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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