Taming uncertainty or why your telephone works

“There are lies, damned lies and statistics” is a remark usually attributed to Benjamin Disraeli, a nineteenth century British Prime Minister and some time novelist. It was popularized by Mark Twain who probably wished he’d said it first himself. Most of you will have heard it and maybe even used it, perhaps accompanied by a wink or a sage shake of the head in an attempt to denigrate a number that is disliked. The trouble is that statistics was not really put onto a sound basis until well after Disraeli’s death. That’s polite speak for great quote, rubbish analysis.

Elsewhere on this blog is a short piece on a few of twentieth century mathematics more accessible achievements. This is going to attempt to make accessible one of the more difficult achievements. Oh dear, I sound like a conjurer announcing his latest sawing-a-lady-in-half trick. Never mind.

Most of today’s statistical tools were either created, or at least plucked from obscure mathematical papers, during the first two decades of the twentieth century. Why? Because a number of things created a demand for them, three of these were inventions that were rapidly adopted, the telephone, the car and the punched card which in its turn had been created to fulfill a need, that of storing and using the data created by the USA’s ten year census. The punched card was to play a big part in the spread of computing many years after its conception, but I digress, that’s a story for another day.

One telephone, by itself, is not a lot of use (the person who sold the first, must have been a brilliant salesperson), but when you’ve got a lot of them you reap the rewards. Any two people can talk to one another provided there’s a line to connect their two ’phones together. That creates a problem, it is just not practical for each one to be connected to every other by a piece of wire: no problem for the first two, only one piece of wire is needed, but ten ’phones need forty five wires and a couple of thousand ’phones would need very nearly a million to join them all together. Back to the drawing board. The solution, which is still the same one used today, is to give each ’phone a label (today that’s the telephone number) and join it to a central hub. At the hub there is set of switches that allows any ’phone to be connected dynamically to any other. OK, there are multiple hubs, and the connection may be by radio, but the principle remains the same.

And now came the big question, what size does the switching mechanism need to be? Today it’s largely done by computers, but in the early days, the switchboard was operated by hand, with wires actually being connected together by plugs, a bit like those used to connect speakers to a stereo system, and at the end of the call, the plugs were removed. Whichever way it’s done, the question about the size of the switch remains, so how do we find the answer? Depends on how many calls are made and how long the calls last. Right then, how long does a telephone call last? Well it varies, it’s not certain. And how often do you make a phone call? Don’t know, it depends.

At first sight those answers do not seem terribly helpful. But please come centre stage Dr. Agner Krarup Erlang. He started by asking another question, how long is someone prepared to wait to get a connection before they give up, and then made the leap that all three questions have a statistical answer. For example there is a mean average length of time for a ’phone call and a measurement of the variation from that length, both calculable from historical data.

He then managed to combine all three questions into a formula that requires wrapping a wet towel round your head before approaching it, but which is (dare I say it) quite easy to use. Much more important, it works, is still used today and has found applications in many other fields, for example the science of epidemics.

It isn’t as catchy or as memorable, but perhaps what Disraeli might say today is “You can tell lies, damned lies over the telephone, but you need statistics to make it work”. Hmmm, I’m not certain that he would though.