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The Pythagorean theorem says that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares on the legs. The sum of squares is not factorable. The squared Euclidean distance between two points, equal to the sum of squares of the differences between their coordinates
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. [1] There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics.
In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable. [ 1 ] It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean of the data), [ 2 ...
In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, y i , i ≤ n {\displaystyle y_{i},i\leq n} , it is defined as the sum over all squared differences between the observations and their overall mean y ...
In statistics, the 68–95–99.7 rule, also known as the empirical rule, and sometimes abbreviated 3sr or 3 σ, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: approximately 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean ...
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 (x 1, x 2, x 3), multiplied by the square root of the number of dimensions of the vector (3 in this case).
More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function. There is also a harmonic average of functions and a quadratic average (or root mean square) of ...
In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being unbiased; that is, the expected value of the estimate is equal to the true value of the underlying parameter.