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Unformatted text preview: Methods and Philosophy of
Statistical Process Control Statistical process control is a collection
of tools that when used together can
result in process stability and variability
reduction Introduction to Statistical Quality Control,
4th Edition The seven major tools are 1) Histogram or Stern and Leaf plot
2) Check Sheet 3) Pareto Chart 4) Cause and Effect Diagram 5) Defect Concentration Diagram
6) Scatter Diagram 7) Control Chart Introduction to Statistical Quality Control,
4th Edition Chance and Assignable Causes of
Quality Variation A process that is operating with only chance
causes of variation present is said to be in
statistical control. A process that is operating in the presence of
assignable causes is said to be out of control. The eventual goal of SPC is reduction or elimination of variability in the process by identiﬁcation of assignable causes. Introduction to Statistical Quality Control,
' 4th Edition 4—2 CHANCE AND ASSIGNABLE CAUSES OF QUALITY VARIATION Assignable cause three /’6 > 5 IS present; process \ 4/
outofcontrol / I” \‘A
/.._ x Assignabie cause two /
i is present; process is //" i
i out—of—control ‘ ' I2 w/ Assrgnabie cause one ,/ ' ’ I e " is present; process is ,,,, /
outofcontrol ;’/1’ Only chance causes of variation present; /”/ .
process is in A/ ‘70 l ,/,. control ,v/ / ,. . ,f/f
,/ x /
./ , ,r' Process quality characteristic, ,\' Figure 41 Chance and assignable causes of variation, Statistical Basis of the Control Chart Basic Principles A typical control chart has control limits set at values such
that if the process is in control, nearly all points will lie between the upper control limit (UCL) and the lower control
limit (LCL). l Ce'nterline p.‘ [I 7—“ ‘3 Lower control limit " Sample quality characteristic __i_J__1_L._.J_..r_ I L_._l._J._l__L__.L .... ,J .... ..J.._i_
Sample number or time Figure 4'2 A tl’PlCal control chart. OutofControl Situations 0 If at least one point plots beyond the control limits, the
process is out of control °_ If the points behave in a systematic or nonrandom
manner, then the process could be out of control. Relationship between hypothesis testing and control charts
° We have a process that we assume the true process mean is u .= 74 and the process standard deviation is o = 0.01.
Samples of size 5 are taken giving a standard deviation of the sample average, as — = 0.0045 J5 Introduction to Statistical Quality Control,
4th Edition Relationship between hypothesis testing and control
charts ° Control limits can be set at 3 standard deviations from the
mean. ° This results in “3Sigma Control Limits”
UCL = 74 + 3(0.0045) = 74.0135
CL= 74
LCL = 74  3(0.0045) = 73.9865  Choosing the control limits is equivalent to setting up the critical region for testing hypothesis }%:n==75 Hﬁp¢75 Introduction to Statistical Quality Control,
4th Edition Relationshiy between the Eroeess and the control
chart Distribution of individual
measurements x: i i ,
Normal I D'Stgicbg'on ________ _ IIIIIIIIIIII _._ _________ m
Wilyﬂliﬁj " Normal with
“a: O 01 / mean u = 74
' ’ and  ............... __.___ ................ ._.___ ....................... .__._ UCL = 740135
i ai = 0.0045 ,/
/ t.
r _/’ 2‘
..«L ............ _. / __1__ _;___ .... ......... _. Ce"ter = 74.0000
\ \\ I f,» / Line
\\ N \ l \[ i
‘\ Sample: \ i
\ a ,, = 5 x!  —~    ~  t ~    v ~   —' LCL = 73.9865
\\ l Figure 44 How the control chart works. Important uses of the control chart Most processes do not operate in a state of statistical
control. Consequently, the routine and attentive use of control
charts will identify assignable causes. If these causes can
be eliminated from the process, variability will be reduced
and the process will be improved. The control chart only detects assignable causes.
Management, operator, and engineering action will be
neceSSary to eliminate the assignable causes. Outofcontrol action plans (OCAPs) are an important
aspect of successful control chart usage . Introduction to Statistical Quality Control,
4th Edition Types the control chart Variables Control Charts — These charts are applied to data that fellow a
continuous distribution (measurement data). Attributes Control Charts ~ These charts are applied to data that follow a discrete
distribution. Type of Process Variability  Stationary behavior, uncorrelated data Stationary behavior, autocorrelated data
 Nonstationary behavior Introduction to Statistical Quality Control,
4th Edition 40 _ _ , . . _ i . . i . . . . . m 30 A“. u 10 50 100 1 50 200
(c)
Figure 46 Data from three different processes. (a) Sta tionary and uncorrelated (white noise). (1:) Stationary and
autocorrelated. (c) Nonstationary. Type of Variability ° Shewhart control charts are most effective when the in
control process data is stationary and uncorrelated. Popularity of control charts 1) Control charts are a proven technique for improving
product1v1ty. 2) Control charts are effective in defect prevention.
3) Control charts prevent unnecessary process adjustment. 4) Control charts provide diagnostic information. 5) Control charts provide information about process
capability. Introduction to Statistical Quality Control,
4th Edition Choice of Control Limits General model of a control chart UCL = pw + Low
Center Line = uw
LCL = uw — Low where L = distance ‘of the control limit from the
center line H w = mean of the sample statistic, w.
c7w = standard deviation of the statistic, w. Introduction to Statistical Quality Control,
4th Edition By moving the control limits farther from the center line, we
decrease the risk of a Type I error or a false alarm (i.e., the
risk of a point falling beyond the control limits, indicating an
outof—control condition when no assignable cause is
present) 9 Widening the control limits will also increase the risk of a
Type 11 error (i.e., the risk of a point falling between the
control limits when the process is really out of control. 74.0180 ‘
0.001 UCL = 74.0139 74.0135 MHW 3sigma UCL = 74.0135
74.0090 ‘ HI 74.0045 ‘
Center line = 74.0000 74.0000 W 73.9955 — 73.9910 ~ .
 ' LCL = 73.9865
73.9865 b 0.001 LCL = 73.9861 73.9820 " o    I 1 I ! l__l__i_1. LJ LJ \ I 1..L_3_..l..‘
Sample number Figure 47 Comparison of threesigma and 0.001 probability limits for
the X chart. ThreeSigma Limits The use of 3—sigma limits generally gives good results in
practice. If the distribution of the quality characteristic is
reasonably well approximated by the normal distribution,
then the use of 3—sigma limits is applicable. 4 These limits are often referred to as action limits. Warning Limits on Control Charts Warning limits (if used) are typically set at 2 standard
deviations from the mean. . If one or more points fall between the warning limits and
the control limits, or close to the warning limits the
process may not be operating properly. Introduction to Statistical Quality Control,
4th Edition Warning Limits on Control Charts 0 Good thing: warning limits often increase the sensitivity
of the control chart. Bad thing: warning limits could result in an increased
risk of false alarms. 74.0180
74.0135
74.0090 74.0045 .
74.0000 CenteLllne _ 74.0000 739955 _ LWL — 73 9910
73.9910 ———:—w'————~———~~~—«. 73.9865 LCL= 73.9865
73.9820 — UCL = 74.0135 1 1 1 Sample number I Figure 4—8 An E chart with twosigma warning
limits. Sample Size and Sampling Frequency In designing a control chart, both the
sample size to be selected and the
frequency of selection must be speciﬁed. Larger samples make it easier to detect
small shifts in the process. Current practice tends to favor smaller,
more frequent samples. Introduction to Statistical Quality Control,
4th Edition Average Run Length _
 The average run length (ARL) is a very important way of determining the appropriate sample size and sampling
frequency. Let p = probability that any point exceeds the control
limits. Then, Illustration Consider a problem with control limits set at 3 standard
deviations from the mean. The probability that a point
plots beyond the control limits is again, 0.0027 (i.e., p =
0.0027). Then the average run length is ARL =1/0.0027
=3 70 Introduction to Statistical Quality Control,
4th Edition What does the ARLgtell us? 0 The average run length gives us the length of
.time (or number of samples) that should plot in
control before a point plots outside the control
limits. 0 For our problem, even if the process remains in control, an out—of—control signal will be
generated every 370 samples, on average. Average Time to Signal  Sometimes it is more appropriate to express the
performance of the control chart in terms of the average
time to signal (ATS). Say that samples are taken at ﬁxed intervals, h hours apart. ATS = ARL (h) Introduction to Statistical Quality Control,
4th Edition Rational Subgroups Subgroups or samples should be selected so that if assignable
causes are present, the chance for differences between
subgroups will be maximized, while the chance for differences
due to these assignable causes Within a subgroup Will be
minimized. Selection of Rational Subgroups 0 Select consecutive units of production. — Provides a “snapshot” of the process.
— Effective at detecting process shifts.  Select a random sample over the entire sampling
interval. . — Can be effective at detecting if the mean has wandered outofcontrol and then back incontrol. Innoduction to Statistical Quality Control,
4th Edition 10 c.
.1 r} [a
.\‘ , C .vs—~;:.—_—;I~
45 I g. g: =
Process 1 i C
mean 5; f :3 5
..._..1‘... :————‘ 5. I...
(i six
1 (g U
“MI ..._L ...I__._LH_ L. I . I I .... .. I._._J ._J_.__J___.._
1 2 3 4 5 6 7 8 9 10 11 12
Time
(LI) 
123456789101112
Time (12) Figure 410 The “snapshot” approach to rational subgroups. (a) Behavior
of the process mean. (12) Corresponding E and R control charts. l
x I Process 9
mean —>‘ .
[,i. f
l
l
(a)
UCL'F
CLrIw “C; “1/04.. {LA
“J \G/ w V‘ \
I.) \O
LCL'F F— ~ —  A —
UCLR . .'
CL,H  R.. _
J \/‘/ \
I I I I I i __l__i ___l___ ’ l
123456789101112
Time
(/2) Figure 411 The random sample approach to rational sub groups. (a) Behavior of the process mean. ([7) Corresponding E
and R control chairs. Analysis of Patterns on Control Charts Nonrandom patterns can indicate outof—control conditions ‘ Patterns such as cycles, trends, are often of considerable diagnostic value  Look for “runs”  this is a sequence of observations of the same type (all
above the center line, or all below the center line)  Runs of say 8 observations or more could indicate an out—ofcontrol situation. —— Run up: a series of observations are increasing
— Run down: a series of observations are decreasing I t L
1 3 5 7 9 11 13 15 17
Sample number 19 21 Figure 412 An X control chart. Center
line LCL i i l l  L_.L. ..... .L .... .J ...... .L.._.._J_._..I.~..._L "3; """" "EM—éwr—Ws 9 10 11 12 13 14 15 Sample number Figure 413 An 3 chart with a cyclic pattern. Western Electric Handbook Rules
Should be used carefully because of the increased risk of false alarms! A process is considered out of control if any of the following occur: V 1) One point plots outside the 3sigma control limits.
2) Two out of three consecutive points plot beyond the 2—sigma warning limits.
3) Four out of ﬁve consecutive points plot at a distance of 1
sigma or beyond from the center line. 4) Eight consecutive points plot on one side of the center line. Introduction to Statistical Quality Control, 4th Edition 11 ...
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 Spring '08
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