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c© Heldermann Verlag Economic Quality ControlISSN 0940-5151 Vol 16 (2001), No. 1, 49 – 63

Statistical Process Control Methods from the

Viewpoint of Industrial Application

Constantin Anghel

Abstract: Statistical process control is a major part of industrial statistics and consists notonly of control charting, but also of capability analysis, of design of experiments and otherstatistical techniques. This paper reviews and comments some available techniques for theunivariate as well as for the multivariate case from the viewpoint of industrial application.

1 Distribution of a Quality Characteristic

Many statistical process control activities assume that the quality characteristic understudy exhibits a stable performance or in other words that the process is stable and capa-ble. The stability of a process can be defined as the stability of the underlying probabilitydistribution over time and very often this can be described as the stability of the distri-bution parameters over time. Only if the stability assumption is met by the process, thecalculation of capability indices is meaningful and may be used in practice for evaluat-ing process performance. Mathematically, we can, of course, calculate always capabilityindices, but for an unstable process these indices have no real significance, because thereare not identified assignable causes in the process. Thus, a correct identification of thetype of probability distribution is not sufficient without the assurance that it is stableover time. In the case that the process is not stable, the probability distribution of thequality characteristic may vary from time-point to time-point. In such a case one coulduse a mixture of probability distributions as model, which, therefore, can be looked uponas an indication for an unstable process.

Each quality characteristic has a “proper nature” defined by physical and technical condi-tions. For instance a meaningful flatness-characteristic is defined as a parameter naturallybounded by null. In such a case the normal distribution cannot describe reality sufficientlywell and a different model has to be selected. One of many possibilities would be to selectthe folded normal distribution [1] and [8]. Of course, any model should always be justifiedby experiments, i.e. by samples. If a sample leads to reject the hypothesis of the assumedmodel, then the situation must be analyzed again, with respect to various features asthe resolution of the measurement gauge, mixture appearance, etc. Table 1 gives somerecommendations for the selection of an appropriate model for some frequently relevantquality characteristics (see also [1]).

A decision for a specific model to be used to describe the random variations exhibited by aquality characteristic, should be based on any available information about the physical and

50 Constantin Anghel

technical conditions on the one hand and on a sufficiently large sample on the other hand,collected under “ideal” conditions (one operator, one supplier, optimal resolution of thegauge). Subsequent samples of the quality characteristic should be utilized for verifyingthe hypothesis that the initial distribution has not changed. In case of a rejection of thenull hypothesis, an explanation has to be found by an additional analysis.

Table 1: Recommended Models for Various Quality Characteristics

Characteristic Model Characteristic Model

geometricdimensions:straightness folded normal parallelism folded normalevenness folded normal rectangularity folded normalround form folded normal inclination (angularity) folded normalcylinder form folded normal position eccentric Rayleighline form folded normal coaxiality (concentricity) eccentric Rayleighsurface form folded normal symmetry folded normalcircularity flatness folded normal

a) form folded normalb) position eccentric Rayleigh

life length Weibull, Hjorth resistance normalvoltage normal capacitance normalpressure normal viscosity normal

These models should be looked upon as a first and preliminary attempt which, in anycase, must be justified or discarded by a subsequent analysis of the situation at hand.

2 Data Collection and the Resolution of the Gauge

Any data collection is the result of a measurement process and, therefore, the gauge hasa high importance beginning with its resolution and its capability. A resolution below2% of the tolerance is considered as consistent with industrial practice. Above this levelthe measurement values are assigned to classes by the measurement procedure and therandom character of the sample will partly be lost. If we take samples of size two, thenthe expectation of range is 1.128 times the process standard deviation. This means thatthe expected distance between any two randomly selected measurements is about 1.1sσ,where σ denotes the process standard deviation. Whenever the difference between twosuccessive measurements is less than the resolution of the gauge, the difference is set tozero. Thus, a larger resolution yields a larger number of zeros, and consequently resultsin a too small value of the average range (process variability) leading to action limits foran X-chart as well as for an R-chart, which are too tight (Figure 1). The consequencesare too many false alarms, and often the wrong decision that the process is not stable[13].

Statistical Process Control Methods 51

1 Example Consider a quality characteristic with tolerance interval [L,U ] = [28.7, 31.3],which is evaluated by a gauge with resolution r = 0.01 (≈ 4% of the length of the toleranceinterval).

50 samples of size n = 2 are taken the results of which are represented in Table 1 andFigures 1.

Table 2: Observed values: 50 samples of size n = 2.

observed value x frequency29.98 129.99 330.00 2830.01 4130.02 2330.03 230.04 2

Figure 1: X-chart and R-chart for the 50 samples of size n = 2.

The uncertainty about the measurement result determines the gauge capability, whichoften is specified by four times the standard deviation 4σ.

Clearly, the gauge capability affects the determination of the actual value of the processcapability given by Cp. If the ratio of the measurement uncertainty (4σ) and the toler-ance U − L, with L lower tolerance limit and U upper tolerance limit, is 25%, then ameasured process capability (with measurement variability added to process variability)of 1.33 means for the actual index (without measurement variability) a value of 1.55 anda measured value of 1.67 corresponds to a real value of 2.2 (Figure 2).


Figure 2: The effect of measurement variability (uncertainty) on the determination ofprocess capability Cp.

Conversely, for a given actual process capability (without measurement variability) of1.33, the capability with added measurement variability (4σ

T= 25%) reaches only 1.2 and

an actual value of 1.67 is decreased to 1.41 (Figure 3).

Figure 3: The effect of measurement variability (uncertainty) on the verification of agiven process capability Cp.

Thus, information about gauge capability is very important to avoid false decisions withrespect to process capability and with respect to releasing false alarms.


3 Capability Indices

As mentioned above, it makes sense to define capability indices for a stable process. Ofcourse, it is possible to use the definitions and to calculate these indices, even if thedistribution of the relevant quality characteristics change in time. However, only a stableprocess assures the predictions based on the stated capability index and, therefore, usingthe knowledge of what happened today for predicting what we expect tomorrow makessense only if the situation does not change. From the viewpoint of application, it isdesirable to define capability indices being independent of the special type of distributionallowing the comparison of two or more processes with different underlying distributions.

The most widely used capability indices for the normal distribution are the following:

Cp =U − L

6σ(1)

Cpk =1

3min

(U − µ

σ,µ − L

σ

)(2)

where µ is the process mean and σ2 the process variance.

Introducing the probability pU of exceeding the upper specification limit U and pL of notreaching the lower specification limit L, another representation of the capability indicesis obtained.

pU = 1 − Φ

(U − µ

σ

), pL = Φ

(L − µ

σ

)(3)

and therefore

Cp =1

6

(Φ−1(1 − pU) − Φ−1(pL)

)(4)

Cpk =1

3min

(Φ−1(1 − pU),−Φ−1(pL)

)=

1

3min

(Φ−1(1 − pU), Φ−1(1 − pL)

)=

1

3Φ−1(1 − p) (5)

where Φ−1 denotes the inverse distribution function of the standard normal distributionand p = max(pU , pL).

The representation (4) and (5) also suggests how to define process capability indices forthe general case given by an arbitrary distribution function F (x). In the general case the

probability of exceeding the upper specification limit denoted by p(F )U and the probability

of not exceeding the lower specification limit denoted by p(F )L are given by the following

expressions:

p(F )U = 1 − F (U) , p

(F )L = F (L) (6)

Replacing pU and pL in (4) and (5) by the expressions (6) yields the following formulasfor Cp and Cpk in the general case.


Cp =1

6

(Φ−1(1 − p

(F )U ) − Φ−1(p

(F )L )

)(7)

Cpk =1

3min

(Φ−1(1 − p

(F )U ),−Φ−1(p

(F )L )

)=

1

3Φ−1(1 − p) (8)

We will call (8) the classical definition of Cpk, as it defines process capability by meansof the maximum nonconformance probability max(pU , pL) on the one or the other side ofthe specification interval.

In the relevant literature (e.g. [5]) a different proposal for defining Cp and Cpk for thegeneral case is made. The definition is based on the following representation of thedenominator of (1) which is valid under the normal model.

6σ = (3σ + µ) − (−3σ + µ) = q0.99865 − q0.00135 (9)

where q0.99865 = µ+3σ is the 0.99865-quantile and q0.00135 = µ−3σ is the 0.00135-quantileof the normal distribution N(µ, σ2). Replacing the denominators in (1) and (2) by theexpressions obtained in (9), we arrive at the following formula for the general case.

C∗p =

U − L

q(F )0.99865 − q

(F )0.00135

(10)

C∗pk = min

(U − µ

q(F )0.99865 − µ

,µ − L

µ − q(F )0.00135

)(11)

where q(F )0.99865 and q

(F )0.00135, respectively, are quantiles of an arbitrary distribution function

F (x).

Clearly in the normal case the two capability indices (11) and (8) are equivalent. However,this is not true in the general case as illustrated by the following example.

2 Example Consider a quality characteristic X with specification limits L = 1.0 andU = 15.2. Moreover, let X follow a log-normal distribution [3] given by its density functionfX(x):

fX(x) =1

σ(x − θ)√

2πe−

12(

ln(x−θ)−µσ )

2

(12)

with θ = −7.5, µ = 2.56, σ = 0.27. The expectation of X is given by

E[X] = θ + eµ+ σ2

2 (13)

and utilizing the fact that ln(X − θ) ∼ N(µ, σ2), the p-quantile of the log-normal distri-

bution q(F )p may be determined by the following formula.

q(F )p = θ + eqpσ+µ (14)

where qp is the p-quantile of the standardized normal distribution. Thus, we obtain:


Table 3: Distribution characteristic of X

characteristic actual valueprocess mean E[X] = 5.9160.99865-quantile q0.99865 = 21.5790.00135-quantile q0.00135 = −1.745

With the values given in Table 3, we obtain the capability index C∗pk

U − E[X]

q0.99865 − E[X]= 0.592 ,

E[X] − L

E[X] − q0.00135

= 0.642 (15)

and thus:

C∗pk = 0.592 (16)

where (16) refers to process quality with respect to exceeding the upper specification limitU .

Next, the classical definition of Cpk, as given by (8), is used for calculating the processcapability. To this end the probability pU of exceeding the upper specification limit andthe probability pL of not reaching the lower specification limit have to be determined.

pU = 1 − FX(U) = 1 − Φ

(ln(U − θ) − µ

σ

)= 1 − Φ(2.08) = 0.01876 (17)

pL = FX(L) = Φ

(ln(L − θ) − µ

σ

)= Φ(−1.56) = 0.05938 (18)

The probability of not reaching the lower specification limit is more than three timesthe probability of exceeding the upper specification limit. Thus, process capability withrespect to meeting the upper specification is much better than process capability withrespect to meeting the lower specification and this fact should be reflected by any mean-ingful capability index. Unfortunately, as shown by (15), C∗

pk violates this requirement.

For the classical capability index Cpk, we obtain:

Φ−1(1 − pU) = Φ−1(0.98124) = 2.08 (19)

Φ−1(1 − pL) = Φ−1(0.94062) = 1.56 (20)

and therefore

Cpk = 0.520 (21)

which refers to the process capability of meeting the lower specification limit L.


It is generally accepted in industry that the requirement for the classical Cpk in thenormal case should be Cpk ≥ 1.33 with the meaning that on an average there are notmore than 31.7 ppm of nonconforming items produced on either side of the specificationinterval. Taking C∗

pk with C∗pk ≥ 1.33 in the case of an exponentially distributed quality

characteristic may lead to an out of specification proportion of 212 ppm on one side ofthe specification interval, which is about seven times the tolerated number. Therefore,C∗

pk should not be used in the non-normal case.

4 Median Versus Expectation

For non-normal distributions it may be more appropriate to use the median rather thanthe expectation for controlling the process location, since the expectation of a non-normaldistribution does not necessarily correspond to a quantile of fixed order as in the case ofthe normal distribution (50%). Figure 5 shows the action lines and the operation of 500simulated samples of size n = 3 of an X-charts for an exponential distribution with =0.3and action limit (99%) determined according to a Shewhart-chart and determined basedon the distribution of the sample mean. For either chart a number of false alarms occurwhich is considerably larger for the Shewhart chart.

Figure 4: Performance of X-charts for controlling the process mean in the exponentialcase.

Figure 5 shows the operation of a median chart for the same simulated samples. As canbe seen, there is no false alarm for the median chart.


Figure 5: Performance of median chart for controlling the process mean in theexponential case.

As can be seen from Figure 5, there is no false alarm for the median chart.

Schneider et al.[11] pointed out that for the traditional three-sigma control limit (She-whart) the probability for a false alarm is 0.27% for the normal distribution but 1.8% foran exponential distribution with µ = 1 (Figure 6).

Figure 6: False alarm probability using three-sigma control limit in the normal and theexponential case.

During process control with control charts it is not necessary to check the distribution andto calculate capability indices as long as the control charts do not indicate a change. Thedistribution as well as the capability indices as determined at the latest process capabilitystudy of the output characteristic are valid until an alarm is released and the process isanalyzed and a new process capability study is carried out.


5 Multivariate Process Analysis

In industrial practice product quality is generally determined not by only one characteris-tic but by a vector of several characteristics which may be correlated. For such situationsit is generally not appropriate to decide about each variable, or to use quality charts, orto calculate capability indices for each characteristic separately by means of the marginaldistributions and thus neglecting the correlations. Therefore, multivariate techniques arenecessary taking into account the correlations between quality characteristics for calcu-lating quality charts, tolerances (see e.g. [6]), and capability indices.

Figure 8 and 9 illustrate independent control of two variables with respect to locationand variability compared with a joint control (Figure 9) using quality charts for themultivariate mean variation. The better performance of the joint charts is evident.

Figure 7: Independent control of locations of the two quality characteristics X, Y .

Figure 8: Independent control of variability of two quality characteristics X, Y .

Figure 9: Joint control of location of (X,Y ) and variability of (X,Y ).

5.1 Reduction of a Multivariate to a Univariate Case

For some problems is possible to reduce the multivariate case to a one-dimensional one, sothat the established one-dimensional capability indices can be used or the correspondingone-dimensional control charts can be implemented for process control.


3 Example Consider the location of the center of a hole defined by its 2-dimensionalcoordinates (X,Y ). Assume that X and Y are independent and normally distributedrandom variables with the same variance σ2.

Let the deviation of X and of Y from given targets x0 and y0 be the quality characteristics.Setting x0 = 0 and y0 = 0 leaves the two-dimensional quality characteristic (X,Y ).

The pair (X,Y ) has a two-dimensional normal distribution with density function f(X,Y ).

f(X,Y ) =1

2πσ2e−

1σ2 (x2+y2) (22)

The deviation from target determines the quality of the product. Thus, instead of (X,Y )one can introduce the deviation Z as single quality characteristic defined by:

Z =√

(X)2 + (Y )2 (23)

Of course, the random variable Z is not normally distributed but has a so-called Raleighdistribution with density function fZ , [2].

fZ(z) =z

σ2e−

z2

2σ2 (24)

By means of Z the two-dimensional problem is transferred to an equivalent one-dimensionalone. A process capability analysis based on Z by means of the normal distribution yieldsinconsistent results [9]. Moreover, because of the equivalence one can take the two or theone dimensional characteristic for a capability analysis, without priority [7]. A graphicalillustration of the problem is presented in Figure 10.

Figure 10: Illustration of the quality characteristics (X,Y ), f(X,Y ) and fZ .

5.2 Multivariate Quality Analysis

Let �X = (X1, . . . , Xm) be a m-dimensional quality characteristic. The multi-dimensional

tolerance region for �X is often given by a hypercube or an ellipsoid. The first step of thequality analysis consists of calculating the non-conformance probability p (see for instance[10]).

p = 1 −∫

. . .

∫tol.region

f �X(x1, . . . , xm)dx1 . . . dxm (25)


Next, a ‘tolerance ellipsoid’ ET is introduced, i.e. an ellipsoid with center at the targetvalue �t = (t1, . . . , tm) representing an event which occurs with probability of (1 − p).

ET is defined as solution of the following quadratic equation:

(�x − �t)T Σ−1(�x − �t) = χ21−p;m−1 (26)

where χ21−p;m−1 is the (1 − p)-quantile of the χ2-distribution with (m − 1) degrees of

freedom.

For m = 1, i.e. the one-dimensional case, and the normal distribution the above introducedtolerance ellipsoid reduces to the given tolerance interval.[

Φ−1(pL), Φ−1(1 − pU)]σ = [L,U ] (27)

The relation of the tolerance region and the tolerance ellipsoid is illustrated for twodimensions in Figure 11.

Figure 11: Tolerance region and tolerance ellipsoid in a two-dimensional case.

The tolerance ellipsoid enables a straightforward generalization of capability indices tocover the multivariate case.

Cp =volume of tolerance ellipsoid

volume of 99.73%-ellipsoid(28)

where the 99.73%-ellipsoid is defined as solution of

(�x − �t)T Σ−1(�x − �t) = χ20.99.73;m−1 (29)

Taam et.al. use instead of (28) a different definition. They introduce instead of thetolerance ellipsoid a modified tolerance ellipsoid, which is an ellipsoid within the toleranceregion, centered at target and having maximum volume. Clearly, the modified toleranceellipsoid is independent of the underlying multivariate distribution and, thus, in particularindependent of the correlation coefficients.

Cp(Taam) =volume of modified tolerance ellipsoid

volume of 99.73%-ellipsoid(30)


If Cp(Taam) = 1 then the modified tolerance ellipsoid contains 99.73% of the possibleprocess space and the tolerance itself in general more than 99.73% which is not in linewith the definition of a Cp-index.

Let �U be the vector of the upper specification limits, then the second index taking intoaccount the actual value E[ �X] of a multi-dimensional location parameter is denoted byCpk and Cpm(Taam), respectively.

Cpk = (1 − k)Cp where k =

√√√√√√(E[ �X] − �t

)T (E[ �X] − �t

)(

�U − �t)T (

�U − �t) (31)

Cpm =Cp(Taam)√

1 +(E[ �X] − �t

)T

Σ−1(E[ �X] − �t

) (32)

where it is assumed here that the target value �t is the center of the tolerance region.

4 Example Consider a two-dimensional quality characteristic distributed according toa bivariate normal distribution. The input parameters are the following:

�t =

(0.00.0

), �L =

( −3.2−1.6

), �U =

(3.21.6

)(33)

E[ �X] =

(0.20.3

), V [ �X] =

(1.000.25

)(34)

The joint density function is given by:

f(X,Y )(x, y) =1

2πσXσY

√1 − ρ2

e−(x−0.02

1.0 )2−2ρ(x−0.02

1.0 )( y−0.030.5 )+( y−0.03

0.5 )2

2(1−ρ2) (35)

In Figure 12 the capability indices Cp and Cpk and Cp(Taam) and Cpm are displayed asfunctions of the correlation coefficient.

6 Conclusions

In many industrial instances product quality depends on a multitude of dependent char-acteristics and, therefore, there is an urgent need for appropriate multivariate models andmethods. One problem is to define suitable tolerance regions taking into account thecorrelation structure among the variables. The problem of tolerance regions is closelyconnected to the problem of deriving process capability indices. Another problem refersto process monitoring for detecting changes in process distribution. Assuming the normalmodel might lead to inferior methods and wrong decisions.


Figure 12: Cp and Cpk and Cp(Taam) and Cpm as functions of ρ.

References

[1] C. Anghel, H. Hausberger, W. Streinz (1992): Unsymmetriegroßen erster undzweiter Art richtig auswerten (Teil 1). QZ 37, 755-758.

[2] C. Anghel, H. Hausberger, W. Streinz (1993): Unsymmetriegroßen erster undzweiter Art richtig auswerten (Teil 2). QZ 38, 37-40. 2)

[3] C. Anghel (1998): Die Prozesszentrierung richtig beurteilen. QZ 43, 1088-1092.

[4] Castagliola, P. (1996): Evaluation of non-normal process capability indices usingBurrs distribution. Quality Engineering 4, pp. 587-593.

[5] Clement, J.A. (1989): Process capability calculation for non-normal distributions.Quality Progress Sept., 95-96.

[6] Collani, E.v. and Killmann, F. (2001): A note on the convolution of uniform andrelated distributions and their use in quality control. EQC, 16, ...

[7] Dietrich, E., Schulze, A. (1996): Fahigkeitsbeurteilung bei Positionstoleranzen. QZ41, 812-814.

[8] Leone, F.C., Nelson, L.S. and Nottingham, R.B. (1996): The folded normal distri-bution. Technometrics 3, 543-550.

[9] Littig, S.J. and Pollock, S.M. (1992): Capability measurements for multivariateprocesses: Definitions and an example for a gear carrier. Technical Report 92-42,Dept. Industrial and Operational Engineering, University of Michigan, Ann Arbor.


[10] Hamilton, D.C. and Lesperance, M.L. (1995): A comparison of methods for univari-ate and multivariate acceptance sampling by variables. Technometrics 37, 329-339.

[11] H. Schneider, W.J. Kasperski, T. Ledford and W. Kraushaar (1995,1996): Controlcharts for skewed and censored data. Quality Engineering 8, 263-274.

[12] W. Taam, P. Subbaiah and J.W. Liddy (1993): A note on multivariate capabilityindices. Journal of Applied Statistics 20, 339-351.

[13] D.J. Wheeler, D.S. Chambers (1989): Understanding Statistical Process Control.Statistical Process Controls, Inc.

Dr. C. AnghelQuality DepartmentBMW AGD-84130 DingolfingGermany

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