PRODUCTS WITH MULTIPLE CHARACTERISTICS

Extremely low ppm levels are imperative for producing high quality products possessing many characteristics (or components ). Table I.1 compares the probability of manufacturing a product with all characteristics inside their respective specifications when each is produced with ±4 sigma (C _p = 1.33) versus ±6 sigma (C _p = 2.00) capability. The processes producing the features are assumed to be dynamic, with up to a 1.5-sigma shift in average possible.

Table I.1: Probability of a Completely Conforming Product

	With 1.56 Shift
Number of Characteristics	C, = 1.33 ( ±46)	C,. = 2.00 ( ±6a)
1	99.3790	99.99966
2	98.7618	99.99932
5	96.9333	99.9983
10	93.9607	99.9966
25	85.5787	99.9915
50	73.2371	99.9830
100	53.6367	99.9660
150	39.2820	99.9490
250	21.0696	99.9150
500	4.4393	99.8301

Suppose a product has only one feature, which is produced on a process having ±4 sigma potential capability. We can then calculate that a maximum of .6210 percent of these parts will be non-conforming under the dynamic model. Conversely, at least 99.3790 percent will be conforming, as is listed in the first line of Table I.1. If this single characteristic is instead produced on a process with ±6 sigma potential capability, at most .00034 percent of the finished product will be out of specification, with at least 99.99966 percent within specification.

If a product has two characteristics, the probability that both are within specification ( assuming independence) is .993790 times .993790, or 98.7618 percent when each is produced on a ±4 sigma process. If they are produced on a ±6 sigma process, this probability increases to 99.99932 percent (.9999966 times .9999966). The remainder of the table is computed in a similar manner.

When each characteristic is produced with ±4 sigma capability (and assuming a maximum drift of 1.5 sigma), a product with 10 characteristics will average about 939 conforming parts out of every 1000 made, with the 61 nonconforming ones having at least one characteristic out of specification. If all characteristics are manufactured with ±6 sigma capability, it would be very unlikely to see even one nonconforming part out of these 1000.

For a product having 50 characteristics, 268 out of 1000 parts will have at least one nonconforming characteristic when each is produced with ±4 sigma capability. If these 50 characteristics were manufactured with ±6 sigma capability, it would still be improbable to see one nonconforming part. In fact, with ±6 sigma capability, a product must have 150 characteristics before you would expect to find even one nonconforming part out of 1000. Contrast this to the ±4 sigma capability level, where 60.7 percent of these parts would be rejected, and the rationale for adopting the six sigma philosophy becomes quite evident.