Outline

Quality control

Quality control chart

Control charts are used to routinely monitor quality.

Two major types:

  • univariate control: a graphical display (chart) of one quality characteristic
  • multivariate control: a graphical display of a statistic that summarizes or represents more than one quality characteristic

The control chart shows

  • the value of the quality characteristic versus the sample number or versus time
  • a center line (CL) that represents the mean value for the in-control process
  • an upper control limit (UCL) and a lower control limit (LCL)

The control limits are chosen so that almost all of the data points will fall within these limits as long as the process remains in-control.

Example

library(qcc)
data(pistonrings)
head(pistonrings,3)
##   diameter sample trial
## 1   74.030      1  TRUE
## 2   74.002      1  TRUE
## 3   74.019      1  TRUE

Piston rings for an automotive engine are produced by a forging process. The inside diameter of the rings manufactured by the process is measured on 25 samples(sample=1,2,..,25), each of size 5, for the control phase I (trial=TRUE), when preliminary samples from a process being considered ‘in-control’ are used to construct control charts. Then, further 15 samples, again each of size 5, are obtained for phase II (trial=FALSE).

Reference:

Montgomery, D.C. (1991) Introduction to Statistical Quality Control, 2nd ed, New York, John Wiley & Sons, pp. 206-213

Example

We shall treat different methods for determining LCL,CL and UCL. In that respect, it is crucial that we have

  • phase I data, where the process is in-control.
  • These data are used to determine LCL,CL and UCL.

The simple six sigma model

Assume that measurements

  • is a sample, i.e they are independent
  • they have a normal distribution
  • we know the mean \(\mu_0\) and standard deviation \(\sigma_0\).

In this case we dont need phase I data.

  • CL=\(\mu_0\).
  • LCL=\(\mu_0-k\sigma_0\).
  • UCL=\(\mu_0+k\sigma_0\).

The only parameter to determine is \(k\).

We dont want to give a lot of false warnings, and a popular choise is

  • k=3, known as the 3*sigma rule.
  • The probability of a measurement outside the control limits is then 0.27%, when the proces is in-control.

This means that the span of allowable variation is \(6\sigma_0\).

The concept ``Six Sigma’’ has become a mantra in many industrial communities.

Average Run Length (ARL)

Let \(\mathtt{pOut}\) denote the probability that a measurement is outside the control limits. On average this means that we need \(1/\mathtt{pOut}\) observations before we get an outlier.

This is known as the the Average Run Length: \[AVL=\frac{1}{\mathtt{pOut}}\] An in-control process with the 3*sigma rule has AVL

round(1/(2*pdist("norm", -3, plot = FALSE)))
## [1] 370

An in-control process with AVL=500 has k*sigma rule, where k equals

-qdist("norm", (1/2)*(1/500), plot = FALSE)
## [1] 3.090232

Types of quality control charts.

Depending on the type of control variable, there are various types of control charts.

chart distribution statistic example

xbar

normal

mean

means of a continuous process variable

S

normal

standard deviation

standard deviations of a
continuous process variable

R

normal

range

ranges of a continuous
process variable

p

binomial

proportion

percentage of faulty items

c

poisson

count

number of faulty items
during a workday

Continuous process variable

Continuous process variable

Phase I data:

  • \(m\) samples with \(n\) measurements in each sample.
  • For sample \(i=1,2,\ldots m\) calculate mean \(\bar{x}_i\) and standard deviation \(s_i\).
  • Calculate \[\bar{x}=\frac{1}{m}\sum_{i=1}^m \bar{x}_i \quad\text{and}\quad \bar{s}=\frac{1}{m}\sum_{i=1}^m s_i\]

When the sample is normal, it can be shown that \(\bar{s}\) is a biased estimate of the true standard deviation \(\sigma\):

  • \(E(\bar{s})=c_4(n)\sigma\)
  • \(c_4(n)\) is tabulated in textbooks and available in the qcc package.

Unbiased estimate of \(\sigma\): \[\hat{\sigma}_1=\frac{\bar{s}}{c_4(n)}\] Furthermore \(\bar{s}\) has estimated standard error \[se(\bar{s})=\bar{s}\frac{\sqrt{1-c_4(n)^2}}{c_4(n)}\]

xbar chart

\[ \begin{aligned} \text{UCL:}\quad &\bar{x}+3\frac{\hat{\sigma}_1}{\sqrt{n}}\\ \text{CL:}\quad &\bar{x}\\ \text{LCL:}\quad &\bar{x}-3\frac{\hat{\sigma}_1}{\sqrt{n}} \end{aligned} \] This corresponds to

  • The probability of a measurement outside the control limits is 0.27%.

If we want to change this probability, we need another z-score. E.g if we want to lower this probability to 0.1%, then \(3\) should be substituted by \(3.29\).

Example

phaseI  <- matrix(pistonrings$diameter[1:125]  , nrow=25, byrow=TRUE)
phaseII <- matrix(pistonrings$diameter[126:200], nrow=25, byrow=TRUE)
h <- qcc(phaseI, type = "xbar", std.dev = "UWAVE-SD",
         newdata = phaseII, title = "xbar chart: pistonrings")
  • phaseI is a matrix with \(m=25\) rows, where each row is a sample of size \(n=5\).
  • Similarly phaseII has 15 samples.

The function qcc calculates the necessary statistics and optionally makes a plot.

  • phaseI and type= are the only arguments required.
  • We want that the limits are based on the unweighted average of standard deviations - UWAVE-SD. This is not the default.
  • We also want to evaluate the phase II data: newdata=phaseII.
  • Optionally, we can specify the title on the plot.

Example

Besides limits we are also told whether the process is above/below CL for 7 or more consecutive samples (yellow dots).

run.length=7 is default, but may be changed. If we e.g. want this to happen with probability 0.2%, then we specify run.length=10.

S chart: Monitoring variability

In most situations, it is crucial to monitor the process mean.

But it may also be a problem if the variability in “quality” gets too high.

In that respect, it is relevant to monitor the standard deviation, which is done by the S-chart:

\[ \begin{aligned} \text{UCL:}\quad&\bar{s}+3se(\bar{s})\\ \text{CL:}\quad &\bar{s}\\ \text{LCL:}\quad &\bar{s}-3se(\bar{s})\\ &se(\bar{s})=\bar{s}\frac{\sqrt{1-c_4(n)^2}}{c_4(n)} \end{aligned} \]

Where \(3\) may be substituted by some other z-score depending on the required confidence level.

h <- qcc(phaseI,type="S", newdata=phaseII, title="S chart: pistonrings")

S chart example

Remark that the plot does not allow values below zero.

Quite sensible when we are talking about standard deviations.

R chart: Range statistics

If the sample size is relatively small (\(n\leq 10\)), it is custom to use the range \(R\) instead of the standard deviation. The range of a sample is simply the difference between the largest and smallest observation.

When the sample is normal, it can be shown that:

  • \(E(\bar{R})=d_2(n)\sigma\), where \(\bar{R}\) is the average of the \(m\) sample ranges.
  • \(d_2(n)\) is tabulated in textbooks and available in the qcc package.

Unbiased estimate of \(\sigma\): \[\hat{\sigma}_2=\frac{\bar{R}}{d_2(n)}\] Furthermore \(\bar{R}\) has estimated standard error \[se(\bar{R})=\bar{R}\frac{d_3(n)}{d_2(n)}\] \(d_3(n)\) is tabulated in textbooks and available in the qcc package.

Charts based on R

xbar chart based on \(\bar{R}\): \[ \begin{aligned} \text{UCL:}\quad&\bar{x}+3\frac{\hat{\sigma}_2}{\sqrt{n}}\\ \text{CL:}\quad &\bar{x}\\ \text{LCL:}\quad &\bar{x}-3\frac{\hat{\sigma}_2}{\sqrt{n}} \end{aligned} \] This is actually the default in the qcc package.

R chart to monitor variability: \[ \begin{aligned} \text{UCL:}\quad & \bar{R}+3se(\bar{R})\\ \text{CL:}\quad & \bar{R}\\ \text{LCL:}\quad & \bar{R}-3se(\bar{R})\\ \end{aligned} \]

R chart example

h <- qcc(phaseI, type="R", newdata=phaseII, title="R chart: pistonrings")

Binomial process variable

Binomial variation

Let us suppose that the production process operates in a stable manner such that

  • the probability that an item is defect is \(p\).
  • successive items produced are independent

In a random sample of n items, the number D of defective items follows a binomial distribution with parameters \(n\) and \(p\).

Unbiased estimate of \(p\): \[\hat{p}=\frac{D}{n}\] which has standard error \[se(\hat{p})=\sqrt{\frac{p(1-p)}{n}}\]

p chart

Data from phase I:

  • \(m\) samples with estimated proportions \(\hat{p}_i,\ \ i=1,\ldots,m\)
  • \(\bar{p}\) is the average of the estimated proportions.

p chart: \[ \begin{aligned} \text{UCL:}\quad & \bar{p}+3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}\\ \text{CL:}\quad & \bar{p}\\ \text{LCL:}\quad & \bar{p}-3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}\\ \end{aligned} \]

Example

data(orangejuice)
head(orangejuice, 3)
##   sample  D size trial
## 1      1 12   50  TRUE
## 2      2 15   50  TRUE
## 3      3  8   50  TRUE

Production of orange juice cans.

  • The data were collected as 30 samples of 50 cans.
  • The number of defective cans D were observed.
  • After the first 30 samples, a machine adjustment was made.
  • Then further 24 samples were taken from the process.
with(orangejuice,
     qcc(D[trial], sizes=size[trial], type="p",
         newdata=D[!trial], newsizes=size[!trial]))

Example

The machine adjustment after sample 30 has had an obvious effect.

The chart should be recalibrated.

Poisson process variable

Poisson variation

Let us suppose that the production process operates in a stable manner such that

  • defective items are produced at a constant rate

The number D of defective items over a time interval of some fixed length follows a poisson distribution with mean value \(c\).

Unbiased estimate of \(c\): \[\hat{c}=D\] which has standard error \[se(\hat{c})=\sqrt{c}\]

c chart

Data from phase I:

  • \(m\) sampling periods with mean estimates \(\hat{c}_i,\ \ i=1,\ldots,m\)
  • \(\bar{c}\) is the average of the estimated means.

c chart: \[ \begin{aligned} \text{UCL:}\quad & \bar{c}+3\sqrt{\bar{c}}\\ \text{CL:}\quad & \bar{c}\\ \text{LCL:}\quad & \bar{c}-3\sqrt{\bar{c}}\\ \end{aligned} \]