Control Charts for Variables

𝒙 and 𝑹 Control Charts

Montgomery, D.C. (2013) Statistical Quality Control: A Modern Introduction. 7th ed., Wiley.

Control Charts for Variables

Control Charts for 𝑥 and 𝑅

If 𝑥1, 𝑥2, … , 𝑥𝑛 is a sample of size 𝑛, then the average of this sample is

𝑥 =𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛

𝑛

Suppose that 𝑚 samples are available, each containing 𝑛 observations. Let 𝑥 1, 𝑥 2, … , 𝑥 𝑚 be the average of each sample. Then the best estimator of 𝜇, the process average, is the grand average:

𝑥 =𝑥 1 + 𝑥 2 + ⋯ + 𝑥 𝑚

𝑚

Let 𝑅1, 𝑅2, … , 𝑅𝑚 be the ranges of the 𝑚 samples (𝑅 = 𝑥𝑚𝑎𝑥 −𝑥𝑚𝑖𝑛). The average range is

𝑅 =𝑅1 + 𝑅2 + ⋯ + 𝑅𝑚

𝑚

Control Charts for 𝑥 and 𝑅

When setting up 𝑥 and 𝑅 control charts, it is best to begin with the 𝑅 chart. Because the control limits on the 𝑥 chart depend on the process variability, unless process variability is in control, these limits will not have much meaning.

Control Charts for 𝑥 and 𝑅: Phase I Application

In phase I control chart usage, when preliminary samples are used to construct 𝑥 and 𝑅 control charts, it is customary to treat the control limits as trial control limits. They allow us to determine whether the process was in control when the 𝑚 initial samples were selected.

• Plot the values of 𝑥 and 𝑅 from each sample on the charts. If all points plot inside the control limits and no systemic behavior is evident, we conclude that the process was in control in the past, and the trial control limits are suitable for controlling current or future production.

• It is highly desirable to have 20-25 samples or subgroups of size 𝑛 (typically 𝑛 is between 3 and 5) to compute the trial control limits.

• Suppose that one or more of the values of either 𝑥 or 𝑅 plot out of control when compared to the trial control limits. It is necessary to revise the trial control limits.

– Examine each of the out-of-control points , look for an assignable cause.

– If an assignable cause is found, the point is discarded and the trial control limits are recalculated, using only the remaining points.

– These remaining points are reexamined for control.

– This process is continued until all points plot in control, at which point the trial control limits are adopted for current use.

When many of the initial samples plot out of control against the trial limits, it is better to concentrate on the patterns on control charts formed by these points.

Control Charts for 𝑥 and 𝑅

Control Charts for 𝑥 and 𝑅

It is customary to define the upper and lower natural tolerance limits as 3𝜎 above and below the process mean.

Specification limits may be set by management, the manufacturing engineers, the customer, or by product developers/designers.

Control Charts for 𝑥 and 𝑅: Estimating Process Capability

• The 𝑥 and 𝑅 charts provide information about the performance or capability of the process.

• From the 𝑥 chart, we may estimate the mean flow width of the resist in the hard-bake process as 𝑥 = 1.5056 microns.

• The process standard deviation may be estimated as

𝜎 =𝑅

𝑑2=

0.32521

2.326= 0.1398 microns

Control Charts for 𝑥 and 𝑅: Estimating Process Capability

• The specification limits on flow width are 1.50 ± 0.50 microns.

• Assuming that flow width is a normally distributed random variable, with mean 1.5056 and standard deviation 0.1398, we may estimate the fraction of nonconforming wafers produced as

𝑝 = 𝑃 𝑥 < 1.00 + 𝑃 𝑥 > 2.00

= Φ1.00−1.5056

0.1398+ 1 − Φ

2.00−1.5056

0.1398

= Φ −3.61660 + 1 − Φ 3.53648

≅ 0.00015 + 1 − 0.99980 ≅ 0.00035

• About 0.035% [350 parts per million (ppm)] of the wafers produced will be outside of the specifications.

Control Charts for 𝑥 and 𝑅: Estimating Process Capability

Another way to express process capability is in terms of the process capability ratio (PCR), 𝐶𝑝

𝐶𝑝 =𝑈𝑆𝐿 − 𝐿𝑆𝐿

6𝜎

For the hard-bake process

𝐶 𝑝 =2.00 − 1.00

6 0.1398= 1.192

This implies that the ‘natural’ tolerance limits in the process (three-sigma above and below the mean) are inside the lower and upper specification limits. Consequently, a moderately small number of nonconforming wafers will be produced.

Control Charts for 𝑥 and 𝑅: Estimating Process Capability

The percentage of the specification band that the process uses up:

𝑃 =1

𝐶𝑝100%

For the hard-bake process

𝑃 =1

𝐶 𝑝100% =

1

1.192100% = 83.89%

The process uses up about 84% of the specification band.

Control Charts for 𝑥 and 𝑅

• Phase II operation of the 𝐱 and 𝐑 charts: Once a set of reliable

control limits is established, we use the control chart for

monitoring future production. This is called phase II control

chart usage.

• 20 additional samples of wafers from the hard-bake process

were collected after the control charts were established and the

sample values of 𝑥 and 𝑅 are plotted on the control charts

immediately after each sample was taken.

Control Charts for 𝑥 and 𝑅

Control Charts for 𝑥 and 𝑅

Guidelines for Control Chart Design

• Control chart design requires specification of sample size, control limit width, and sampling frequency

• Some general guidelines that will aid in control chart design: – For 𝑥 chart, choose as small a sample size is consistent with

magnitude of process shift one is trying to detect.

– For moderate to large shifts, 2𝜎 or larger, relatively small samples of size 𝑛 = 4, 5, or 6 are effective.

– For small shifts, larger sample sizes are needed.

– If a shift occurs while a sample is taken, the sample average can obscure this effect.

– If you are interested in small shifts, use the CUSUM or EWMA charts.

– For small samples, R chart is relatively insensitive to changes in process standard deviation. For larger samples (n > 10 or 12), s or s2 charts are better choices.

– Current industry practice favors small, frequent samples.

Charts based on Standard Values When it is possible to specify standard values for the process mean and standard deviation, we may use these standards to establish the control charts for 𝑥 and 𝑅 without analysis of past data.

Suppose that the standards given are 𝜇 and 𝜎.

Parameters of the 𝑥 chart:

Charts based on Standard Values

Parameters of the 𝑅 chart:

Interpretation of 𝑥 and 𝑅 Charts

Operating Characteristic Curve for an 𝑥 Control Chart

If the mean shifts from the in-control value 𝜇0 to another value 𝜇1 = 𝜇0 + 𝑘𝜎, the probability of not detecting this shift on the first subsequent sample (𝛽-risk) is

𝛽 = 𝑃 LCL ≤ 𝑥 ≤ UCL 𝜇 = 𝜇1 = 𝜇0 + 𝑘𝜎

𝛽 = Φ 𝐿 − 𝑘 𝑛 − Φ −𝐿 − 𝑘 𝑛

where Φ denotes the standard normal cumulative distribution function.

Since 𝑥 ~𝑁 𝜇, 𝜎2 𝑛 and UCL = 𝜇0 + 𝐿𝜎 𝑛 , LCL = 𝜇0 − 𝐿𝜎 𝑛 :

Operating Characteristic Curve for an 𝑥 Control Chart

Example: Suppose that we are using an 𝑥 chart with 𝐿 = 3 (the usual three-sigma limits), the sample size 𝑛 = 5, and we wish to determine the probability of detecting a shift to 𝜇1 = 𝜇0 + 2𝜎, on the first sample following the shift.

Then, since 𝐿 = 3, 𝑘 = 2, and 𝑛 = 5:

𝛽 = Φ 3 − 2 5 − Φ −3 − 2 5

= Φ −1.47 − Φ −7.37

≅ 0.0708

The probability that such a shift will be detected on the first subsequent sample is

1 − 𝛽 = 1 − 0.0708 = 0.9292

Operating Characteristic Curve for an 𝑥 Control Chart

If the shift is 1.0𝜎 and 𝑛 = 5, we have 𝛽 = 0.75, approximately.

The probability that the shift will be detected on the first sample is

1 − 𝛽 = 0.25

The probability that the shift is detected on the second sample is

β 1 − 𝛽 = 0.75 0.25 = 0.19

The probability that it is detected on the third sample is

𝛽2 1 − 𝛽 = 0.752 0.25 = 0.14.

Operating-characteristic curves for the 𝑥 chart with three-sigma limits. 𝛽 = 𝑃(not detecting a shift of 𝑘𝜎 in the mean on the first sample following the shift)

The probability that the shift will be detected on the 𝑟𝑡ℎ subsequent sample is

𝛽𝑟−1 1 − 𝛽

The Average Run Length for the 𝑥 Chart

In general, the expected number of samples taken before the shift is detected is the average run length

𝐴𝑅𝐿 = 𝑟𝛽𝑟−1 1 − 𝛽

∞

𝑟=1

=1

1 − 𝛽

In the example, we have

𝐴𝑅𝐿 =1

1 − 𝛽=

1

1 − 0.75= 4

The expected number of samples taken to detect a shift of 1.0𝜎 with 𝑛 = 5 is 4.

The Average Run Length for the 𝑥 Chart

For any Shewhart control chart,

ARL =1

𝑃(one point plots out of control )

In-control ARL:

ARL0 =1

𝛼

Out-of-control ARL:

ARL1 =1

1 − 𝛽

If the samples are taken at equally spaced intervals of time ℎ, then the average time to signal (ATS) is

ATS = ARL ℎ

Expected number of individual units sampled to detect a shift:

ATS = ARL 𝑛

Example: Samples of 𝑛 = 5 units are taken from a process every hour. The 𝑥 and 𝑅 values for a particular quality characteristic are determined. After 25 samples have been collected, we calculate 𝑥 = 20 and 𝑅 = 4.56.

a) What are the three-sigma control limits for 𝑥 and 𝑅?

b) Assume that both charts exhibit control. Estimate the process standard deviation.

c) Assume that the process output is normally distributed. The specifications are 19 ± 5. Compute the process capability ratio 𝐶𝑝.

d) How much reduction in process variability would be required to make this process a Six Sigma process?

e) If the process mean shifts to 24, what is the probability of not detecting this shift on the first subsequent sample?

Example: An 𝑥 chart is used to control the mean of a normally

distributed quality characteristic. The 𝑥 control chart is

established with standard values 𝜇 = 200 and 𝜎 = 6 . The

sample size is 𝑛 = 4 . The specifications on the quality

characteristic are LSL=185 and USL=215.

a) Find the parameters of the 𝑥 control chart.

b) Assume that the process exhibits statistical control. What is

the fraction nonconforming produced by this process?

c) If the process mean shifts to 188, find the probability that

this shift is detected on the first subsequent sample.