makes a positive contribution to S/N_{T }when
_{ }is less than 1.0, and a negative contribution when it is greater than 1.0. Thus, high values of S/N_{T} require the mean to be "ontarget" and the variation between replicate measurements to be small.
Because these data have large differences between means and small differences between replicates,
can be expected to dominate the second term. The differences between the mean and the target range from –2.7 to +0.5, and
the values of S_{x} range from 0.4 to 1.4. If S_{x}
^{2} = 0.5, the difference between the mean and the target has to be less than 0.7 for the second term to make a positive contribution.
Because
_{ }has a large effect on the value of S/N_{T}, there is little difference between the two control charts in Figures 1 and 2. There is only a small chance that the chart
of Xbar will lead to the rejection of good batches that could be prevented by using a S/N_{T} chart. From Equation (2), we can see that high values of S/N_{T}, particularly those that exceed the upper limit of the control chart, are "good" because they result from means that are
on target with very small differences between the replicates.
As McAllister pointed out, signalto noise ratios are particularly useful in robust engineering, where the goal is to design
products and processes that consistently deliver "on target." But combining control chart information on replicate variability
with the extent to which the mean is offtarget is not necessarily beneficial. A very high or low batch mean indicates a shift
in the process, which could result from equipment, raw material, or the operator. A high range or standard deviation is more
likely to result from a large measurement error. The combined information in an S/N_{T} chart can tell us that there is a problem, but identifying the possible causes requires an examination of the mean and standard
deviation used to calculate S/N_{T}.
We also need to be aware that when individual responses or measurements are combined, it is possible for opposite differences
to cancel and for one of the terms to dominate. The dominance of the
term in S/N_{T} makes it impossible to detect repeatability problems, and some of the acceptable S/N_{T} values may have very high standard deviations of the replicates. This is easy to demonstrate using Equation (2). For example,
the very low S/N_{T} of 13.97 for McAllister's batch EE results from a mean of 83.67 and a range of 2. We can calculate that a range of 0 for
batch EE would only increase S/N_{T} to 13.98. However, a mean around the overall average of 93.8 and a range of 12, fifteen times the average, will result in
S/N_{T }= 19.67, a value so far within the limits that no warning bell would ring.
