Sigma Story [5]: Surveyor

She was a looker. For her debut in 1820, she was the first to sail under the New London Bridge—celebrating the ascent to the throne of King George IV. But there was no use for her at the time, so she laid low for about five years. Then, she was retrofitted as a survey vessel; giving up four of her cannons and gaining an extra mast.

By 1830 she completed her first voyage, four and a half years surveying the coasts of Patagonia and Tierra del Fuego. She was a good ship, but the surveying task was daunting and ultimately boring. After about two years at sea, her captain had slid into deep waters of depression and had committed suicide—that boring.

She was due to return to South America for another five years. Her new captain took her to a dock, arranged for a new deck, retrofitted whatever he could, and loaded her with the latest and greatest surveying gizmos of the time.

Determined not to suffer the fate of his predecessor, the captain decided to look for an interesting companion—a gentleman interested in sciences—who would  accompany him and provide stimulating conversation at the dinner table. Eventually, he settled for a “naturalist” from a respectable family of doctors and scientists. And so, once more, on December 27, 1831, she set sail towards South America. Onboard, Charles Darwin was getting seasick.

To be continued.

Sigma story [4]: Friends in high places

The year is 1807: the French army is approaching the city of Göttingen, Prussia. Napoleon, the Emperor, orders the city to be spared, because “the greatest mathematician of all times is living there”. When the French decide to charge the Germans for the generosity, Carl Friedrich Gauss, age 30, is supposed to pay a fine of 2000 francs, way above his means. Fortunately, mathematicians’ solidarity kicks in; Count Laplace pays the fine on his behalf.

In 1818 Gauss begins a geodesic survey of Hannover, the idea being to use the curvature of the earth to improve accuracy of geographic measurements; it takes him almost 30 years to finish. He notices that as the number of measurements grows they seem to cluster around a central point, and the error distribution looks like the bell curve. Sometime during this period, Gauss figures out the equation of the curve: the probability density function of the normal distribution. The equation has two variables: the mean and the sigma.

To be continued.

Sigma story [3]: An officer and an astronomer

Paris, January 21, 1793: it is all over for the King Louis XVI, as the guillotine blade approaches his neck. In the months to follow, tens of thousands will be guillotined during the Reign of Terror, by the new French republic.

The King’s Royal Artillery examiner, with his wife and two children, is leaving Paris, not a healthy place to be right now. It was a cushy, well-paid job for a well-known mathematician. Pierre Simon Laplace, who got the job in 1784, did not enjoy the boring part of writing reports on cadets he examined, but it gave him access to government officials and others in power. Things are not going to be easy for a member of bourgeoisie.

He remembers a young artillery cadet with “thorough knowledge of mathematics and geography”; whom he examined eight years ago. Since then, the young officer joined the Revolution and made it to the rank of general. As it turns out, Napoleon Bonaparte remembers him too. Next year Laplace returns to Paris, his head still firmly attached to the rest of his body. In 1806 the Emperor Napoleon makes him a Count.

In 1812 Laplace publishes Théorie Analytique des Probabilités, in which he discusses the method of least squares. The method is nothing new: Legrende formulated it in 1805; Gauss published it in 1809. Laplace is trying to predict trajectories of Jupiter and Saturn; known position of celestial bodies is crucial to navigation on open seas (that’s what the sextant is for).

The problem is in fitting a curve through a series of points, or deriving a single value from several measurements where the values have a spread. The idea is to minimize the sum of squared distances between observed and predicted values. To do that:

  • Calculate squared differences between observed and predicted values.
  • Sum all of these.
  • Divide by number of points.

Ring any bells? What if the “predicted values” is replaced by the “mean”? Sure sounds like variance, and the positive square root of that one is called standard deviation or sigma.

To be continued.

Sigma story [2]: Mathematician with no diploma

France 1685: King Luis XIV revokes the edict of Nantes. The edict, issued by Henry IV, 87 years earlier after a series of five civil wars, kept an uneasy peace between the Catholics and the Protestants. Hundreds of thousands flee to neighboring Protestant countries; Abraham de Moivre, age eighteen, settles in England.

London 1738: a Fellow of the Royal Society, buddy-buddy with Isaac Newton and Edmund Halley, Moivre publishes his second edition of The Doctrine of Chances. Near the end, Moivre expands on his earlier paper, in which the concept of normal distribution was introduced for the first time, although it was not called that explicitly.

It does take a mathematician with lots of patience to read something titled as:

A Method of approximating the Sum of the Terms of the Binomial (a + b)n
expanded into a Series, from whence are deduced some practical Rules to estimate the
Degree of Assent which is to be given to Experiments.

Laplace later expands on his result and today we have the de Moivre-Laplace theorem. The following paragraph (from the Doctrine of Chances) is particularly interesting:

Again, as it is thus demonstrable that there are, in the constitution of things,
certain Laws according to which Events happen, it is no less evident from Observation,
that those Laws serve to wise, useful and beneficent purposes; to preserve
the stedfast Order of the Universe, to propagate the several Species of Beings, and
furnish to the sentient Kind such degrees of happiness as are suited to their State.
But such Laws, as well as the original Design and Purpose of their Establishment,
must all be from without; the Inertia of matter, and the nature of all created
Beings, rendering it impossible that any thing should modify its own essence, or
give to itself, or to any thing else, an original determination or propensity. And
hence, if we blind not ourselves with metaphysical dust, we shall be led, by a short
and obvious way, to the acknowledgment of the great Maker and Governour of
all; Himself all-wise, all-powerful and good.

Whoa, it seems he had a clear understanding that the laws governing games of chance are somehow present “in the constitution of things” and “preserve the order of the universe”. All this two hundred years before quantum mechanics. Not bad for a guy who never actually had a college degree; another dropout success story.

To get a feeling for the theorem check out the bean machine (Galton board).

To be continued.