APPENDIX
Calculation of twosided multipliers (M
_{
UL
}
) by Howe's method and onesided multipliers (MU/ML) by Natrella's method
Howe's method calculates a variable M_{UL} to provide us with a level of confidence of C% that the limits (x mean – M_{UL} * S) and (x mean + M_{UL} * s) will contain D% of the distribution.
The sigma multiplier M_{UL} is given by
in which N is the number of values in the sample, Z_{(1 – D)/2} is the critical value of the Normal distribution that is exceeded with probability (1 – D)/2 and X
^{
2
}
_{
(N – 1),C
}is the critical value of the chisquare distribution with (N – 1) degrees of freedom that is exceeded with probability C%.
Equation [1] looks complex, and the statement, "We are 99% confident that 95% of the measurements will fall within the calculated
tolerance limits" requires a bit of thought, but it is not difficult to calculate M_{UL} using Excel. For example, if cell J24 contains N, if C = 99% and D = 99.25%, the value of M_{UL} is given by
= SQRT((J24 – 1)*(1 + (1 / J24))* (NORMINV(0.00375,0,1)^2) / (CHIINV(0.99,J24 – 1)))........... [2]
Some values of the sigma multipliers calculated by Howe's method appear in Table 1. The entries in Table 1 were calculated
with C set to 99% and D set to 99.25%.
These values of C and D were selected by noting that the usual practice for specifications calculated from the mean (x mean)
and standard deviation (s) is to set the limits at +/– 3 times the standard deviation either side of the mean. C and D were
thus selected so that the twosided tolerance interval limits would be (x mean – 3 * s) and (x mean + 3 * s) when the sample
size is around 250.
Natrella's method for probabilistic onesided tolerance intervals calculates variables M_{U} and M_{L} to provide us with a level of confidence of C% that D% of the values will be less than an upper limit of (x mean + M_{U} * s) or that D% of the values will be more than a lower limit of (x mean – M_{U} * s). The calculations are the same for M_{U} and M_{L}
and N is the number of values in the sample, ZD is the critical value of the Normal distribution that is exceeded with probability
D and ZC is the critical value of the Normal distribution that is exceeded with probability C.
M_{U} can be calculated using Excel. For example, if cell B4 contains N, if C = 99% and D = 99.625%, the values of a and b are
given by:
a = 1 – (NORMINV(0.99,0,1))^2 / (2*(B4 – 1))........... [6]
and
b = (NORMINV(0.99625,0,1))^2 – (NORMINV(0.99,0,1))^2 / B4........... [7]
