Replacing once again we find the solution: Using a square root calculator we find √2 equals 1.4142 and using this calculator we find erf -1(0.8) = 0.9062. Knowing that the top decile represents the top 10 percent, we need to find the cut-off at 90%, or 0.9. For example, if we want to find the cut-off for the top decile of a normal distribution with mean μ = 0 and standard deviation σ = 3, we would use the equation: The inverse error function can be used to compute the quantile function (inverse distribution function) of a normal distribution. Table of commonly used numbers: Common values of erf(x) and erfc(x) x Error function tableĪn erf table contains tabulated values of real numbers and their corresponding error function values. The solution can easily be found by various root-finding methods. While a true inverse function would be multivalued and thus would not have a unique solution, for values of y between −1 and 1 there is a unique real number solution to the equation: The inverse error function, denoted erf -1(y) takes as input the result of y = erf(x), and produces the corresponding x value. Its solution is a simple subtraction from one. The equation for the complementary error function is given by: In this calculator we use a polynomial approximation with a maximal error of 1.2 × 10 7 for any real argument as per reference. The equation has no closed-form solution and various approximations are in use. The formula can therefore be expressed by the following integral equation: To calculate erf(x) one performs an integration from minus infinity to x of the equation e -t 2. Since e -t 2 is an even function and erf(-x) = -erf(x), the error function is an odd function. The function plot illustrates its sigmoid shape: the Gauss error function or just erf, is a complex function of a complex variable defined as :Īside from applied mathematics where it is used to solve differential equations and in physics in solutions of the heat equation in the case where boundary conditions are given by the Heaviside step function, it also sees use in statistics where the inverse error function is used in the calculation of critical values, p-values, confidence intervals which are all related to statistical hypothesis testing and estimation. In mathematics and statistics, the error function a.k.a. The input y needs to be a real number between minus one and one (y ∈ ). In inverse error function mode the output contains the inverse of erf and its complement. erf(x) returns a result between zero and one for any real value of x. ![]() The output also contains the complementary error function for the same number as well as a function plot showing where erf(x) lies relative to other possible function values. The erf calculator can be used to compute the error function of any number on the real line.
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