How did Carl Friedrich Gauss use the normal distribution in the method of least squares?
Swiss commemorative stamp of mathematician Jakob Bernoulli, issued 1994, displaying the formula and the graph for the law of large numbers, first proved by Bernoulli in 1713.
In 1809, the German mathematician Carl Friedrich Gauss provided a theoretical foundation for the method of least squares by linking it to the normal distribution.
While Adrien-Marie Legendre had previously proposed minimizing the sum of squared deviations as a practical technique for fitting data, Gauss demonstrated that this method yields the best possible estimate of a parameter if the errors in the measurements are assumed to follow a normal distribution.1
Gauss developed this approach while studying astronomical observation errors, formulating what became known as the law of observational error. By incorporating the normal distribution—often referred to as the "astronomer’s curve of error"—into his derivation, Gauss provided a rigorous probabilistic justification for the least squares method. This work was decisive for the development of probability and statistics, establishing least squares as a primary tool for estimation and the measurement of uncertainty in scientific fields like astronomy and geodesy.1