Control Charts for Variables Terminology Variables refers to - - PowerPoint PPT Presentation

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Control Charts for Variables

Terminology “Variables” refers to quantitative variables, like physical dimensions, as opposed to “attributes”, which refers to qualitative variables, like acceptable/unacceptable. Control charts for variables and for attributes are constructed differently, but with the same general goal of monitoring performance. Both process mean and process variability need to be monitored.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Control Charts for ¯ x and R

Statistical basis of the charts Suppose that a process is in control, and the monitored variable X ∼ N(µ, σ2) with independent measurements. Suppose that charts are based on samples of size n (typically n = 3, 4, or 5). Then a sample mean ¯ X ∼ N(µ, σ2/n), and “three-sigma” control limits µ ± 3σ/√n would result in the probability of a point falling

• utside the limits being 0.0027 .

2 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Unknown µ and σ When the parameters are unknown, it is customary to take m preliminary samples, also of size n, where m is at least 20 to 25. Conventionally, the center line is set to the “best estimator” of µ, the grand mean ¯ ¯ x = ¯ x1 + ¯ x2 + · · · + ¯ xm m . Also conventionally, the control limits are based on the sample ranges r1, r2, . . . , rm, using the average range ¯ r = r1 + r2 + · · · + rm m .

3 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Monitoring the process mean Note that R, the range of a random sample of size n, does not directly estimate σ, but needs to be scaled: E(R) = d2σ, where d2 is tabulated in Appendix VI. The scaling factor is combined with the multiplier 3 and the divisor √n in tabulated constants A2 = 3/(d2 √n), and the control limits are: UCL = ¯ ¯ x + A2¯ r Center line = ¯ ¯ x LCL = ¯ ¯ x − A2¯ r.

4 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Monitoring process variability Process variability is monitored by a control chart of the ranges, with three-sigma control limits: UCL = D4¯ r Center line = ¯ r LCL = D3¯ r. The calculation of A2, D3, D4, and similar constants is described by Montgomery, and also at the National Institute of Standards and Technology.

5 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Note The control limits for the R chart are symmetric around the center line: D4 = 1 + 3d3 d2 , D3 = 1 − 3d3 d2 except for n ≤ 6 where D3 would be negative, and instead D3 = 0. That is, D3 = max

• 1 − d3

d2 , 0

• .

These limits are not based on the sampling distribution of R.

6 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Development of ¯ x and R charts Example 6.1: flow width in photolithography. Specification limits are (1.5 ± 0.5) microns. In R:

flow <- read.csv("Data/Table-06-01.csv", row.names = 1) flowDev <- flow[1:25,]

Development of the charts First set up the R chart:

library(qcc) summary(qcc(flowDev, "R"))

7 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Variability appears to be in control, so continue with the ¯ x chart:

summary(qcc(flowDev, "xbar"))

The ¯ x chart also shows process is in control, so control limits are acceptable.

8 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Process capability On the ¯ x chart, the “Center” is shown as 1.50561 microns; this is the grand mean ¯ ¯

• x. Also the “StdDev” is shown as 0.1398143 microns;

this is the estimated standard deviation of individual measurements (not the estimated standard error of sample means). If X is normally distributed with this mean and standard deviation, P(X is outside specs) = P(X < LSL) + P(X > USL).

9 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Now P(X < LSL) = Φ 1.00 − 1.50561 0.1398143

• = 0.0001494239

= 0.0149%, P(X > USL) = 1 − Φ 2.00 − 1.50561 0.1398143

• = 0.0002030809

= 0.0203%. So 0.0352% of parts would be outside specifications, or 352 per million.

10 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Process Capability Ratio The natural tolerance limits of the process (LNTL and UNTL) are conventionally defined as µ ± 3σ. For normally distributed measurements, they cover 99.73% of measurements. The process capability ratio (PCR) compares the specification limits with the natural tolerance limits: Cp = USL − LSL UNTL − LNTL = USL − LSL 6σ . In the example, Cp = 1.19.

11 / 12 Control Charts for Variables Control Charts for ¯ x and R

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

In R: The function process.capability() takes a qcc() object of type “xbar” and carries out all these calculations:

process.capability(qcc(flowDev, "xbar"), spec.limits = c(1, 2))

The graph displays Cp, and “Exp<LSL” and “Exp>USL”, the expected percentages below and above the specification limits, and the corresponding observed percentages. The console output adds the 95% confidence interval for Cp. The alternatives Cpl, Cpu, Cpk, and Cpm are used for off-center processes.

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