# Control Charts for Variables. Control Chart for X and R

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1 Control Charts for Variables X-R, X-S charts, non-random patterns, process capability estimation. 1 Control Chart for X and R Often, there are two things that might go wrong in a process; its mean or its variance (or both) might change. 2

2 Statistical Basis for the Charts A normally distributed variable x with known µ and σ, can be controlled: average: x = x 1+x x n n with µ ± Z α σ 2 n where ±Zα is the distance from µ ( in # of σ), 2 that would capture (1- α)% of the normal. n standard deviation: Σ (x i -µ) 2 s 2 i=1 = ~ σ2 2 χ (n-1) with: n-1 n-1 σ 2 2 χ 1- α 2,(n-1) < s 2 < σ2 2 χ α 2, (n-1) n-1 n-1 2 where χ 1- α 2 2,(n-1) and χ α 2, (n-1) are the 2 numbers that capture between them 1-α of the χ (n-1) distribution Control Chart for x and s (with mean and variance known). UCL.977 (+3 σ) µ = LCL.514 (-3 σ) 8.1 UCL.78 σ 2 = LCL.3 8 4

3 Statistical Basis for the Charts (cont.) In practice we do not know the mean or the sigma. The mean can be estimated by the grand average. If the sample size is small, we can use the range to describe spread. x= x 1+x x m m R=x max -x min Range R is related to the sigma in terms of a constant (depending on sample size) that is listed in statistical tables: R= R 1+R R m m σ = R d 2 5 Statistical Basis for the Charts (cont.) The control limits for the X-R chart are as follows: UCL = x + A 2 R center at x LCL = x - A 2 R UCL = RD 4 center at R LCL = RD 3 A 2 = 3 D 3,4 = 1±3 d 3 d 2 n d 2 (d 2 and d 3 are tabulated constants that depend on n) 6

4 Range and Mean charts for Photoresist Control Range n=5 and from table, D 3 =. and D 4 = Average Range is , so the range center line is , the LCL is. and the UCL is These control limits will give us the equivalent of 3 sigma control. (α =.27). X-bar The global average is and from table, A 2 is.577, so LCL is and UCL is These control limits will give us 3 sigma control. (α =.27). 7 Example: Photoresist Coating (cont) Range and X-bar chart for all wafer groups UCL = LCL =. UCL = LCL = Wafer Groups 8

5 The Grouping of The Parameters is Crucial Known as rational subgrouping, the choice of grouping is very important. In general, only random variation should be allowed within the subgroup. (i.e. grouping wafers within the boat of a diffusion tube is inappropriate - gas depletion effect is systematic.) The range of the appropriate group should be used to estimate the variance of the process. (i.e. the range across a lot should not be used to estimate the variance of a parameter measured between lots - within lot statistics are different from between lot statistics). 9 Rational Subgrouping Rule of thumb: use only groups with IIND data, if possible. (Independently, Identically, Normally Distributed). Wafer Lot Batch The natural grouping of semiconductor data might not lead to IIND subgroups! 1

6 Example: charts for line-width control.3 Range Chart, n=5 D3=., D4= UCL X chart, n=5, A2=.577 LCL..8.7 UCL LCL what is the problem? Lot No 11 Rational Subgrouping (cont.) Remember, we are using R to estimate the global sigma. The rational subgroups we chose might bias this estimation. In this case, the within lot variation is much less than the global variation. If we estimate the sigma from the global range, we get: X chart for Global Range UCL LCL If this looks too good, it might be because of the "mixture" patterns in it! 6 12

7 Specification Limits vs. Control Limits The specification limits of a process reflect our need. These limits are set by the management as objectives. The control limits of a process tell us what the process can do when it is operating properly. These limits are set by the quality of the machinery and the skills of the operators. Process Capability is a figure of merit that tells us whether a process is suitable for our manufacturing objectives. 13 Process Capability Process specifications and control limits are, in general, different concepts. A process might be in control without meeting the specs, or might be meeting the specs without being is control... Figure 7.2 pp 139 Kume 14

8 Process Capability Estimation Calculate what the process (when in control) can do and compare it with specifications. A control chart provides good estimates of σ, so it can be used for process capability evaluation. Process Capability Ratio (PCR, CP) Cp = (USL-LSL) / 6 σ Example The line-width control data show a process capability of 1.8, since the specs for DLN are.5 to 1. µm and σ is.77. The symbol CPK is used for the process capability when the spec limits are not symmetrical around the process spread: CPK = min { (USL-x)/3σ, (x-lsl)/3σ } 15 LSL Process Capability (cont) Cp = (USL-LSL) / 6 σ USL 6 σ LSL Cpk = min { (USL-x)/3σ, (x-lsl)/3σ } USL 3 σ 16

9 Never Mix Specification and Control Limits! Only individuals should be plotted against the specs... Relationship of natural tolerance, control and spec limits: 17 Never Attempt to Estimate Cp or Cpk from small groups! Plot of CP estimate, n=5 (true CP=.67) CP CP 18

10 Special patterns in charts While a point outside the control limits is a good indicator of non-randomness, other patterns are possible: Cyclic (periodic signal) Mixtures (two or more sources) Shifts (abrupt change) Trends (gradual change) Stratification (variability too small) Additional rules can be used to detect them. (Watch for an increase in false alarm rates). 19 Special Patterns 2

11 A B C C B A The Western Electric Rules UCL (+3σ) Center Line LCL (-3σ) 1. Any point beyond 3s UCL or LCL. 2. 2/3 cons. points on same side, in A or beyond 3. 4/5 cons. points on same side, in B or beyond. 4. 9/9 cons. points on same side of center line. 5. 6/6 cons. points increasing or decreasing /14 cons. points alternating up and down /15 cons. points on either side in zone C. 21 Robustness of the X-R control chart So far we have assumed that our process is fluctuating according to a normal distribution: X ~ N(µ, σ 2 ) This assumption is not important for the X chart (thanks to the central limit theorem). The R chart is much more sensitive to this assumption. If the underlying distribution is not normal, watch for signs of correlation between X and R. 22

12 The X-R Operating Characteristic Function What is the probability of not detecting a shift of µ by k σ? β = Φ UCL-(µ o+k σ) - Φ LCL-(µ o+k σ) σ / n σ / n or, for 3 σ control linits: β = Φ ( 3 - k n ) - Φ ( -3 - k n ) The probability that the shift will be detected on the mth sample is: β m-1 (1 - β) And the average number of samples that we need for detecting this shift is: ARL = β also known as the Average Run Length 23 The X-R Operating Characteristic Function (cont.) 24

13 The X-R Operating Characteristic Function (cont.) The OC for the R part of the chart shows that it cannot catch small shifts in sigma: 25 X and S control charts When n is larger (>1), then using the standard deviation s gives better results. Although s is a good, unbiased estimator of the variance, n Σ (x i -x) 2 s 2 i=1 = n-1 s is a biased estimator of sigma. A correction factor is a function of n and can be found is tables. In summary: s= 1 m m Σ i=1 s i and s C 4 unb. estim of sig CL s =B 3,4 s B 3,4 =1± 3 C 4 1-C 4 2 CL x = x±a 3 s A 3 = 3 C 4 n 26

14 X and S Control Charts (when n is large).15 S Chart, assuming sigma=.6 B 5 =.276, B 6 =1.669 for n= S Chart, no standard B 3 =.281, B 4 =1.716 for n=1 UCL.11 sigma S.528 LCL.166 UCL.96 S LCL.148 Lot No 27 The control limits for x can now be calculated from s. 1. S=.53, n=1, A 3 = UCL LCL Lot No 28

15 Control Charts for Individual Units Moving Range chart for Temp. Control: 5 Moving Range Graph, n=2, D 3 =., D 4 = Temp Samples, n=2, d 2 =1.128 UCL LCL. UCL 3.2e+ 4.5e LCL -3.19e+ Time 29 Questions Most Frequently Asked Q How many points are needed to establish control limits? A Typically 2-3 in-control points will do. Q How do I know whether points are in-control if limits have not been set? A Out-of-control points should be: a) explained and excluded. b) left in the graph if cannot be explained. When done, we should have about 1/ARL unexplained outof-control points (for 3σ control ~1/37 samples.) These points are accepted as false alarms. Q How often limits must be recalculated? A Every time it is obvious from the chart that the process has reached a new, acceptable state of statistical control. 3

16 Choosing the proper control chart Variables new process or product old process with chronic trouble diagnostic purposes destructive testing very tight specs decide adjustments Attributes reduce process fallout multiple step process evaluation cannot measure variables historical summation of the process Individuals physically difficult to group full testing or auto measurements data rate too slow 31 Summary, so far Continuous variables need two (2) charts: chart monitoring the mean chart monitoring the spread Rational subgrouping must be done correctly. Charts are using Control Limits - Spec limits are very different (conceptually and numerically). The concept of process capability brings the two sets of limits together. A chart is a hypothesis test. It suffers from type I and type II errors. Violating the control limits is just one type of alarm, out of many. 32

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