information:extraction, linguistic:translation
Geometric interpretation

To gain some geometric insights and clarification, we will start with a population of three values, x, x, x. This defines a point P = (x, x, x) in R. Consider the line L = {(r, r, r) : r ∈ R }. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and P would lie on L. So it is not unreasonable to assume that the standard deviation is related to the distance of P to L. That is indeed the case. To move orthogonally from L to the point P, one begins at the point:

M = (x ¯, x ¯, x ¯) {\displaystyle M=\left({\bar {x}},{\bar {x}},{\bar {x}}\right)}

whose coordinates are the mean of the values we started out with.

A little algebra shows that the distance between P and M (which is the same as the orthogonal distance between P and the line L) ∑ i (x i − x ¯) 2 {\textstyle {\sqrt {\sum _{i}\left(x_{i}-{\bar {x}}\right)^{2}}}} is equal to the standard deviation of the vector (x, x, x), multiplied by the square root of the number of dimensions of the vector (3 in this case).