Pourquoi la Moyenne Minimise la Somme des Carrés des Écarts ? (Démonstration Statistique)

This source offers a mathematical proof explaining why the mean is the value that minimizes the sum of squared deviations in a data set. The author first uses a numerical and graphical approach to illustrate that the resulting function takes the form of a parabola whose vertex corresponds to the mean. He then proceeds with a rigorous algebraic proof by expanding the analytical expression to reach a second-degree polynomial form. By calculating the x-coordinate of the vertex using the formula -b/2a, he confirms that this critical point is precisely equivalent to the arithmetic mean. This presentation also allows us to define the variance as the minimum value reached by this specific function. Explain to me why the mean minimizes the sum of squared deviations. How is the variance function related to a parabola? What are the key steps in the mathematical proof presented?