Control Charts for x and s Introduction The R chart, based on the - - PowerPoint PPT Presentation

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

Control Charts for ¯ x and s

Introduction The R chart, based on the sample range, is sensitive in detecting an assignable cause of variation that perturbs only a single value in a sample. In most other cases, the sample standard deviation s is a better measure for tracking process variability. The control charts are constructed in much the same way as for ¯ x and R.

1 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

Construction and operation of ¯ x and s charts The ¯ x chart with three-sigma control limits: UCL = ¯ ¯ x + A3¯ s Center line = ¯ ¯ x LCL = ¯ ¯ x − A3¯ s. Note: S2 is an unbiased estimator of σ2, but S is biased for σ. The tabulated constant A3 is of the form A3 = 3 √n × 1 c4 , where c4 is the unbiasing factor: E

• S

c4

• = σ.

2 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

The s chart with three-sigma limits: UCL = B4¯ s Center line = ¯ s LCL = B3¯ s. As for the R chart, the control limits for the s chart are symmetric around the center line except for n ≤ 5 where B3 would otherwise be negative: B4 = 1 + 3

• 1 − c2

4

c4 , B3 = max

• 1 − 3
• 1 − c2

4

c4 , 0

• .

3 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

Example 6.3: inside diameters of piston rings. In R:

pistonrings <- read.csv("Data/Table-06-03.csv") # convert from single-column to row-per-sample: diams <- qcc.groups(pistonrings$ID, pistonrings$Sample) # could be: diams <- with(pistonrings, qcc.groups(ID, Sample))

Begin with the s chart:

summary(qcc(diams, type = "S"))

4 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

Process variability looks stable, so continue with the ¯ x chart, specifying that the control limits are based on ¯ s (the unweighted average of the sample standard deviations):

summary(qcc(diams, type = "xbar", std.dev = "UWAVE-SD"))

Process mean also looks stable.

5 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

¯ x and s charts with variable sample size Suppose the sample sizes are n1, n2, . . . , nm, not all equal. For the center line on the ¯ x chart, just use a weighted average: ¯ ¯ x = m

i=1 ni ¯

xi m

i=1 ni

. To estimate σ, use (the square root of) a pooled variance: ¯ sp = m

i=1(ni − 1)s2 i

m

i=1(ni − 1)

Note: this does not simplify to ¯ s when all ni = n.

6 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

The ¯ x chart with three-sigma control limits: UCL = ¯ ¯ x + A3¯ sp Center line = ¯ ¯ x LCL = ¯ ¯ x − A3¯ sp where now A3 varies from sample to sample. Note: the formula for A3 is A3 = 3 × 1 c4 × 1 √ni where the unbiasing factor 1/c4 in qcc() reflects the degrees of freedom in s2

p, m i=1(ni − 1); using tables, we use ni − 1.

7 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

For the center line on the s chart, we could use ¯ sp or the weighted average: ¯ sw = m

i=1 nisi

m

i=1 ni

. Montgomery suggests using ¯ sp, but qcc() uses ¯ sw. The s chart with three-sigma limits: UCL = B4ˆ σ Center line = ˆ σ LCL = B3ˆ σ where ˆ σ is either ¯ sp or ¯ sw, and now B3 and B4 vary from sample to sample.

8 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

Example 6.4: piston rings with some data excluded. In R:

diamsV <- diams n <- c(5, 3, 5, 5, 5, 4, 4, 5, 4, 5, 5, 5, 3, 5, 3, 5, 4, 5, 5, 3, 5, 5, 5, 5, 5) for (i in 1:nrow(diamsV)) for (j in 1:ncol(diamsV)) if (j > n[i]) diamsV[i, j] <- NA

Begin with the s chart, necessarily based on ¯ sw:

summary(qcc(diamsV, type = "S"))

9 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

Continue with the ¯ x chart, specifying that the control limits are based on ¯ sp (the root mean square of the sample standard deviations weighted by degrees of freedom):

summary(qcc(diamsV, type = "xbar", std.dev = "RMSDF"))

Note: the default for std.dev is "UWAVE-R", unless some ni > 25, in which case it is "RMSDF".

10 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

s2 control chart As an alternative way of presenting s, we could make a chart of s2 instead of s. The center line is just the square of either ¯ s or ¯ sp; Montgomery appears to suggest ¯ s2

p (“an average sample variance”).

Control lines could similarly be squares of those for the s chart. But since s2 has a χ2-distribution, it is more natural to specify α and use percent points χ2

1−α/2,n−1 and χ2 α/2,n−1.

Of course, the square roots of those percent points could also be used on the s chart.

11 / 15 Control Charts for Variables Control Charts for ¯ x and s

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

Shewhart Control Chart for Individual Measurements

The charts discussed above are all based on samples of size n, typically 3 to 5. In some situations, we prefer to base charts on individual measurements: n = 1. Track the process mean using the individual measurements xi. Track the process variability using the moving range: MRi = |xi − xi−1|.

12 / 15 Control Charts for Variables Shewhart Control Chart for Individual Measurements

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

Control chart for individual measurements with three-sigma limits: UCL = ¯ x + 3MR d2 Center line = ¯ x LCL = ¯ x − 3MR d2 . Control chart for moving range is the same as the R chart.

13 / 15 Control Charts for Variables Shewhart Control Chart for Individual Measurements

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

Example 6.5: Loan processing costs In R:

qcc() does not make a moving range chart, so we construct a matrix

with overlapping samples of n = 2:

mortgage <- read.csv("Data/Table-06-06.csv") nr <- nrow(mortgage) mortgagePairs <- with(mortgage, cbind(Cost[1:(nr-1)], Cost[2:nr])) summary(qcc(mortgagePairs, type = "R"))

Montgomery: “the moving range chart cannot really provide useful information about a shift in process variability.”

14 / 15 Control Charts for Variables Shewhart Control Chart for Individual Measurements

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

Continue with the x chart:

summary(qcc(mortgage$Cost, type = "xbar.one"))

Note: the default control limits use the moving range of k = 2

• values. That can be over-ridden:

summary(qcc(mortgage$Cost, type = "xbar.one", std.dev = sd.xbar.one(mortgage$Cost, k = 3)))

15 / 15 Control Charts for Variables Shewhart Control Chart for Individual Measurements