An Accidental Statistician

by George E P Box

  I've forgotten our Company Sergeant Major's name, but we called him “The Bull.” Everyone was afraid of him, but I think he had some admirable traits. In particular, he always wanted to be absolutely sure that he was familiar with everything that the Company needed to do. People were frequently sent to courses. If there were a new type of trench mortar, for example, a sergeant might be sent on a course of training and would return to instruct our company on how it was operated and how you could take it apart and put it together again. Or there might be a new anti-tank mine: How, for example, was this to be set? Without delay, the Bull required anyone returning from such a course to teach him the new techniques, and while being instructed, he would forget all about rank. He would smartly carry out instructions until he could do the task as well as his instructor. He felt he must know all the things that everybody in the Company knew and be able to complete a drill better than anybody else.

  When the Bull asked you a question, he expected a quick and plausible answer, and if he got this, then he was satisfied. For example, for a time we were stationed at a beautiful seaside resort on the south coast of England, partially evacuated at the time because of the possibility of invasion. One day there wasn't much to do so three of us goofed off and went for a swim. The senior of our small gang was a Corporal Cornford, so it was to him that the Bull, who unexpectedly appeared, addressed his remarks. “What are you on, Corporal?” he demanded. With great presence of mind, Cornford replied, “Teaching Lance Corporal Box to swim, Sir.” I think the Bull knew as well as we did that this wasn't true, but because it was said smartly and without hesitation, he accepted it and gave us a detailed lecture then and there on the best way to teach a beginner to swim.

  There was a great hill that dominated the resort town where the company was stationed. Our commanding officer must have been at a dinner party with the local bigwigs one night because the following morning he got us all together to tell us about an iron gun that was on top of the hill that the city fathers wanted removed. It must have been 200 years old and was very large, close to 20 feet long. He asked us all to write a reconnaissance report on how it might be removed. Very quickly it became obvious that we didn't have appropriate equipment to move such a heavy object. All we had were explosives. I had a scheme to first blow a very large hole into the ground and then roll the gun into it with explosives attached to it. We would then fill in the hole and blow up the gun. My suggestion was not adopted.

  Someone claimed that they could put explosives on the gun as it sat on the hill and so finely calculate the charge that the resulting explosion would merely crack the gun. The gun could then be broken into parts and hauled away. When we did this, however, our calculations must have been a bit off. There was an enormous explosion and bits of gun fell from the sky all over the town, and greatly surprised some fishermen who were out to sea in their boat. A lady with a very proper accent stated emphatically that her piano had been shot completely through with little holes from the falling debris. By some miracle, there were no casualties, but for some days, we went about the town mending things.

  The Company consisted of platoons, each divided into four sections of 10 to 12 men. In the summer, we had lectures during which we sat on the ground in an open field and an NCO lectured about some aspect of our duties. It might be about the names of parts of a machine gun and how these parts could be assembled or any number of other things. One morning we had for our instructor a particular sergeant who was what we called an “old sweat”—that is, a “lifer” in the military. He knew very little except for one topic that he knew very well and that was “knots and lashings.” One day I remember he was making a hash of some topic or other when in the middle of a sentence, he abruptly switched to talking about how to tie a bowline. When we looked around, we saw that he had seen the colonel who had come over to listen. As the sergeant was well aware, knots and lashings were also the colonel's favorite topic. “Very good sergeant,” the colonel remarked approvingly. He then recounted his way of remembering how to tie the knot. All I recall is that “the rabbit ran around the tree.” There was a saying in the Army: “Bullshit baffles brains” and it was very true.

  The commanding officer for our company worked very hard at getting us properly trained. In particular he taught us how to avoid “booby traps.” Once, for example, we were working in the trees putting up camouflage nets when along came a truck that we thought was the tea wagon bringing us a well-earned cup of tea and a bun. We left off working and gathered around the truck in expectation. Then out jumped four men with submachine guns trained on us. We did not need to be told that we would all have been dead had this been the enemy. He and his crew also kept a supply of “thunder flashes,” which made a tremendous flash and a deafening crack. In the middle of the night, if you were a member of the guard and he suspected that you were not sufficiently alert, you could expect a rude awakening when one of these was thrown at you.

  We used to go on exercises that usually lasted three nights. We called them “stunts.” The company, in 30 or so 1500-weight trucks, would drive onto Salisbury Plain to follow a route described in the operation order. When (and if) we got where we were supposed to be going, we carried out a mission, such as a demolition, and found our way home. We had a detailed ordinance map showing the proposed route, but we moved only at night and the only light you were allowed was a very faint one underneath the truck in front of you. We frequently got lost, finding ourselves in a farm yard with everybody swearing and the Sergeant saying, “Well the map must be wrong.” Even worse, when it rained, one or more of our trucks might get bogged down and had to be dug out.

  On one occasion, I had just returned from leave and found our vehicles all lined up to go on a stunt. I was too late for the briefing, but being a lance corporal, I put on my equipment and took my place in the front of the truck. I asked my driver, “What are we supposed to be doing tonight?” He said, “I don't know Corporal, but I'll tell you one thing I'm absolutely certain of: it's bound to be a balls up.”4

  Strangely enough, during the war, I was never in any danger except when I was on leave! Salsbury was far from the bombing, but Gravesend was on the direct route of the bombers going to London. As it turned out, our house was never hit, but a number of neighbors' houses were. The fact that a house was bombed did not mean necessarily, or even usually, that people were killed. When people were “bombed out” after a raid, the neighbors rescued them from the ruins, gave them tea, fed them, provided shelter, and helped in every way they could. The war seemed to bring out the best in people. After it was over, I heard it said many times, “Why can't people be like they were during the war?”

  Everything was rationed or, if not rationed, very hard to get. On one occasion, when I was returning from leave, I saw a queue outside a shop, so I joined in and asked what we were queuing for. It was for alarm clocks, so I waited my turn and bought one. I set the clock to the correct time and put it back in its cardboard box and put the box in my kit bag.

  On the train, I sat opposite an old lady. After we had traveled some distance, she asked if I could tell her the time. I said, “Yes, certainly,” and standing, I removed my kit bag from the overhead rack, fiddled about inside it, eventually retrieved the cardboard box, opened it, and looked at the clock. “It's ten minutes to three,” I told her. Then I put the clock back in the box, put the box back in the kit bag, and put the bag back up on the rack. My companion appeared a little dazed.

  England had imported about two thirds of its food before the war, and these supplies were cut off by the Germans once war broke out. The rationing that began in January 1940 was severe, and the British Army was included in this, although soldiers fared somewhat better than citizens. Among the most sought after food items that were on the restricted list were sources of protein. Weekly rations per person consisted of one egg, three pints of milk, and a very small portion of meat, which was rationed according to its value.5

  The American soldiers who were in England du
ring the war had abundant food, as far as we were concerned. Before coming to England, all U.S. GIs had been issued a small book advising them of British customs and of how the war had affected England since 1939. One passage read, “If you are invited to eat with a family, don't eat too much. Otherwise you may eat up their weekly rations.”6

  One year I became friendly with a GI and we used to sometimes meet at a pub in Salisbury. When Christmas came, he invited me to the Christmas party that the American soldiers were having. When I went to the gathering, I was simply amazed: They had everything that you could possibly want and anything that could not be bought in England. So for that one day, I lived in Arcady.

  Food scarcity, and especially meat shortages, continued long after the war ended. Things were so bad that by the early 1950s, it became legal to sell sausage that had very little meat content at all. One member of the House of Commons famously asked, “Would the right honorable gentleman be prepared to say at what point a sausage becomes a cream bun?”

  I sometimes like to play the piano. I'm not very good, but how I arrived at what I can do is of some interest. When I was about eight years old, I was made to take piano lessons from my cousin. That involved a two-mile walk to her house and a two-mile walk back. I never really made any progress because Vera taught me piano as if I were learning to type, but my mind was, and is, such that I can never learn anything unless I know why.

  After I had taken the intermediate science exams, and before the war broke out, I was looking for something to do in my new-found free time, and I took up the guitar, using the one that Uncle Willy had bequeathed us. My instructor was from the merchant navy. (He was killed early in the war, as were so many in that service.) He explained to me about chords and chord progressions and I became quite interested in this, and particularly in composers like George Gershwin and Cole Porter who used clever chord sequences. Later, when I started tentatively playing the piano, I transferred the guitar chords to the piano notes, and this at last made sense to me.

  When I was stationed in Salisbury, I asked my father to send my guitar to me. I went down to the railway station each day, and sure enough, it eventually turned up. He had built a guitar case that looked a bit like a small coffin, but it worked. There were a number of us who were interested in playing jazz. In addition to a trumpet, a trombone, drums, and my guitar, there was a pianist named Smudger Smith. He was quite small and could play the piano in almost any key, drunk or sober. We got some music and started playing together.

  Smudger Smith had a friend who looked like a heavyweight boxer. This friend was not very bright and did not play an instrument himself, but he believed that his duty was to supply Smudger with plenty of beer. So while we played, he would make sure that at any given time, there were two pint glasses on top of the piano appropriately filled.

  After a bit we got hired to play at dances. All went well until one evening we were to play at a temperance society dance. Smudger's friend could not, of course, find any beer at the dance, so he walked up the road and bought two pints at the nearest pub. When he returned with the beer, the men at the door tried to stop him, but he elbowed them away and delivered his goods atop the piano in the usual way. As I recall the supply was regularly replenished during the session—an unusual temperance meeting!

  On my 21st birthday, I was sharing a barrack room with seven other guys. One of them and I felt the need to celebrate my birthday, so we caught the bus into Salisbury and visited a few pubs. The beer in some was better than in others. Before catching the bus home, we became somewhat morbid thinking of our friends in the barrack room who didn't have any beer, so we bought a good supply to take back with us. Fortunately the large pockets in our great coats accommodated several bottles. When we got back, we announced to our colleagues that we had brought them beer, but when we emptied our pockets, we were surprised to see that the bottles were drained. In our celebratory state, we had forgotten that the bus had been delayed and that we had drunk them on the way home.

  One morning I was a bit late on parade, and when I got there, I found we had been told to form a single line. An NCO divided the line into two halves. The front half was marched off to where we did not know. The half I was in remained. I discovered later that the first half had been shipped off to Singapore, which fell to the Japanese on February15, 1942. All had become prisoners of war, and most of them were never heard from again.

  Early on in the army, I trained with the engineers and, in particular, learned how to demolish bridges. You don't actually blow up a bridge; you use explosives to cut the girders that are holding it up. There are simple formulas that tell you how much explosive to use, and it was because I understood these that I got to calculate the charges and work the exploder in our practice demolitions. There was no doubt that I was better at demolitions than I was at knots and lashings.

  The Experimental Station at Porton Down

  Before I could employ my demolition skills in real warfare, someone in the Army found out that I had a background in chemistry, and I received orders to report to the Chemical Defence Experimental Station at Porton Down. This was near Salisbury, in the south of England. It was fully expected that the Germans would eventually use poison gas, and the purpose of the experimental station was to find out what to do about it. Some of England's best scientists were there. My boss, Dr. Harry Cullumbine, for example, was a professor of physiology dressed up as a colonel. I became a lab assistant dressed up as a staff sergeant (Figure 2.1).

  Figure 2.1 Porton, Colonel Collumbine center. I am to his left.

  The army decided that every army unit should be issued a small sample of liquid mustard gas so that volunteers could have a drop put on their skin and see how a painful blister appeared. In army operation orders, you always had to sign off on a line that said, “All Informed,” but I'm afraid this protocol was not followed with mustard gas samples. Cullumbine, who was the expert on the treatment of mustard gas casualties, was kept very busy by the misuse of these samples. In one instance, for example, someone must have decided that the black liquid in the can was for painting the stove. The result was close to disastrous for the people who slept in that barrack room.

  Initially my job was to make biochemical determinations in experiments on small animals. The results I was getting were very variable, and I told Cullumbine that what we needed was a statistician to analyze our data. He said, “Yes, but we can't get one. What do you know about it?” I told him I had once tried to read a book about it by someone called R.A. Fisher, but I hadn't understood it. He said, “Well you read the book so you'd better do it.” So I said, “Yes Sir.”

  I wrote to the Army Education Corps, which sent me a list of books on statistics that I carefully studied.7 I soon realized that it was not just a question of statistical analysis, but also that we needed to carefully design our experiments using statistical principles. Before long I was given some assistants, and for the rest of the war, I spent my time planning, supervising, and helping to run experiments both in the lab and outside on the range where we simulated warfare. Thus, I changed my plan from becoming a chemist to becoming a statistician. For the rest of the war, I was in fact the only one at the station responsible for statistics, and Cullumbine and I wrote several articles describing the results of our experiments.8

  To illustrate the power of experimental design, I show here a simple experimental arrangement that we used in the lab to find the best treatment for mustard gas blisters on soldier-volunteers. A very small drop of mustard gas liquid applied to the arm of a volunteer caused a blister about three quarters of an inch in diameter. It was similar to that produced by an ordinary burn, but it was more difficult to treat and took longer to heal. There were difficulties in making comparisons between different treatments because the ability of the body to heal is different for different people, and for different parts of the body. There was a further problem in that we needed to get valid results quickly.

  The response measure that we used was healing ti
me in days. For each experiment, we had six volunteers (labeled 1–6 in the arrangement), and six drops of mustard gas were placed at six different places on each arm of each volunteer (labeled A–F). There also were six different treatments (a–f) allocated as indicated in the table. You will see that each treatment was tested once on each of the six volunteers at each of the six positions on the arms. So each treatment occurred once with each volunteer and once in each position. Thus, differences among the volunteers and differences resulting from positions on the arm could be calculated and eliminated. This arrangement, which Fisher invented, is called a “Latin Square” design. The use of similar but sometimes much more complicated arrangements is part of the general science of “statistical experimental design.”

  There are many ways in which you can rearrange the letters in this diagram and still get a Latin Square, so Fisher enumerated all possible rearrangements and proposed that the experimenter pick one at random. He also pointed out that then, without making any assumptions about probability distributions, you can make a statistical significance test to reveal what treatments are best by considering what actually happened in relation to what might have happened with all other possible random arrangements of the same data. Later, tests of this kind were rediscovered by mathematical statisticians who called them “nonparametric” tests.

  One day at the lab, I was having trouble with a particular statistical problem, and a senior medical scientist there suggested that I write to R. A. Fisher about it.9 I thought Fisher would be much too busy to talk to me, but he replied, asking me to come to see him and to bring my data. The Army, however, did not have a procedure to send a sergeant to see a professor at Cambridge, so they made out a railway warrant that said I was taking a horse there.

  When I arrived at Fisher's house, it was a beautiful day. He said, “Let's go and sit under that tree in the orchard. I'll look up the probits and you look up the reciprocals, and we'll plot the data.” I didn't know that that was what we were going to do, but in doing it, my problem was quickly solved. Fisher was extremely kind and spent the whole day with me.