6.3.1. What are Control Charts?

The control limits as pictured in the graph might be 0.001
probability limits. If so, and if chance causes alone were
present, the probability of a point falling above the upper limit
would be one out of a thousand, and similarly, a point falling
below the lower limit would be one out of a thousand. We would be
searching for an assignable cause if a point would fall outside
these limits. Where we put these limits will determine the
risk of undertaking such a search when in reality there is no
assignable cause for variation.

Since two out of a thousand is a very small risk, the 0.001 limits
may be said to give practical assurances that, if a point falls
outside these limits, the variation was caused be an assignable
cause. It must be noted that two out of one thousand is a purely
arbitrary number. There is no reason why it could not have been set to
one out a hundred or even larger. The decision would depend on the
amount of risk the management of the quality control program is
willing to take. In general (in the world of quality control) it
is customary to use limits that approximate the 0.002 standard.

Letting X denote the value of a process characteristic, if
the system of chance causes generates a variation in X that
follows the normal distribution, the 0.001 probability limits will
be very close to the 3σ

limits. From normal tables we glean that the 3σ

in one direction is 0.00135, or in both directions 0.0027. For
normal distributions, therefore, the 3σ

limits are the practical equivalent of 0.001 probability limits.