Measurement Uncertainty effects on Process Capability

[Daniel]

Hello Everybody!

This seems to me a very simple question, so I'm frustrated because I
couldn't find a clear answer so far.

Introduction:

I work in the field of Quality Engineering. As you may know, every
measurement has some uncertainty associated with it, usually described
as a standard deviation (u) that produces a "confidence interval"
around the value measured.

On the other side, the concept of Process Capability refers to the
total variability of a certain process, calculated from a group of
measurements, and how this variability relates to the tolerance of the
product. Usually this relationship is expressed as a Quality Index
called Cp. The formula for Cp is:

Cp=(Tolerance) / (6 sigma )

The denominator is six times the value of estimated total standard
deviation, representing the "process width".

Question:

It is argued that as the "sigma" in the denominator contains the
uncertainty of the measurements, that are exogenous to the process,
"inflating" therefore the process variability with a variance component
u^2. Thus, in order to find the true "process variability", without the
influence of the uncertainty in the measurements, we should "deflate"
sigma in the denominator, using sqr(sigma^2 - u^2)

1) So according to this the <process width> should be considered
"shorter" due to the effect of uncertainty in measurements.

but...

If I look at the histogram, made up with the individual results of the
data set, each one of the extreme values should be considered V +/- 3u,
so actually the histogram representing the process will be wider (3u on
each side).
Widening the histogram this way, could intuitively represent the
confidence interval of what I can consider the process variability.

2) So according to this the <process width> should be considered
"wider" due to the effect of uncertainty.

(1) and (2) are contradictory, but if I look at the arguments, I can't
find the flaw that makes one of them the valid answer. Can you help me?

Thanks
Daniel

[Scott Brogan]

(1) is correctly reasoned & intuition is at fault in (2). If I
understand you right I think I know why. Measurement error already
forms part of the results. When you take a data point at random, then
nearly all the time its real value will lie +/- 3u on each side of the
measured value : when you take 100 data points at random and pick out
the maximum, then more often its real value will lie below the measured
value-rarely will the maximum have a negative measurement error;
commonly it will be the maximum precisely because it has a large
positive one. It's picking out extreme points & considering them as
typical that misleads intuition. If I haven't explained this well look
up regression towards the mean or do a little simulation.

[Daniel]

Thanks Scott ! Excellent !

What I find interesting is that the probable value of the extreme point
will depend on my knowledge of how many points I already have in the
data set.

question: So the probable width of the histogram should be reduced as
the dataset size increases, and maybe this "width reduction" is related
to the formula in (1) sqr(sigma^2 - u^2) ?

Until now, I have allways regarded Regression in the usual "lineal
models" arena, and this is the first time I find the use in a way
related to the original meaning by which Galton defined it.

Can you point to a book with a good explanation of this concept and its
implications for the analysis of groups of data?

Thanks again !

.

Or even more: the actual width could be considered

[Scott Brogan]

If that seems odd, remember that classical statistics is concerned with
procedures having defined success rates, not with probabilities of the
true values (which are not random variables, but fixed & unknown).
"Take a random sample from a Normal distribution & calculate +3u & -3u
from that value" is one procedure; "Take n random samples, find the
maximum & calculate +3u & -3u from that value" is quite another :
while the first covers the true value of the data point 99.7% of the
time, the second does not.

I didn't quite understand your question about the probable/actual width
of histograms.

Regression towards the mean gets a mention in Lindgren, & I'm sure is
covered in many books, but you might as well Google for it. A nice
interactive simulation is here :
http://www.ruf.rice.edu/~lane/stat_sim/reg_to_mean/