Quality Control Charts Synopsis



Quality Control Charts Synopsis

DS101, D. Freeman

One widely used application of statistical analysis is in the field of statistical quality control. A basic tool in that field is the control chart. Control charts are created based on the presumption that we have a sample from a stable process (all variation is common cause). The control chart then provides a visual display to test this presumption.

There are, in fact, many types of control chart designed for use with particular types of data. In DS101 we will focus on three of these types.

XBAR and R charts are designed for use with continuous variable data. These are actually two charts which are always used in combination to view variability in both averages and in the range of data in small disjoint subgroups. The XBAR chart plots the means of subgroups of data, usually of size 5 to 10. The idea is that by plotting the small group means instead of the raw data we filter out most of the white noise and leave behind specific cause variation which may then be identified visually. The more trials per subgroup the stronger the filter, so that we can look for both short term and longer term variation by adjusting the subgroup size. The range chart is based on the notion that the subgroup ranges should show little specific cause variation if the system is stable. Again we can vary the subgroup size to apply different filter strengths to the data. A nominal subgroup size is 5 for general application. These control charts are constructed assuming the normal distribution applies to the data, and so we are concerned with the mean and standard deviation of the data. In Statgraphics use the special/quality control/variable control charts/xbar and R… menu to use these charts.

P-charts are designed for use with proportional data. Proportional data represents the proportion of a fixed sample size in which a specified event (such as rejection or failure of a manufactured part) occurred. This type of event is best described using a binomial distribution with the number of trials equal to the sample size and the event probability estimated by the mean value of the proportional data. The control chart is constructed using the binomial distribution in this way. In Statgraphics use the special/quality control/attributes control charts/p-chart… menu to use these charts.

C-charts are designed to use raw count data collected over a fixed interval. Often an interval of time is used, but it may also be a fixed amount of input to a process. An example might be a count on the number of red light violations during a peak traffic hour on week days. This type of random occurrence is best described using a Poisson distribution with the mean arrival rate determined by the average count divided by the interval, e.g. violations per hour. The c-chart is constructed using a Poisson distribution in this way. In Statgraphics use the special/quality control/attributes control charts/c-chart… menu to use these charts.

Each of these chart types has two main modes of operation, an initial study mode and a control to standard mode. In an initial study data is used to represent the capability of the process under study. In control to standard mode we are using the results of past studies as a baseline for comparison. In Statgraphics the default mode is initial study. To switch modes right click on the window and select analysis options from the pop-up menu. Selection of control to standard enables you to specify process parameters to use in constructing the control chart, instead of using the current data set.

Exercises:

The file QCDATA.sf, contained in the 131data.exe file, includes samples of variable and attribute data.

1) Use the variable cereal and construct an xbar and R chart (initial study) with a subgroup size of 5.

a) What is the process mean?

b) What is the process standard deviation?

c) The upper an lower bounds on the xbar chart are indicated to be +/- 3 sigma (sigma refers to standard deviation). How is this consistent with the process standard deviation indicated? Hint: what is the standard deviation of the sampling distribution? Remember the central limit theorem.

d) Is the process in statistical control?

e) Examine the variable Cereal (using Special/Time Series/Descriptive). Does this process look stable?

f) Is the cereal variable a good choice for an initial study? Why?

g) Notice that several subgroups are painted red, indicating some problem may exist. Under tabular options select runs tests and review the statadvisor. Subgroup 20 is not on the list for runs test violations. Why is it painted red?

h) What do you notice in the xbar and range charts before subgroup 20 that might be an indication of a process drifting out of control?

i) The variable standards includes two numbers, the process mean and the process standard deviation to be used in running cereal samples against the process standard. Run the variable cereal against these standards (use control to standard analysis option). The runs test violations have mostly disappeared. Why?

2) Use the variable Tprops and construct a p-chart with a subgroup size of 50 (initial study). Assume Tprops represents a proportion of rejects from each sample, found be counting the rejects from each sample and dividing by the sample size.

a) What does the subgroup size represent in this chart?

b) Is the process in statistical control?

c) Are the upper and lower bounds symmetrical about the center line?

d) Should we be concerned that some points lie on the lower control limit? Why?

e) How many Bernoulli trials are represented in each row of Tprops?

f) What is the mean probability of rejection for each Bernoulli trial?

g) The binomial distribution for this data is represented by what two numbers?

h) Change the analysis mode of the chart to “control to standard” and enter a value of 0.032 for the parameter, p. Is the process in statistical control against the standard?

i) You have a prospective customer whose requirement is for a rejection rate less than 0.0375. Based on the data in Tprops can you meet these specifications?

The data for Tprops, it turns out was based on a variable output sample size, given in the column tsizes. Create a new variable, Tcount, and generate data for the column using the formula, round(tprops * tsize). (To create the variable double click on the first available column header and enter the name Tcount. To generate the data select the column and right click anywhere in the column and select generate data from the pop up menu. In the formula bar type in the formula.) The variable Tcount represents the raw count data. Assume it was for a fixed amount of input to the process ( or a fixed amount of operating time ).

3) Using the variable Tcount create a C-chart (initial study). Assume data was taken at a fixed input rate for one hour.

a) What is the mean arrival rate of rejects for the process?

b) Is this sufficient information to define a Poisson distribution?

c) Is the process in statistical control?

d) Can you state, based in this data, that the number of rejects per hour will be 5 or less at the 2 sigma confidence level?

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