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Many computer languages support the ISO C standard function hypot(x,y), which returns the value above. [11] The function is designed not to fail where the straightforward calculation might overflow or underflow and can be slightly more accurate and sometimes significantly slower. Some languages have extended the definition to higher dimensions.
Both the "compatibility" function STDEVP and the "consistency" function STDEV.P in Excel 2010 return the 0.5 population standard deviation for the given set of values. However, numerical inaccuracy still can be shown using this example by extending the existing figure to include 10 15 , whereupon the erroneous standard deviation found by Excel ...
The hypotenuse c of a right triangle with sides a and b is the Pythagorean sum of a and b.. In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides.
An alternative formulation avoiding this problem (Golub & Van Loan 1996, §5.1.8) is implemented as the hypot function in many programming languages. The following Fortran code is a minimalistic implementation of Givens rotation for real numbers.
from the formula for the tangent of the difference of angles. Using s instead of r in the above formulas will give the same primitive Pythagorean triple but with a and b swapped. Note that r and s can be reconstructed from a , b , and c using r = a / ( b + c ) and s = b / ( a + c ) .
All superlative indices produce similar results and are generally the favored formulas for calculating price indices. [14] A superlative index is defined technically as "an index that is exact for a flexible functional form that can provide a second-order approximation to other twice-differentiable functions around the same point." [15]
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...