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# Error Function Calculator Erf(x)

The error function, denoted erf, is defined by the integral

erf(x) = (2/√π)∫^{x}_{o} e^{-t2} dt.

Erf(x) is closely related to the normal probability curve; the cumulative distribution function of a normally distributed random variable X is

CDF(X) = 0.5 + 0.5erf[(X-μ)/σ√2],

where μ is the mean and σ is the standard deviation of the distribution. The error function integral cannot be evaluated in terms of elementary function, so one must use numerical algorithms. The error function is an odd function whose limit is -1 for negative values of x, and 1 for positive values of x. The function rapidly converges to its asymptotic values; erf(3) = 0.99998 and erf(-3) = -0.99998.

### Properties and Equations

The values of x for which x = erf(x) are approximately 0.6175 and -0.6175. If you don't have access to an error function calculator such as the one above, you can approximate the function with the formula The error function can also be expressed with infinite series and continued fractions: You can approximate the inverse of the error function, erf^{-1}(x), by inverting the approximation formula above. The inverse of erf has several interesting derivative and integral properties.

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