Mean control chart

Synonyms are X-control chart, mean control chart or average control chart. 

Control chart methods are very valuable in monitoring and controlling microbial contamination within a food establishment [24,25]. A control chart is a simple graph on which the microbial population numbers at a specific environmental site (sink, countertop, etc.) are recorded over time [24,25], The control chart s vertical axis most commonly is scaled to microbial population numbers (Fig. 2). It is termed a mean control chart because the center horizontal line represents the average target value of the process being measured. 

Mean control chart values drift and, eventually, if corrective action is not taken, may eventually exceed tolerance limits range chart data do not shift (no trend apparent) (Fig. 5). 

The mean control chart does not show a shift (trend), but the range chart does (Fig. 6). 

Use the standard deviation of a mean control chart (use of a 10 pg/l Br03 control solution)... 

Operation of a mean control chart, concentration of the control solution e.g. 10 ag/l Br03 ... 

Some measure of dispersion of the subgroup data should also be plotted as a parallel control chart. The most reliable measure of scatter is the standard deviation. For small groups, the range becomes increasingly significant as a measure of scatter, and it is usually a simple matter to plot the range as a vertical line and the mean as a point on this line for each group of observations. 

The principal tool for performance-based quality assessment is the control chart. In a control chart the results from the analysis of quality assessment samples are plotted in the order in which they are collected, providing a continuous record of the statistical state of the analytical system. Quality assessment data collected over time can be summarized by a mean value and a standard deviation. The fundamental assumption behind the use of a control chart is that quality assessment data will show only random variations around the mean value when the analytical system is in statistical control. When an analytical system moves out of statistical control, the quality assessment data is influenced by additional sources of error, increasing the standard deviation or changing the mean value. 

Control charts were originally developed in the 1920s as a quality assurance tool for the control of manufactured products.Two types of control charts are commonly used in quality assurance a property control chart in which results for single measurements, or the means for several replicate measurements, are plotted sequentially and a precision control chart in which ranges or standard deviations are plotted sequentially. In either case, the control chart consists of a line representing the mean value for the measured property or the precision, and two or more boundary lines whose positions are determined by the precision of the measurement process. The position of the data points about the boundary lines determines whether the system is in statistical control. 

Construction of Property Control Charts The simplest form for a property control chart is a sequence of points, each of which represents a single determination of the property being monitored. To construct the control chart, it is first necessary to determine the mean value of the property and the standard deviation for its measurement. These statistical values are determined using a minimum of 7 to 15 samples (although 30 or more samples are desirable), obtained while the system is known to be under statistical control. The center line (CL) of the control chart is determined by the average of these n points... 

Property control charts can also be constructed using points that are the mean value, Xj, for a set of r replicate determinations on a single sample. The mean for the ith sample is given by... 

To construct the control chart, ranges for a minimum of 15-20 samples (preferably 30 or more samples) are obtained while the system is known to be in statistical control. The line for the average range, R, is determined by the mean of these n samples... 

Interpreting Control Charts The purpose of a control chart is to determine if a system is in statistical control. This determination is made by examining the location of individual points in relation to the warning limits and the control limits, and the distribution of the points around the central line. If we assume that the data are normally distributed, then the probability of finding a point at any distance from the mean value can be determined from the normal distribution curve. The upper and lower control limits for a property control chart, for example, are set to +3S, which, if S is a good approximation for O, includes 99.74% of the data. The probability that a point will fall outside the UCL or LCL, therefore, is only 0.26%. The... 

The statistical techniques applicable to control charts are thus restricted to those of Section 1.5, that is detecting deviations from the long-term mean respectively crossing of the specified limits. 

The conventional control chart is a graph having a time axis (abscissa) consisting of a simple raster, such as that provided by graph or ruled stationary paper, and a measurement axis (ordinate) scaled to provide six to eight standard deviations centered on the process mean. Overall standard deviations are used that include the variability of the process and the analytical uncertainty. (See Fig. 1.8.) Two limits are incorporated the outer set of limits corresponds to the process specifications and the inner one to warning or action levels for in-house use. Control charts are plotted for two types of data ... 

Figure 4.10. At the top the raw data for dry residue for 63 successive batches is shown in a standard control chart format. The fact that as of batch 34 (arrow ) a different composition was manufactured can barely be discerned, see the horizontals that indicate the means DRi 33 resp. DR34 g3- A hypothesis that a change occurred as of batch 37 would find support, though. Cusum charts for base period 1. .. 63 resp. base period 1. .. 37 make the change fairly obvious, but the causative event cannot be pinpointed without further information. Starting with batch 55 (second arrow ), production switched back to the old composition.

No averaging has taken place (option 5 in the menu) the individual average is equal to the over-all mean y ,ean which is displayed as a horizontal line this corresponds to the classical use of the Cusum technique. By this means, slight shifts in the average (e.g., when plotting process parameters on control charts) can be detected even when the shift is much smaller than the process dispersion, because the Cusum trace changes slope. 

One can apply a similar approach to samples drawn from a process over time to determine whether a process is in control (stable) or out of control (unstable). For both kinds of control chart, it may be desirable to obtain estimates of the mean and standard deviation over a range of concentrations. The precision of an HPLC method is frequently lower at concentrations much higher or lower than the midrange of measurement. The act of drawing the control chart often helps to identify variability in the method and, given that variability in the method is less than that of the process, the control chart can help to identify variability in the process. Trends can be observed as sequences of points above or below the mean, as a non-zero slope of the least squares fit of the mean vs. batch number, or by means of autocorrelation.106... 

Appropriately designed, Cusum control charts give sensitive and instructive impressions on process changes. Cumulative stuns S = YT,= ( — 0) also contain information on actual as well as on previously obtained values. Therefore their display enables one to perceive earlier changes leading to OCS than by means of the chart of original values (see Woodward and Goldsmith [1964] Marshall [1977] Doerffel [1990]). 

Here the concentration range of the analyte in the ran is relatively small, so a common value of standard deviation can be assumed. Insert a control material at least once per ran. Plot either the individual values obtained, or the mean value, on an appropriate control chart. Analyse in duplicate at least half of the test materials, selected at random. Insert at least one blank determination. 

Longer (e g. n > 20) frequent runs of similar materials Again a common level of standard deviation is assumed. Insert the control material at an approximate frequency of one per ten test materials. If the run size is likely to vary from run to run it is easier to standardise on a fixed number of insertions per run and plot the mean value on a control chart of means. Otherwise plot individual values. Analyse in duplicate a minimum of five test materials selected at random. Insert one blank determination per ten test materials. 

It is often helpful to record the results of control samples in a visible manner not only because of the greater impact of a visual display but also for the relative ease with which it is possible to forecast trends. A variety of styles of quality control charts have been suggested but the most commonly used are those known as Levey-Jennings or Shewart charts, which indicate the scatter of the individual control results about the designated mean value (Procedure 1.7). 

Figure 10.2 Statistical process control charts for clearings. Top panel runs chart showing clearings as a function of measurement number. Middle panel x-bar chart with dashed upper control limit (UCL) and lower control limit (LCL) solid horizontal line is the grand mean, X. Bottom panel range chart with dashed upper control limit (UCL) solid horizontal line is the average range, r.

In preparing a control chart, the mean upper control limit (UCL) and lower control limit (LCL) of an approved process and its data are calculated. A control chart with mean UCL and LCL with no data points is created data points are added as they are statistically calculated from the raw data. (See also the chapter on control charts)... 

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