Qual. Reliab. Engng. Int. 2002; 18: 111–116 (DOI: 10.1002/qre.450)
ON THE DISTRIBUTION OF THE ESTIMATED PROCESS
YIELD INDEX S
pkJ. C. LEE1∗, H. N. HUNG1, W. L. PEARN2AND T. L. KUENG1
1Institute of Statistics, National Chiao Tung University, 1001 TA Hsueh Road, Hsinchu, Taiwan 30050, Republic of China
2Department of Industrial Engineering and Mangement, National Chiao Tung University, 1001 TA Hsueh Road, Hsinchu,
Taiwan 30050, Republic of China
SUMMARY
This paper considers an asymptotic distribution for an estimate ˆSpkof the process yield index Spkproposed by
Boyles (1994). The asymptotic distribution of ˆSpkis useful in statistical inferences for Spk. An illustrative example
is given for hypothesis testing and for interval estimation on the yield index Spk. Copyright2002 John Wiley &
Sons, Ltd.
KEY WORDS: central limit theorem; illustrative example; normal distribution
1. INTRODUCTION
Process capability indices, establishing the relation-ship between the actual process performance and the manufacturing specifications, have been the focus of recent research in quality assurance and capability analysis. Those capability indices, quantifying process potential and process performance, are essential to any successful quality improvement activities and quality program implementation. Some basic capability in-dices that have been widely used in the manufacturing
industry include Cp, Ca and Cpk. These indices are
explicitly defined as follows [1–3]:
Cp= USL− LSL 6σ Ca= 1 − |µ − m| d Cpk= min USL− µ 3σ , µ − LSL 3σ
where USL and LSL are the upper and the lower specification limits, respectively, µ is the process mean, σ is the process standard deviation, m =
(USL + LSL)/2 is the midpoint of the specification
interval and d = (USL − LSL)/2 is half the length of specification interval. We will focus on the situation in which the specification interval is two-sided with the target value T at m, which is most common in practice.
The index Cpmeasures the overall process variation
relative to the specification tolerance and, therefore, only reflects process potential (or process precision). ∗Correspondence to: J. C. Lee, Institute of Statistics, National Chiao
Tung University, 1001 TA Hsueh Road, Hsinchu, Taiwan 30050, Republic of China.
The index Ca measures the degree of process
centering, which alerts the user if the process mean deviates from its target value. Therefore,
the index Ca only reflects process accuracy. The
index Cpk takes into account the magnitudes of
process variation as well as the degree of process centering, which measures process performance based on yield (proportion of conformities). For a normally
distributed process with a fixed value of Cpk, the
bounds on process yield are given by
2(3Cpk) − 1 ≤ %Yield < (3Cpk)
where (·) is the cumulative distribution function of N(0, 1), the standard normal distribution. For
example, if Cpk = 1.00, then it guarantees that
the %Yield will be no less than 99.73%, or no greater than 2700 ppm (parts per million) of
non-conformities. We note that the index Cpkonly provides
an approximate rather than an exact measure of the process yield.
To obtain an exact measure, Boyles [4] considered
a yield index, referred to as Spk, for processes with
normal distributions. The index Spkis defined as:
Spk= 1 3 −11 2 USL− µ σ +1 2 µ − LSL σ (1.1) It can be written as Spk= 1 3 −11 2 1− Cdr Cdp +1 2 1+ Cdr Cdp (1.2) Received 27 June 2001
Table 1. Various Spkvalues and the corresponding process yield Spk Process yield 1.00 0.997 300 204 1.24 0.999 800 777 1.33 0.999 933 927 1.50 0.999 993 205 1.67 0.999 999 456 2.00 0.999 999 998
where −1is the inverse function of , Cdr = (µ −
m)/d and Cdp = σ/d. For a process with Spk = c,
we can obtain %Yield = 2(3c) − 1. Obviously,
there is a one-to-one relationship between Spk and
the process yield. Thus, the yield index Spk provides
an exact measure of the process yield. For normal processes, the expected number of non-conformities
corresponding to a capable process with Spk = 1.00
is 2700 ppm, a satisfactory process with Spk = 1.33
is 63 ppm, an excellent process with Spk = 1.67
is 0.6 ppm and a super process with Spk = 2.00 is
0.002 ppm, as summarized in Table1.
2. AN ESTIMATOR AND ITS DISTRIBUTION
Let X1, . . . , Xnbe a random sample from the normal
process, a natural estimator of Spkis
ˆSpk= 1 3 −11 2 1− ˆCdr ˆCdp +1 2 1+ ˆCdr ˆCdp (2.1)
where ˆCdr and ˆCdp are estimates of Cdr and Cdp,
respectively and are defined as
ˆCdr = ( ¯X − m)/d
ˆCdp= S/d
(2.2)
with ¯X = 1/n ni=1Xi and S2 = 1/(n −
1)ni=1(Xi − ¯X)2. The distribution of ˆSpk is
non-trivial as it is a complex function of the statistics ¯X
and S2. However, a useful approximate distribution
of ˆSpk can be furnished by considering the following
asymptotic expansion of ˆSpk. Let Z =
√
n( ¯X − µ), Y = √n(S2− σ2), then Z and Y are independent
and since the first two moments of ¯X and S2exist, by
the Central Limit Theorem they converge to N(0, σ2)
and N(0, 2σ2), respectively, as n goes to infinity.
Consequently, ˆSpkcan be expressed as
ˆSpk= Spk+ 1 6√n(φ(3Spk)) −1W + O p(n−1) (2.3) where W = − d 2σ3Y (1 + Cdr)φ 1+ Cdr Cdp +(1 − Cdr)φ 1− Cdr Cdp − 1 dCdpZ φ 1− Cdr Cdp − φ 1+ Cdr Cdp (2.4) which is normally distributed with a mean of zero and a variance of a2+ b2, a = √d 2σ (1 − Cdr)φ 1− Cdr Cdp +(1 + Cdr)φ 1+ Cdr Cdp b = φ 1− Cdr Cdp − φ 1+ Cdr Cdp (2.5)
and φ is the probability density function of the standard normal distribution. It is noted that
the asymptotic expansion of ˆSpk, as given in
(2.3), indicates that asymptotically ˆSpk is normally
distributed with mean Spk and variance (a2 +
b2)/36n(φ(3Spk))2. In (2.3), Cdr and Cdp appear in
the asymptotic expression of ˆSpkas a consequence of
the Taylor expansion used in the asymptotic expansion of ˆSpkaround the true values Cdr and Cdp. In practice,
ˆCdr and ˆCdp are used instead because they will
converge to Cdr and Cdp, respectively. Also, the
remaining terms Op(n−1) represent the error of the
expansion having a leading term of order n−1 in
probability.
The first-order approximation of ˆSpk, as given
in (2.3), can produce an adequate approximate
distribution for a large enough sample size. Figures1
and 2 depict approximate and exact distributions,
obtained by simulations, with a sample size of n = 100, 200, 300, 400, 500, 1000. It is clear that as the sample size n reaches 1000, the approximate and exact distributions are almost indistinguishable. In fact, even with n = 100 the approximation is quite reasonable for practical purposes.
3. INFERENCE BASED ON ˆSpk
From (2.3) and (2.4), it is clear that ˆSpk is an
asymptotically unbiased estimator of Spk. Also, thanks
to the asymptotic distribution of ˆSpk given in (2.3)–
0.50 0.55 0.60 0.65 0.70 0.75 samplesize=100 0. 0 0 .2 0. 4 0 .6 0. 8 1 .0 exact approx. 0.55 0.60 0.65 0.70 samplesize=200 0. 0 0 .2 0. 4 0 .6 0. 8 1 .0 exact approx. 0.55 0.60 0.65 samplesize=300 0 .00 .2 0 .40 .6 0 .81 .0 exact approx. 0.54 0.56 0.58 0.60 0.62 0.64 0.66 samplesize=400 0 .00 .2 0 .40 .6 0 .81 .0 exact approx. 0.54 0.56 0.58 0.60 0.62 0.64 0.66 samplesize=500 0. 0 0 .2 0. 4 0 .6 0. 8 1 .0 exact approx. 0.56 0.58 0.60 0.62 0.64 samplesize=1,000 0. 0 0 .2 0. 4 0 .6 0. 8 1 .0 exact approx.
Spk can be constructed. For example, the following
null hypothesis
H0: Spk≤ c, a specified value (3.1)
versus alternative hypothesis
H1: Spk> c (3.2)
can be executed by considering the testing statistic
T = 6( ˆSpk− c)
√
nφ(3 ˆSpk)
ˆa2+ ˆb2 (3.3)
whereˆa and ˆb are estimates of a and b, with Cdr, Cdp
and σ replaced by ˆCdr, ˆCdp and S, respectively. The
null hypothesis H0 is rejected at α level if T > zα,
where zα is the upper 100α% point of the standard
normal distribution.
An approximate 1− α confidence interval for Spkis
ˆSpk− ˆa2+ ˆb2 6√nφ(3 ˆSpk) zα/2, ˆSpk+ ˆa2+ ˆb2 6√nφ(3 ˆSpk) zα/2 (3.4) An illustrative example is given in the next section.
4. AN ILLUSTRATIVE EXAMPLE
Consider the following example taken from a supplier manufacturing high-end audio speaker drivers including 3-inch tweeters, 3-inch full-range drivers, 5-inch mid-range drivers, 6.5-inch woofers and 8-inch, 10-inch and 12-inch subwoofers. A standard woofer driver consists of components including an edge, cone, dustcap, spider (damper), voice coil, lead wire, frame, magnet, front plate and back plate. The edge (on the top) and the spider (on the bottom) are glued onto the frame to hold the cone for the piston movement and the dustcap (glued onto the cone to cover the top of the voice coil) decouples the noise from the musical signals. One characteristic, which reflects the bass performance, musical image, clarity and cleanness of the sound, transparence and compliance (excursion movement) of the mid-range,
full-range or subwoofer driver units, is F0 (the
free-air resonance frequency). Some key factors, which
determine F0values, include the hardness, thickness,
weight of the damper, weight of the edges and the
weight of the cone. Typical ranges of F0 values are
25–40 Hz for subwoofers, 40–60 Hz for woofers, 50–100 Hz for full-range, and 500–5000 Hz for mid-ranges.
One particular model of the 3-inch full-range drivers, designed particularly for the central and
Table 2. F0measures for the 3-inch drivers
81 80 82 79 78 76 78 78 76 81 83 78 81 85 81 78 79 79 80 82 79 79 82 78 82 80 75 85 80 80 80 75 81 78 82 84 76 78 80 79 82 82 78 78 82 78 82 80 82 83 81 78 83 81 82 79 80 79 81 82 79 80 82 77 81 80 81 81 75 76 83 86 82 79 82 85 80 80 77 75 78 85 81 79 81 83 78 78 80 80 79 76 77 74 85 83 76 80 75 82
background channels of home-theater applications, has used the specially designed Pulux edge, Pulux dustcap and PP-mica cone. This model of 3-inch driver
requires the F0 value to be 80 Hz with ±10 Hz
tolerance. The production specification limits for this particular model of drivers are therefore set to
(LSL, T , USL) = (70, 80, 90) for F0. The quality
requirement was predefined as Spk≥ 1.00 (equivalent
to USL− LSL = 6σ). A total of 100 samples of data
were collected from the factory, which are displayed in Table2.
Thus, statistical inferences on the index, Spk, such
as hypothesis testing and interval estimation can be
considered. For testing the null hypothesis H0 as
given in (3.1) with c = 1 against the alternative
hypothesis as given in (3.2), the testing statistic T ,
as given in (3.3), yields 3.1389. Since 3.1389 >
z0.05 = 1.96, the null hypothesis H0 is rejected at
α = 0.05. We may conclude that the process satisfies
the capability requirement Spk ≥ 1.00. Moreover, an
approximate 95% confidence interval for Spkis easily
obtained from (3.4) as
(1.1078, 1.4664)
which is consistent with the hypothesis testing result.
REFERENCES
1. Kane VE. Process capability indices. Journal of Quality
Technology 1986; 18(1):41–52.
2. Pearn WL, Kotz S, Johnson NL. Distributional and inferential properties of process capability indices. Journal of Quality
Technology 1992; 24(4):216–231.
3. Pearn WL, Lin GH, Chen KS. Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics: Theory and Methods 1998; 27(4):985–1000.
4. Boyles RA. Process capability with asymmetric tolerances.
Communications in Statistics: Computation and Simulation
Authors’ biographies:
Jack C. Lee is a Professor at the Institute of Statistics, National Chiao Tung University, Taiwan. He received a BA in Business from the National Taiwan University, an MA in Economics from the University of Rochester and a PhD in Statistics from SUNY at Buffalo.
H. N. Hung is an Associate Professor at the Institute of Statistics, National Chiao Tung University, Taiwan. He received a BS in Mathematics from the National Taiwan University, an MS in Mathematics from National Tsing Hua University and a PhD in Statistics from the University of Chicago.
W. L. Pearn recieved his MS degree in Statistics and PhD degree in operations Research from the Univer-sity of Maryland at College Park, USA. He worked for Bell Laboratories as a quality engineer before he joined the National Chiao Tung University. Currently, he is a Professor in the Department of Industrial Engineer-ing and Management, National Chiao Tung University, Taiwan.
T. L. Kueng is a Senior Engineer at the Behavior Design Corporation, Taiwan. He received a BA in Business from the National Taiwan University and an MS in Statistics from the National Chiao Tung University, Taiwan.
