Quality control charts

In all production processes, it is necessary to monitor the extent to which the products meet specifications. In the most general terms, there are two opponents to product quality: (1) deviations from target specifications, and (2) excessive variability around target specifications. During the earlier stages of developing the production process, designed experiments are often used to optimize these two quality characteristics. The methods provided in Quality Control are on-line or in-process quality control procedures to monitor an ongoing production process.

The general approach to on-line quality control is straightforward: We simply extract samples of a certain size from the ongoing production process. We then produce line charts of the variability in those samples, and consider their closeness to target specifications. If a trend emerges in those lines, or if samples fall outside pre-specified limits, then we declare the process to be out of control and take action to find the cause of the problem. These types of charts are sometimes also referred to as Shewhart control charts (named after W.A. Shewhart who is generally credited for being the first to introduce these methods).

This procedure has been extended and applied to analytical chemistry and the control of the “production” of data in the laboratory. The principle is the same as described before, but instead of taking samples from the production process, one plots the results of the determination of a given analyte in a specific sample. This practice helps the analytical chemist to determine whether there arise unexpected problems with his analytical procedure and to detect the presence of systematic errors. Result outside the predetermined warning or action limits imply immediate review of the complete methodology and correction for any problem found.

In the chart shown above, a Shewhart or X chart, the horizontal axis represents the results obtained when analysing a given sample at time intervals. The vertical axis represents the content (individual or mean mass fraction or concentration) of the analyte of interest. A typical chart includes four additional horizontal lines to represent the upper and lower warning limits (UWL, LWL, respectively) and the upper and lower action limits (UAL, LAL, respectively).

Daily result for sample A A: 12.9 W: 18.0 M: 28.3 W: 38.5 A: 43.6 1 10 20 30 40 50 60

FIG. 7. Example of a Shewhart or X quality control.

Typically, the individual points in the chart, representing the results for the analyte, are connected by a line. If this line moves outside the upper or lower control limits or exhibits systematic patterns across consecutive samples, then a quality problem may potentially exist. Even though one could arbitrarily determine when to declare a process out of control (that is, outside the UWL-LWL range), it is common practice to set this limits at a 2s (two standard deviations) from the central line. The UAL and LAL are set at 3s (three standard deviations) from the mean (central) line.

Results falling outside the control limits are only one indication of a measurement out of control. Another indication occurs when the values fall into some sort of pattern over time. That is, an analysis in control should result in random errors about the centre line; non-random errors indicate that assignable-cause variability may exist. There are a variety of rules to use when looking for non-randomness, four of which are given here.

(1) Two of three observations in a row beyond two sigma

(2) Eight consecutive observations above or eight consecutive observations below the centre line (3) Seven observations in a row up or down

(4) Four out of five beyond one sigma.

The Shewhart chart provides a way of monitoring the results from the analysis, but it does not monitor the variability of it. Sometimes the chart will indicate that the analysis appears lo be under control, but the variability of it is not in control. More variability means the analysis is not under control because of assignable causes. Because Shewhart charts are designed to monitor the process and not the process variability, an additional control measure is necessary.

In most quality control applications, variability is measured using the range of the items in each sample. Recall that the range is the difference between the highest and lowest values in a sample. The use of the range to measure variability in quality control is partly statistical and partly historical. The statistical part stems from an advantage in the estimation process, especially for small sample sizes. The historic part is a result of the fact that, when statistical process control first originated, it was much easier for QC employees to calculate (and understand) a range rather than a standard deviation.

Small ranges suggest a small variation from results to results. That is, the analysis output is similar from item to item. A large range indicates sample items that tend to differ from one another. Thus, small values for the range of an analysis are desirable, as long as the process is under control.

Variability is monitored with a range chart, which is abbreviated as R chart. An R chart provides a plot, through time, of the range of the observations at a point in time. Even for a process where the Shewhart chart appears to be in control, an R chart may indicate that a process is not in control. As with means, a process in control will result in ranges that fall in a random pattern, within three-sigma limits. Thus, points outside the three-sigma limits and non-random points on an R chart indicate an analysis that appears to be out of control.

The patterns to look for in an R chart are the same as those in a Shewhart chart, except one no longer counts the number of observations outside one or two sigmas (because the R values cannot be assumed to be normally distributed).

Conceptually, an R chart is similar to a Shewhart chart. The average range (_{R) is the centreline.} Ǧ

This is the average of all the ranges. The estimated standard deviation of the range is designated as R, The three-sigma limits are then calculated and designated as UCLR, and LCLR. It may help to think of these control limits as follows:

UCLR = R + 3 R LCLR = R - 3 R