Process Capability Analysis: Normal Plotting and Capability Ratios, Exams of Statistics

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IE 361 Module 17

Process Capability Analysis: Part 1

Prof.s Stephen B. Vardeman and Max D. Morris

Reading: Section 5.1, 5.

Statistical Quality Assurance Methods for Engineers

Normal Plotting for Process Characterization If (by virtue of process monitoring and wise intervention) one is willing to saythat a data set represents a

stable process

, it may be used to characterize

process output.

Section 5.1 discusses several graphical techniques for summa-

rizing a sample and therefore representing the process that stands behind it.Here we emphasize one of these, so called "normal plotting," a tool for invesit-gating the extent to which a data set (and thus the process that produced it)can be described using a normal distribution.Normal plots are made using so called

quantiles

The

p

quantile (or

×

p

th

percentile) of a distribution is a number such that a fraction

p

of the distribution

lies to the left and a fraction

p

lies to the right.

If one scores at the.

quantile (80th percentile) on an exam, 80% of those taking the exam had

(Standard normal quantiles

Q

z

p

can be found by locating values of

p

in

the body of a typical cumulative normal probability table and then readingcorresponding quantiles from the table’s margin. And statistical packages like JMP

provide "inverse cumulative probability" functions and "normal plotting"

functions that can be used to automate this.)

This plot allows comparison

of data quantiles and (standard) normal ones.

A "straight line" normal plot

indicates that a data set has the same shape as the normal distributions, andsuggests that the process that stands behind the data set can be modeledas producing normally distributed observations.

(Section 5.1 has a careful

discussion of interpretation of such

Q

Q

plots for those who need a review of

this Stat 231 material.) Example 17-

Table 5.7 of

SQAME

contains measured "tongue thicknesses"

for

n

steel levers.

Figure 1 shows a

JMP

report including a normal plot

for the data of Table 5.7.

Figure 1:

JMP

Report for the Data of Table 5.7 of

SQAME

Including a Normal

Plot

exactly normal distribution, the slope of the plot is the reciprocal of

σ

and the

horizontal intercept is

μ

. That suggests that for a real data set whose normal

plot is fairly linear,

1. the horizontal intercept of an approximating line is a sensible estimate of

the mean of the process generating the data, and

2. the reciprocal of the slope is a sensible estimate of the standard deviation

of the process generating the data.

The facts that (for bell-shaped data sets) normal plotting provides a simple wayof approximating a standard deviation and that

σ

is often used as a measure

of the intrinsic spread of measurements generated by a process, together lead to

the common practice of basing

process capability analyses

on normal plotting.

The next

fi

gure shows a very common type of industrial form that essentially

facilitates the making of a normal plot by removing the necessity of evaluatingthe standard normal quantiles

Q

z

p

. (On the special vertical scale one may

simply use the plotting position

p

rather than

Q

z

p

, as would be required

when using regular graph paper.) After plotting a data set and drawing in anapproximating straight line,

σ

can be read o

ff

the plot as the di

ff

erence in

horizontal coordinates for points on the line at the "

σ

" and "

σ

" vertical

levels (i.e., with

p

and

p

Forms like the one in the

fi

gure encourage the

plotting

of process data (always

a plus) and also allow even fairly nonquantitative people to easily estimate anddevelop some intuition about "the process spread." Process Capability Measures and Their Estimation Graphical methods provide a visual representation of the pattern of variationassociated with a process. Often, in addition to these tools, it is convenient tohave some numerical "capability index" summary measures to quote.We discuss the "process capability" and the two "capability ratios,"

C

p

and

C

pk

, and methods for making con

fi

dence intervals for them. But it is important

to begin with a disclaimer:

Unless a normal distribution makes sense as a

description of process output, these measures are of dubious relevance. Further,the con

fi

dence interval methods presented here are completely unreliable unless

a normal model is appropriate. So the normal plotting idea just presented is avery important prerequisite for using these methods. It is well known that the majority of a normal distribution is located within threestandard deviations of its mean. The following

fi

gure illustrates this elementary

point, and in light of the picture, it makes some sense to say that (for a normaldistribution)

σ

is a measure of process spread, and to call

σ

the

process

capability

for a stable process generating normally distributed measurements.

The fact that there are methods for estimating the standard deviation of anormal distribution implies that it is easy to give con

fi

dence limits for the process

capability. That is, if one has a sample of

n

observations with corresponding

sample standard deviation

s

, then con

fi

dence limits for

σ

are simply 6 times

the limits for

σ

(met

fi

rst in Stat 231 and used in this course beginning already

in Module 2) namely

s

vuut

n

χ

2 upper

and/or

s

vuut

n

χ

2 lower

where

χ

2 upper

and

χ

2 lower

are upper and lower percentage points of the

χ

distribution with

n

degrees of freedom.

Example 17-

An IE 361 group did some measuring of angles with a

fl

at

surface made in the EDM drilling of holes on a high precision metal part.

The

n

data values they collected are on page 209 of

SQAME

The sample

mean for these data is

¯x

◦

and the sample standard deviation is

s

◦

The next

fi

gure is a

JMP

report for these data.

It includes a

normal plot for the data that (as it is very linear) indicates that a normal modelfor angles produced by this process is quite sensible.

It also includes 95%

con

fi

dence limits for

σ

, namely

◦

and

◦

These limits translate to limits

◦

and

◦

for the "process capability."

Where there are both an upper speci

fi

cation

U

and a lower speci

fi

cation

L

for measurements generated by a process, it is common to compare processvariability to the spread in those speci

fi

cations. One way of doing this is through

process capability ratios.

And a popular process capability ratio is

C

p

U

L

σ

When this measure is 1, process output will

fi

t more or less exactly inside

speci

fi

cations

provided the process mean is exactly on target at

U

L

When

C

p

is larger than 1, there is some "breathing room" in the sense that a

process would not need to be perfectly aimed in order to produce essentially allmeasurements inside speci

fi

cations. On the other hand, where

C

p

is less than

1, no matter how well a process producing normally distributed observations isaimed, a signi

fi

cant fraction of the output will fall outside speci

fi

cations.

The very simple form of

C

p

makes it clear that once one knows how to estimate

σ

, one may simply divide the known di

ff

erence in speci

fi

cations by con

fi

dence

limits for

σ

in order to

fi

nd con

fi

dence limits for

C

p

. That is, lower and upper

con

fi

dence limits for

C

p

are respectively

U

L

s

vuut

χ

2 lower

n

and/or

U

L

s

vuut

χ

2 upper

n

where again

χ

2 upper

and

χ

2 lower

are upper and lower percentage points of the

χ

2

distribution with

n

degrees of freedom.

Example 17-1 continued

Speci

fi

cations on the angles in the EDM drilling

application were

◦

◦

. That means that for this situation

U

L

◦

Based on the measurement of

n

parts, the students found

s

and

95% two-sided con

fi

dence limits for

σ

of

◦

and

◦

Thus, one can

be 95% con

fi

dent that

C

p

is between

and

Another process capability index that does take account of the process mean(and is more a measure of current process performance than of potential per-formance) is

C

pk

. This measure can be described in words as "the number of

σ

’s that the process mean is to the good side of the closest speci

fi

cation."

For example, if

U

L

is

σ

, and

μ

is

σ

below the upper speci

fi

cation, then

C

pk

is

σ/

σ

. On the other hand, if

U

L

is

σ

and

μ

is

σ

above

the upper speci

fi

cation, then

C

pk

is

In symbols,

C

pk

= min

½

U

μ

σ

μ

L

σ

¾

U

L

¯¯¯

μ

U

L

2

¯¯¯

σ

This quantity will be positive as long as

μ

is between

L

and

U

. It will be large

if

μ

is between

L

and

U

(preferably centered between them) and

U

L

is

large compared to

σ

It is always true that

C

pk

C

p

and the two measures are equal only when

μ

U

L

exactly.

The best currently available con

fi

dence interval method for

C

pk

is only appropri-

ate for large samples and provides a real con

fi

dence level that only approximates

the nominal one. The method is based on the natural single number estimateof

C

pk

C

pk

= min

½

U

¯x

s

¯x

L

s

¾

U

L

¯¯¯

¯x

U

L

2

¯¯¯

s

Then for

z

an appropriate standard normal upper percentage point, approximate

con

fi

dence limits for

C

pk

are

C

pk

z

vuut

n

C

(^2) pk

n