defines a point P = (x1, x2, x3) in R3. 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: whose coordinates are the mean of the values we started out with. is on therefore for some . The line is to be orthogonal to the vector from to . Therefore: 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) is equal to the standard deviation of the vector (x1, x2, x3), multiplied by the square root of the number of dimensions of the vector (3 in this case). Chebyshev's inequality An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality ensures that, for all distributions for which the standard deviation is defined, the amount of data within a number of standard deviations of the mean is at least as much as given in the following table. Rules for normally distributed data The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of where μ is the expected value of the random variables, σ equals their distribution's standard deviation divided by n1/2, and n is the number of random variables. The standard deviation therefore is simply a scaling variable that adjusts how broad the curve will be, though it also appears in the normalizing constant. If a data distribution is approximately normal, then the proportion of data values within z standard deviations of the mean is defined by: where is the error function. The proportion that is less than or equal to a number, x, is given by the cumulative distribution function: . If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, μ ± σ, where μ is the arithmetic mean), about