Derivation of Probability pab

Source: http://mathworld.wolfram.com/ConfidenceInterval.html, reference AB30

Assuming Normal Distribution of results:

Pxu = ERF(n/SQRT(2))

where

• Pxu = probability result lies within ± n SD of mean
• ERF = error function
• n = specified number of SD
• SD = Standard Deviation

If, instead of the difference between one result and the mean, we have the difference between any two results then we must use Pab instead of Pxu:

Pab (SQRT(n^2 + n^2)) = ERF(n/SQRT(2)) Taking the RMS value of the two errors

Pab (SQRT(2)n) = ERF(n/SQRT(2)) Simplifying LHS

Pab = ERF(n/2) Subst SQRT(2)n for n in RHS

where

• Pab = probability that two results lie within n SD of each other, as in figure A

Substituting #SD for n and taking absolute value of # SD:

Pab = ERFC(ABS(#SD/2))

where

• Pab = probability two results lie within # SD each other = area under curve in figure B
• # SD = measured difference between two results

Taking complement of error function to select area outside of confidence interval:

pab = ERFC(ABS(#SD/2))

where

• pab = probability two result lie outside # SD each other = area under curve in figure C
• ERFC = complimentary error function = 1 - ERF

pab = ERFC(ABS(#SD/2))

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