APPENDIX

Calculation of two-sided multipliers (M _UL ) by Howe's method and one-sided 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.

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 chi-square 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 two-sided 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 one-sided 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]