| The Gaussian probability density function with mean = 0 and variance =1 is |
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| The error function Erf(x) is defined as: |
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Note that Erf(0) = 0.5, and that Erf∞)=1. |
The complimentary error function Erfc(x) is defined as: |
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The following graph illustrates the region of the normal curve that is being integrated. |
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For large values of x (>3), the complimentary error function can be approximated by: |
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The error in the approximation is about -2% for x=3, and -1% for x=4, and gets progressively better with larger values of x. |
An even closer approximation (about 10x better) is: |
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In standard statistics texts, the error function is typically defined as (note lower case “e”): |
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The relationship between erfc(x) and Erfc(x) is as follows: |
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The following chart illustrates how the Erfc(x) function and the closer approximation formulas converge for larger values of “x.” The chart was generated using the Analysis ToolPak Add-In in Microsoft Excel for the Erfc(x) function (and the modification for the communications version as shown above) and entering the approximation formula in adjacent cells. |
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