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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Control Charts for Variables

Introduction

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, σ 2 ) 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, σ 2 /n), and “three-sigma” control limits µ ± 3σ/ n would result in the probability of a point falling outside the limits being 0.0027 .

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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¯1 + x¯2 + · · · + x¯m x¯ = . m Also conventionally, the control limits are based on the sample ranges r1 , r2 , . . . , rm , using the average range r¯ =

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r1 + r2 + · · · + rm . m

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) = d2 σ, 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.

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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: d3 , d2 d3 D3 = 1 − 3 d2

D4 = 1 + 3

except for n ≤ 6 where D3 would be negative, and instead D3 = 0. That is,   d3 D3 = max 1 − , 0 . d2 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 USL) = 1 − Φ 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: USL − LSL UNTL − LNTL USL − LSL = . 6σ

Cp =

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 “ExpUSL”, 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. 12 / 12

Control Charts for Variables

Control Charts for x¯ and R