If measurement errors are random, how are they distributed? In Theoria motus, Gauss started from the idea that the mean is the best estimate and arrived at the shape of the error curve:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}},$$

where $\mu$ is the mean and $\sigma$ is the standard deviation. It is the normal curve, or Gaussian bell: high in medium, symmetrical, and increasingly lower away from the mean. It appears everywhere, from the heights of the people to the errors of a machine.

A beautiful way to see the bell rise is the Galton machine, invented by Francis Galton in 1873. Marbles fall through a grid of pegs and, on each peg, they go left or right with the same chance. Most end near the middle, few reach the ends:

Drop marbles and watch the Gaussian bell appear. With just a few balls, everything is messy; with hundreds, the form emerges alone.

For a normal curve, the rule 68-95-99,7 applies:

  • around 68% of the values ​​are up to 1 standard deviation from the mean;
  • about 95% is up to 2 standard deviations away;
  • about 99,7% is up to 3 standard deviations away.

Shortly afterwards, Belgian Adolphe Quételet used the normal curve to study people: height, weight, Births, deaths and income roughly followed Gauss’s bell.

The 10 Deutsche Mark banknote (1989–2001): Gauss, the normal curve and the formula, next to Göttingen buildings.
The 10 Deutsche Mark banknote (1989–2001): Gauss, the normal curve and the formula, next to Göttingen buildings.Euler’s Eyes Collection

All topics by Carl Friedrich Gauss · ← Ceres and the least squares method · The Egregium Theorem and curvature →

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