### Control Charts in Factory Management

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Assuming the measured material characteristic is normally distributed, a statistical hypothesis test will show that there is only a 0. Therefore, one can be fairly certain that measurements outside the control limits are due to controllable factors. Third, although sample standard deviation s is an acceptable way of estimating actual population standard deviation s , the sample range R, difference between the highest and lowest sample measurements is adequate for the small sample sizes usually encountered in roadway construction.

Since the sample range R is easiest to calculate, it is often used. Finally, Figures 2 and 3 are typical control charts for x sample average and R sample range. This type of control chart is called a Shewhart control chart after Dr.

## SPC for predictive quality control

Walter S. Shewhart who first proposed the general theory of control charts in This control chart can be used to monitor material quality characteristics such as HMA asphalt content, gradation or compaction, or PCC strength. Often, samples are only taken one at a time as in asphalt content monitoring and therefore require a slightly different control chart — one that tracks individual measurements and a moving range MR. Figures 4 and 5 show a control chart for individual measurements. The general rule-of-thumb is to act when measurements exceed control limits. However, many companies have expanded on this and developed their own rules such as Montgomery, [1] :.

Four popular control charts within the manufacturing industry are Montgomery, [1] : Control chart for variables. In variable sampling, measurements are monitored as continuous variables. Because they retain and use actual measurement data, variable sampling plans retain more information per sample than do attribute sampling plans Freeman and Grogan, [2]. This implies that compared to attribute sampling, it takes fewer samples to get the same information.

Because of this, most statistical acceptance plans use variable sampling. Control chart for attributes. This chart is used when a number cannot easily represent the quality characteristic. These charts look similar to control charts for variables but are based on a binomial distribution instead of a normal distribution. Two of the most common attribute control charts are for fraction nonconforming and defects. Cumulative sum control chart. A disadvantage of control charts for variables and attributes is that they only use data from the most recent measurement to draw conclusions about the process.

This makes it quite insensitive to shifts on the order of 1.

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## Control Charts

The cumulative sum control chart is a more sensitive control chart that can use information from an entire set of points to draw conclusions about the process. Basically the cumulative sum or cusum chart plots the cumulative sum of measurement deviations from an average. Used when identifying the total count of defects per unit c that occurred during the sampling period, the c -chart allows the practitioner to assign each sample more than one defect.

This chart is used when the number of samples of each sampling period is essentially the same. Similar to a c -chart, the u -chart is used to track the total count of defects per unit u that occur during the sampling period and can track a sample having more than one defect. However, unlike a c -chart, a u -chart is used when the number of samples of each sampling period may vary significantly.

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Use an np -chart when identifying the total count of defective units the unit may have one or more defects with a constant sampling size. Used when each unit can be considered pass or fail — no matter the number of defects — a p -chart shows the number of tracked failures np divided by the number of total units n. Notice that no discrete control charts have corresponding range charts as with the variable charts. The standard deviation is estimated from the parameter itself p , u or c ; therefore, a range is not required.

Although this article describes a plethora of control charts, there are simple questions a practitioner can ask to find the appropriate chart for any given use. Figure 13 walks through these questions and directs the user to the appropriate chart.

### Common Problems with SPC Control Charts

A number of points may be taken into consideration when identifying the type of control chart to use, such as:. Subgrouping is the method for using control charts as an analysis tool. The concept of subgrouping is one of the most important components of the control chart method. The technique organizes data from the process to show the greatest similarity among the data in each subgroup and the greatest difference among the data in different subgroups. The aim of subgrouping is to include only common causes of variation within subgroups and to have all special causes of variation occur among subgroups.

When the within-group and between-group variation is understood, the number of potential variables — that is, the number of potential sources of unacceptable variation — is reduced considerably, and where to expend improvement efforts can more easily be determined. The R chart displays change in the within subgroup dispersion of the process and answers the question: Is the variation within subgroups consistent? If the range chart is out of control, the system is not stable. Analytically it is important because the control limits in the X chart are a function of R-bar.

If the range chart is out of control then R-bar is inflated as are the control limit. This could increase the likelihood of calling between subgroup variation within subgroup variation and send you off working on the wrong area.

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Within variation is consistent when the R chart — and thus the process it represents — is in control. The R chart must be in control to draw the Xbar chart. The Xbar chart shows any changes in the average value of the process and answers the question: Is the variation between the averages of the subgroups more than the variation within the subgroup?

### Control Charts behind glass

The between and within analyses provide a helpful graphical representation while also providing the ability to assess stability that ANOVA lacks. Knowing which control chart to use in a given situation will assure accurate monitoring of process stability. It will eliminate erroneous results and wasted effort, focusing attention on the true opportunities for meaningful improvement. This was a nice summary of control chart construction.

Just wanted to share a couple of my thoughts that I end having to emphasize when introducing SPC. Yes, when the conditions for discrete data are present, the discrete charts are preferred. Even with a Range out of control, the Average chart can and should be plotted with actions to investigate the out of control Ranges. Hi Carl, compliments!

A great contribution to clarify some basic concepts in Control Charts. Four comments. First, the limits for attribute control charts are based on discrete probability distributions—which, you know, cannot be normal it is continuous. Thus, no attribute control chart depends on normality. Second, the range and standard deviations do not follow a normal distribution but the constants are based on the observations coming from a normal distribution. Your statement could apply to the MR-, R-, and S-charts.

## Control chart - Wikipedia

There is evidence of the robustness as you say of these charts. Third, the Xbar chart easily relies on the central limit theorem without transformation to be approximately normal for many distributions of the observations. Fourth, even for the I-chart, for many roughly symmetrical or unimodal distributions, the limits are rather robust—as you said. The d2 factor removes the bias of Rbar conversion as does the c4 factor when using the S-chart, so both are unbiased if that is what you meant by accurate. I would use the R-chart over the S-chart regardless of the subgroup size—except possibly if the charts are constructed manually.

The reason is that the R-chart is less efficient less powerful than the S-chart. In addition, as you indicated, the limits are constructed by converting Rbar into an estimate of the standard deviation by dividing by d2. Why estimate it indirectly—especially if software is doing the calculations? For the I- and Xbar-charts, the center line is the process location.

The center line is the average of this statistic across all subgroups. Similarly, for the S-, MR-, and all the attribute charts. Estimating the standard deviation,? Multiplying that number by three 3. Adding 3 x? Again, to be clearer, the average in this formula if applied generically to all control charts is the average of the statistic that is plotted on the chart. It could be the average of means, the average of ranges, average of counts, etc. It is the standard error of the statistic that is plotted.

That is, it is the standard deviation of averages in the Xbar-chart, the standard deviation of counts in the c-chart, the standard deviation of standard deviations in the S-chart, and so on.

There is a specific way to get this?. Because of the lack of clarity in the formula, manual construction of charts is often done incorrectly.

This is why it is recommended that you use software. And if they do, think about what the subgrouping assumptions really are. But what if those samples are correlated, not independent? Then you limits can be off by 2 or 3 x. Where is the discussion of correlated subgroup samples and autocorreleated averages for X-bar charts? Montgomery deals with many of the issues in his textbook on SPC. To successfully do that, we must, with high confidence, distinguish between Common Cause and Special Cause variation.

How would you separate a special cause from the potential common cause variation indicated by the statistical uncertainty? I find your comment confusing and difficult to do practically. As Understanding Statistical Process Control, by Wheeler and Chambers is used as a reference by the author, it is worth noting that this same text makes it clear that:.