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Supplement to Chapter 17 Statistical process control (SPC) 525
Table S17.1 Type I and type II errors in SPC
Actual process state
Decision In control Out of control
Stop process Type I error Correct decision
Leave alone Correct decision Type II error
Upper control limit
Lower control limit
are to the population mean, the higher the likelihood of investigating and trying to rectify a
process which is actually problem-free. If the control limits are set at two standard deviations,
the chance of a type I error increases to about 5 per cent. If the limits are set at one standard
deviation then the chance of a type I error increases to 32 per cent. When the control limits
are placed at ±3 standard deviations away from the mean of the distribution which describes
‘normal’ variation in the process, they are called the upper control limit (UCL) and lower
control limit (LCL).
Critical commentary
When its originators first described SPC more than half a century ago, the key issue was only
to decide whether a process was ‘in control’ or not. Now, we expect SPC to reflect common
sense as well as statistical elegance and promote continuous operations improvement.
This is why two (related) criticisms have been levelled at the traditional approach to SPC.
The first is that SPC seems to assume that any values of process performance which lie
within the control limits are equally acceptable, while any values outside the limits are not.
However, surely a value close to the process average or ‘target’ value will be more acceptable
than one only just within the control limits. For example, a service engineer arriving
only 1 minute late is a far better ‘performance’ than one arriving 59 minutes late, even if the
control limits are ‘quoted time ± one hour’. Also, arriving 59 minutes late would be almost
as bad as 61 minutes late! Second, a process always within its control limits may not be
deteriorating, but is it improving. So rather than seeing control limits as fixed, it would be
better to view them as a reflection of how the process is being improved. We should expect
any improving process to have progressively narrowing control limits.
Quality loss function
The Taguchi loss function
Genichi Taguchi proposed a resolution of both the criticisms of SPC described in the critical
commentary box. 12 He suggested that the central issue was the first problem – namely that
the consequences of being ‘off-target’ (that is, deviating from the required process average
performance) were inadequately described by simple control limits. Instead, he proposed a
quality loss function (QLF) – a mathematical function which includes all the costs of poor
quality. These include wastage, repair, inspection, service, warranty and generally, what he
termed, ‘loss to society’ costs. This loss function is expressed as follows:
L = D 2 C
where
L = total loss to society costs
D = deviation from target performance
C = a constant.
Target-oriented quality
Figure S17.5 illustrates the difference between the conventional and Taguchi approaches
to interpreting process variability. The more graduated approach of the QLF also answers the
second problem raised in the critical commentary box. With losses increasing quadratically
as performance deviates from target, there is a natural tendency to progressively reduce process
variability. This is sometimes called a target-oriented quality philosophy.