Calculation of two-sided multipliers (M
) by Howe's method and one-sided multipliers (MU/ML) by Natrella's method
Howe's method calculates a variable MUL to provide us with a level of confidence of C% that the limits (x mean – MUL * S) and (x mean + MUL * s) will contain D% of the distribution.
The sigma multiplier MUL 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
(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  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 MUL using Excel. For example, if cell J24 contains N, if C = 99% and D = 99.25%, the value of MUL is given by
= SQRT((J24 – 1)*(1 + (1 / J24))* (NORMINV(0.00375,0,1)^2) / (CHIINV(0.99,J24 – 1)))........... 
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 MU and ML 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 + MU * s) or that D% of the values will be more than a lower limit of (x mean – MU * s). The calculations are the same for MU and ML
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.
MU 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
a = 1 – (NORMINV(0.99,0,1))^2 / (2*(B4 – 1))........... 
b = (NORMINV(0.99625,0,1))^2 – (NORMINV(0.99,0,1))^2 / B4...........