open societies are full of usefull information for those that look. 

But as somebody posted, did anybody do anything with it? Probably not. Why, my guess would be the powers that be in the Nazi Party were banking on a short war and finding out the other side was going to far outstrip your production capabilities would put a big hurting on your plans for success. As stupid and short sighted as that may sound, it is most likely accurate. 

The Ostrich Affect is pretty common when the going gets bad for totalitarian regimes.


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## Smokey (Mar 12, 2007)

timshatz said:


> The Ostrich Affect is pretty common when the going gets bad for totalitarian regimes.



This is extreme nitpicking, but



> In popular mythology, the Ostrich is famous for hiding its head in the sand at the first sign of danger. The Roman writer Pliny the Elder is noted for his descriptions of the ostrich in his Naturalis Historia, where he describes the Ostrich and the fact that it hides its head in a bush.
> 
> There have been no recorded observations of this behaviour. A common counter-argument is that a species that displayed this behaviour would not survive very long. The myth may have resulted from the fact that, from a distance, when ostriches feed they appear to be burying their head in the sand because they deliberately swallow sand and pebbles to help grind up their food. However, ostriches that are threatened but unable to run away may fall to the ground and stretch out their necks; as their necks are the same color as sand, within a desert this camouflage could be mistaken for having disappeared or being buried.





Ostrich - Wikipedia, the free encyclopedia


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## bigZ (Mar 12, 2007)

HealzDevo said:


> So did this information get to Germany from Peter Riedel considering the name seems German?



Yes the information did get back. But as Timshatz quite rightly pointed out it was ignored. Ridel describes what a mess the organisation was during Udet's leadership.


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## Jackson (Mar 12, 2007)

the Falaise gap shows what thousands of Jabos could do

Air Superiority is the real statistic.

When the weather broke in the Ardennes Peifer learned how far he could walk in a day, while the victors flew overhead at 400 mph..

The rest is all just burning scrap metal.


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## Jackson (Mar 12, 2007)

Like so


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## DerAdlerIstGelandet (Mar 13, 2007)

The good old Highway of Death...


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## HealzDevo (Mar 14, 2007)

Ok, interesting picture. I didn't really know that information about the ostrich burying its head in the sand. Any WWII destroyed convoy photos to post?


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## Cojimar 1945 (Mar 14, 2007)

I didn't know T34's could penetrate the front armor of Tigers and Panthers. 
The German tanks seem to have fairly thin armor on their sides which could make them even more vulnerable to being destroyed from hits in this area.

The IS-2 and IS-3 weighed about the same as a Panther while the Pershing was slightly lighter. However, the Soviet tanks seem to have better armor protection than the Panther and the IS-3 seems particularly impressive in this regard.


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## plan_D (Mar 15, 2007)

The IS-3 never saw combat in World War II. It only arrived to take part in the Victory Parade in Berlin. The British were shipping their A41 Centurion to Europe with the 22nd Armoured Brigade when the war ended; this would have been something to rattle the Germans. 

The Tiger and Panther were both more than capable of destroying the IS-2 well beyond the range that the IS-2 could react. Especially since the IS-2 had purely made armour; a problem which was discovered but allowed to remain. It was theoretically invincible when taking on Tigers and Panthers head-on but the crews were reporting losses and the German crews were reporting kills at distances up to 1,000 m.


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## HealzDevo (Mar 18, 2007)

Okay, interesting information, that one.


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