Statistical Process Control. Control Charts

Control charts are used widely in order to determine whether a process is or is not under statistical control. In other words, control charts are used to understand how a process is evolving. Basic fundament for control chart is the representation of collected observation on a chart, along with drawing the statistical limits. There is a fundamental difference between statistical limits of control and specification limits (these are usually imposed by customer requirements).

In order to use control charts, some terms must be understood:

• Population – represents the overall number of "items". (e.g. all products made during a working shift);
• Sample – a set of "items" extracted from the population. In theory, in order to get better results for statistical analyze, a larger number of samples should be taken. Most often, this is not practical, either due to high costs, destructive methods for testing, or simply small batches. Recent studies (Central limit Theorem) indicate that a sample size of 30 or more is sufficient for analyzing the population.
• Mean – the average of observation;
• Standard deviation – represents a measure of the variation of data.

Please remember that a batch represent a group of the same kind of goods, made under the same conditions. Conditions include workers, machines, environment, measurements, materials used, and working methods. As an example, if a factory is running with the same machines, at the same parameters, under the same environmental conditions, with materials of the same kind, and verifies the products in the same way, but a change in working shifts occurs, then, the batch is ending upon working shift change.

The result of a sampling process may lead to individual observations or to subgroups. Conditions where samples result in individual data are (no particular order):

• long time between sampling;
• cycle of production is very long;
• no correlation to time of observed data;
• expensive process of sampling;
• destructive method for inspection;
• each sample comes from a distinct batch;
• each sample is uniquely identified with a certain time period.

Where practical, samples may by grouped, resulting what is called subgroups. Analyze can be made working with subgroups. Example: from 20 batches of products, are randomly extracted 5 items. Each group of 5 items, extracted from a batch represents a subgroup.

If the number of subgroups is less than 5, Xbar - R charts must be used (Xbar stand for mean of each subgroup, R stands for Range of Subgroups). For subgroups larger than 5, Xbar - S charts must be used (S stands for Standard Deviation of Subgroups). Both types of charts measure the variability of the process. As a particular observation, S charts are more sensitive for small process mean shifts. S chart is preferred tool when the production output is high and collection of data is simple and/or inexpensive. Based on the high sensitivity, S chart is used when a more rigorous control is needed.

For individual observations, where no grouping is possible, I - MR charts are used (I stands for Individual and MR stands for Moving Range).

Statistical control limits are determined as a principle using the +/-3 sigma from the mean, where sigma is the standard deviation of data. After plotting the data and the control limits in the chart, if there is any point outside control limits, the process is not under statistical control, and there are special causes which lead to it. However, if all the points are within control limits, there may still exist a lack of control when:

• 9 consecutive observations on the same side of center line;
• 6 consecutive observations, all decreasing or increasing;
• 14 consecutive observations alternating up and down;
• 2 out of 3 consecutive observations, more than 2 sigma on the same side of central line;
• 4 our of 5 consecutive observations more than 1 sigma from the center line, on the same side;
• 15 consecutive observations within 1 sigma from center line, no matter which side;
• 8 consecutive observations more than 1 sigma from the center line, mo matter which side.

Mentioned values can be customized depending on the purpose of the study. As an example, for the first mentioned test, increasing the number of observation (let's say 15 instead of 9) will lead to a lower precision of determining the lack of control over the process.

An example of I-MR chart is presented below. A company is manufacturing high precision tubes. Quality Control department is interested to determine if the production process in under control. For simplicity, we assume that from each batch is measured the outer diameter of a random selected tube. Measured data are:

No.	Observation
1	99.82
2	99.63
3	99.89
4	99.45
5	100.03
6	99.76
7	100.23
8	99.81
9	99.91
10	100.12
11	100.05
12	99.78
13	100.01
14	100.04
15	99.95
Mean = 99.90
Sigma I = 0.176507
Sigma MR = 0.169733

Results of needed computations are included on the last row of the above table.

Analyzing I chart and MR chart (above) is observed that all points are within control limits determined as +/-3 sigmas from the center line. This indicates the existence of a control in the process that produces those tubes.

Let's assume that one of the measured values at position 7 is 100.98 instead of 100.23. Graph has changed as follows:

No.	Observation
1	99.82
2	99.63
3	99.89
4	99.45
5	100.03
6	99.76
7	100.98
8	99.81
9	99.91
10	100.12
11	100.05
12	99.78
13	100.01
14	100.04
15	99.95
Mean = 99.95
Sigma I = 0.297536
Sigma MR = 0.286116

In this case, I chart shows that at observation no 7, the measured value falls outside the upper control limit. The MR chart shows a very close value to the upper limit. As a conclusion, measured data indicate a process outside of control, meaning that a special cause appeared. Supplemental investigation is needed in order to determine what caused that value, outside control limits.

Pay special attention that using control charts is investigating the variation on the process, in other words if it's in control or not. Control charts are not used to assess if measured component falls within the specification imposed by customer or not. You noted that within this paper, no values were mentioned regarding the specification limits...