WERELD & DENKEN
 
 

Toelichting bij De Gaussiaanse grafiek

Carl Friedrich Gauss heeft de Gaussiaanse of normale verdeling ontdekt aan de hand van iets dat iedereen zelf kan waarnemen: de variatie in zintuiglijke of instrumentele waarnemingen, zoals het snelst te zien door het schitteren van de sterren: de ster blijft op zijn eigen vaste plaats, maar lijkt heen en weer te bewegen door de storingen in de atmosfeer. Maar je een grote hoeveelheid snelle foto's van de ster, en zet je bijvoorbeeld alle waargenomen horizontale posities en telt hoeveel het er zijn, krijg je een Gaussiaanse grafiek , met als gemiddelde de werkelijk plaats van de ster. Zie onderstaande bron voor een warme versie van dit verhaal:
 

Uit: J. Bronowski, The Ascent of Men:

We are here face to face with the crucial paradox of knowledge. Year by year we devise more precise instruments with which to observe nature with more fineness. And when we look at the observations, we are discomfited to see that they are still fuzzy, and we feel that they are as uncertain as ever. We seem to be running after a goal which lurches away from us to infinity every time we come within sight of it.
    The paradox of knowledge is not confined to the small, atomic scale; on the contrary, it is as cogent on the scale of man, and even of the stars. Let me put it in the context of an astronomical observatory. Karl Friedrich Gauss's observatory at Gottingen was built about 1807. Throughout his lifetime and ever since (the best part of two hundred years) astronomical instruments have been improved. We look at the position of a star as it was determined then and now, and it seems to us that we are closer and closer to finding it precisely. But when we actually compare our individual observations today, we are astonished and chagrined to find them as scattered within themselves as ever. We had hoped that the human errors would disappear, and that we would ourselves have God's view. But it turns out that the errors cannot be taken out of the observations. And that is true of stars, or atoms, or just looking at somebody's picture, or hearing the report of somebody's speech.
    Gauss recognised this with that marvellous, boyish genius that he had right up to the age of nearly eighty at which he died. When he was only eighteen years old, when he came to Gottingen to enter the University in 1795, he had already solved the problem of the best estimate of a series of observations which have internal errors. He reasoned then as statistical reasoning still goes today.
    When an observer looks at a star, he knows that there is a multitude of causes for error. So he takes several readings, and he hopes, naturally, that the best estimate of the star's position is the average - the centre of the scatter. So far, so obvious. But Gauss pushed on to ask what the scatter of the errors tells us. He devised the Gaussian curve in which the scatter is summarised by the deviation, or spread, of the curve. And from this came a far-reaching idea: the scatter marks an area of uncertainty. We are not sure that the true position is the centre. All we can say is that it lies in the area of uncertainty, and the area is calculable from the observed scatter of the individual observations.



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