Shewhart Control Charts

Shewhart charts indicate that a special (assignable) cause of variation is present when the sample data point plotted is outside the control limits. A 

The Range chart R chart), or the standard deviation chart, is used S chart) to monitor within sample process variation or spread (process variability at a given time). The process spread must be in-control for proper interpretation of the x chart. The x chart must be used together with a spread chart. 

If only one observation is available, individual values can be used to develop the X chart (rather than the x chart) and the range chart is developed by using the moving range concept discussed in Subsection 2.2.3. 

Describing Variation The location or central tendency of a variable is described by its mean, median or mode. The spread or scatter of a variable is described by its range or standard deviation. For small sample sizes (n 6, n = number of observations in a sampling time), the range chart or the standard deviation chart can be used. For larger sample sizes, the efficiency of computing the variance from the range is reduced drastically. Hence, the standard deviation charts should be used when n 10. 

Selection of Control Limits Three parameters affect the control limit selection  

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Shewhart Control Charts , ISO 8258 1991, International Organization for Standardization (ISO), Geneva, Switzerland, 1991. 

Both the EC50 values and the 3-pM point of the 2,3,7,8TCDD ealibration curve serve as quality criteria. For each participant, the results for both data points from all 96-well plates analyzed during the presented study were collected and reeorded in Shewhart control charts. The Shewhart control chart is used to identify variations on performanee of the DR CALUX bioassay brought about by unexpected or unassigned causes. The Shewhart eontrol chart shows the mean of the EC50 and 3-pM control point and the upper and lower eontrol limits. In Figure 2, a typical Shewhart control chart is shown. Over the analysis period, none of the participants exceeded the aetion levels (AVG 3 S). 

Precision data can be documented in bar charts or control charts such as Shewhart control charts (see the discussion of internal quality control in Section 8.2.3.5). Bar charts plot %RSD values with their corresponding confidence interval. Control charts plot the individual measurement results and the means of sets of measurements with their confidence level (or with horizontal lines representing limits, see below) as a function of the measurement number and the run number, respectively [15,55,56, 58,72, 85]. 

Two aspects are important for IQC (1) the analysis of control materials such as reference materials or spiked samples to monitor trueness and (2) replication of analysis to monitor precision. Of high value in IQC are also blank samples and blind samples. Both IQC aspects form a part of statistical control, a tool for monitoring the accuracy of an analytical system. In a control chart, such as a Shewhart control chart, measured values of repeated analyses of a reference material are plotted against the run number. Based on the data in a control chart, a method is defined either as an analytical system under control or as an analytical system out of control. This interpretation is possible by drawing horizontal lines on the chart x(mean value), x + s (SD) and x - s, x + 2s (upper warning limit) and x-2s (lower warning limit), and x + 3s (upper action or control limit) and x- 3s (lower action or control limit). An analytical system is under control if no more than 5% of the measured values exceed the warning limits [2,6, 85]. 

Shewhart control charts enable average process performance to be monitored, as reflected by the sample mean. It is also advantageous to monitor process variability. Process variability within a sample of k measurements can be characterized by its range, standard deviation, or sample variance. Consequently, control charts are often used for one of these three statistics. 

Western Electric Rules Shewhart control charts can detect abnormal process behavior by comparing individual measurements with control chart limits. But the pattern of measurements can also provide useful information. For example, if 10 consecutive measurements are all increasing, then it is very unlikely that the process is in a... 

Internal quality control (e.g., using Shewhart control charts)... 

Fig. 1 Shewhart control chart for a hypothetical powder fill process.

Two alternatives to the Shewhart control chart, which are more complicated to calculate but generally more effective to detect small shifts, are the Cumulative Sum (or Cusum) control chart and the Exponentially Weighted Moving Average (EWMA) control chart. These control charts will not be discussed here, but are described in standard references. ... 

Champ, C.W. Woodall, W.H. Exact results for Shewhart control charts with supplementary runs rules. Technometrics 1987, 29 (4), 393-399. 

French Standard, 1995. NFX 06-031-1, Application of statistics, Control Charts - Part 1 Shewhart control charts by variables. 

Another important characteristic is that of precision. This becomes evident only when repeat measurements are made, because precision refers to the amount of agreement between repeated measurements (the standard deviation around the mean estimate). Precision is subject to both random and systematic errors. In industrial quality control and chemical analysis, Shewhart Control Charts provide a means of assessing the precision of repeat measurements but these approaches are rarely used in ecotoxicity testing. The effect is that we generally understand little about either the accuracy or the precision of most bioassays. 

A conventional response to issues of variability in bioassays is to construct Shewhart Control Charts based on the results achieved in repeat tests within a laboratory using a reference toxicant. This effectively describes the range of results typically found within the laboratory and hence can be used to define limits within which the laboratory normally expects to operate. However, there is a flaw in such internal quality control because the more variable a laboratory s reference toxicant test results are, the wider the limits of acceptability will be. Indeed, it can serve merely to reinforce high variability or bias. 

Figure la. Shewhart (control chart) of control samples for the determination of lead in filter material. Concentration in fig/L, filters from NBS (currently NIST). Limits set at 2 s of reference value. 

Besides providing the basic concepts, Shewhart also provided a tool, the Shewhart control chart, for determining whether a process is dominated by common or specieil causes. The control chart is the means to operationally define the concept of a stable process. There ace many different types of control charts. The appropriate chart to use in a particular application depends in part on the type of data obtained from the process or product. 

The control chart (discussed extensively below) is an extension of the run chart. The control chart method provides a more formal way to learn from variation and guide the development of changes for improvement. The Shewhart control chart is a fundamental tool to guide improvement of processes. 

Nelson, L. S., The Shewhart Control Chart—Test for Special Causes, Journal cf Quality Technology, Vol. 16, No. 4, 1984, pp. Til-239. 

Normality assumption Most traditional SPC tools are based on the assumption that the process output characteristic is normally distributed, among which Shewhart control charts and multivariate control charts. In some cases, the central limit theorem can be used to justify approximate normality when monitoring means, but in numerous cases normality is an untenable assumption, and one is unwilling to use another parametric model. A number of nonparametric methods are available in these cases. As data availability increases, nonparametric methods seem especially useful in multivariate applications where most methods proposed thus far rely on normality. 

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