R Chart

R-charts are control charts used to track the ranges of samples. The range of a sample is the difference between the largest observation and the smallest. The range is not recommended for sample sizes greater than 10 since it becomes less effective than the standard deviation as a detector of causes (American Society for Testing and Materials 1995). To construct an R-chart, data are divided into subgroupings of (usually) 

The upper and lower control limits can be found with the following formulas (American Society for Testing and Materials 1995)  

Dj is the value from the table in Appendix A D, is the value from the table in Appendix A 

R is the average range for the measurements UCL is the upper control limit LCL is the lower control limit 

Dg and can be found in the table in Appendix A. An excerpt from the table follows  

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The general procedures oudined previously for handling fluids involving the friction factor, f, and the R, chart are used with the above relations. This is applicable to compressible flow systems under the following conditions [3]. 

Consequently, to be confident in a design, those parts of the I r chart where slopes are steep should be avoided, irrespective of FT > 0.757. A simple method to achieve this is based upon the fact that for any value of R there is a maximum asymptotic value for P, say Pmax, which is given as I f tends to — oo, and is given by7 ... 

The validation process begun in Phase I is extended during Phase II. In this phase, selectivity is investigated using various batches of drugs, available impurities, excipients, and samples from stability studies. Accuracy should be determined using at least three levels of concentration, and the intermediate precision and the quantitation limit should be tested. For quality assurance evaluation of the analysis results, control charts can be used, such as the Shewart-charts, the R-charts, or the Cusum-charts. In this phase, the analytical method is refined for routine use. 

This is the Range Chart. It can be nsed only as a repeatability precision check, becanse the target valne for the analysis is not known. This chart only has npper limits. It can be nsed with the range itself (R-chart) or with the percentage difference (R%-chart). 

CSD on Rosin-Rammler Chart. lognogn/RjVlog p. Figure 1 shows some examples of CSD on R-R chart. As mentioned above, R based on Equation 4 does not obey the R-R s expression except for Rn at j=l and Rn, Rw at j=. Nevertheless, as can be seen from Figure 1, it was found that there exists an approximate linear relationship between R and p (or L) on R-R graph paper in the... 

From Equation 5 (in Table I), the slope of CSD on R-R chart at the dimensionless median size, S is given by,... 

CSD on R-R chart and j, p Values estimated by different ways (number indicates sample no. in Table IV), respectively. 

Figure 1. Some examples of CSD based on equation 2 on R-R chart. (Reproduced with permission from ref. 1. Copyright 1984 Elsevier Sdenoe Publishers B. V.)...

Fig.6 Some examples of CSD data on R-R chart O weight basis, number basis...

Figure 4.13 — (A) Serially arranged ISEs for the simultaneous determination of sodium, potassium, calcium and chloride ions. (B) Serially and parallelly arranged ISEs for the simultaneous determination of T, Br", Cl and F . WE working electrode PC personal computer Ej-E r, Cr, Br" and F ISEs, respectively C C3 amalgamated-lead columns (1.5 and 2.5 cm long, respectively) AgCl column RE reference electrode PHM pH/mV-meter S sample injection CS carrier stream P pump R chart recorder W waste. (Reproduced from [126] and [127] with permission of Elsevier Science Publishers and Pergamon Press, respectively).

Each hour, also check on three ampoules the length of the ampoules, sealing, particles, break ring presence, or any printing and note down in X and R charts for parenterals. 

Review and verify the fill volume/weight (X and R Chart) and hlling machine speed. 

In this chapter, several types of control charts for the analysis of historical data are discussed. Explanations of the use of x and R charts, for both two or more measurements per batch and only one measurement per batch, are give, along with explanations of modified control charts and cusum charts. Starting with a brief exposition on the calculation of simple statistics, the construction and graphic analysis of x and R charts are demonstrated. The concepts of under control and out of control, as well as their relationship to test specifications, are included. The chapter concludes with consideration of the question of robustness of x and R charts. 

Draw dotted horizontal lines for the UCL and LCL on x and R charts, respectively. 

Example Suppose there are 50 batches of retrospective data, with two potency values recorded for each batch. How would the x and R charts be constructed ... 

The results of the construction of the 3c and R charts may resemble the top two graphs in Figs. 2-6. The points in Fig. 2 show little evidence of trends (i.e., a rising, falling, and rising distribution of points). In such a situation, the process is said to be in control. 

Two or more consecutive points on the x or R charts fall outside control limits. 

Eight or more consecutive points on the 3c or R charts fall on the same side of the central line, even if none of the points exceed the control limits. 

In the analysis of retrospective data, the use of x and R charts has advantages and disadvantages. If no data points exceed the x or R control limits, then it is reasonable to say the process has been in control and that the standard operating procedures are fulfilling their functions. While not explicidy discussed here, data obtained from new batches can be plotted on new x and R charts using the same control limits. This new plotted data can help to warn the operator when the process is close to being or is out of control. 

While a range for any batch cannot be computed, the control limits for the x chart depend on finding R. The procedure for constructing x and R charts needs to be modified and is described below in stepwise fashion, using an example. 

Here is discussed the situation in which the R chart shows that the within batch variation is under control, but the x chart suggests the between-batch variation is out of control. When the specifications are wide, a modified control chart can be employed. Example In the following, each batch has two determinations. The upper specification for an individual determination is 15 mg/g. (Lower specification can be considered similarly.)... 

The factors A2, D3, and D4 used in the construction of x and R charts were derived from the assumption that all the retrospective data follow a normal distribution. However, random variation occurs in other nonsymmetrical forms. The term robustness refers to the extent to which the charts are still useful when the random variation of retrospective data is not normal. 

So, what happens if the random variation of the retrospective data is not normal, but has some other distributional form Are x and R charts useful in such a situation The x chart is probably useful, but the R chart is not. 

R charts. Many published studies [5-7] show that for small sample sizes per batch the factors />, and l)4 used in setting control limits are... 

As an example, 80 batches with four observations per batch were each simulated for the following random variation forms normal, exponential, and lognormal, (see Fig. 5 A-C). x and R charts were constructed for each set as if the true random variation were normal. The charts appear in Figs. 2-4. The results appear in Table 2. This table shows that roughly the same number of points falls outside the x control limits, regardless of the form of the random variation. However, the lognormal distribution has many more R values outside the control limits than the other four distributions. The operator of the process would mistakenly think this process was frequently out of control. The R chart shows greater susceptibility to nonnormality in the random error structure. 

Apart from the standard Shewart charts, the analyst can also apply X-charts, on which the mean of several replicate measurements is plotted, or R-charts, where the difference between two replicate measurements is plotted. X- and R-charts give an indication of the reproducibility of the method. Drift in analytical procedure, for example, slows changes in the system caused by the aging of parts of instruments, decalibration in wavelength, or the aging of calibration stock solutions, can be detected early when a Cusum chart (cumulative sum) is applied. In Cusum charts, the analyst reports the cumulative sum of the differences between delivered and reference values. If this reference value is certified (CRM), the Cusum chart allows the accuracy of the determination to be monitored. 

By far the most famous implementations of Shewhart s basic logic come where the plotted statistic is either the mean, the range, or, less frequently, the standard deviation. Such charts are commonly known by the names x-bar charts, R charts, and 5 charts, respectively. As a basis of discussion of Shewhart charts, consider the data given in Table 5.4. These... 

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