Statistical Process Control
Site:  Saylor Academy 
Course:  BUS606: Operations and Supply Chain Management 
Book:  Statistical Process Control 
Printed by:  Guest user 
Date:  Sunday, July 21, 2024, 7:35 AM 
Description
Read this chapter on the basics of statistical process control (SPC). SPC is a standard tool for monitoring whether a process is performing as expected and, if not, where problems occur. While reading, consider how this kind of tool factors in process capacity management.
Introduction
Control charts are one of the most commonly used methods of Statisical Process Control (SPC), which monitors the stability of a process. The main features of a control chart include the data points, a centerline (mean value), and upper and lower limits (bounds to indicate where a process output is considered "out of control").They visually display the fluctuations of a particular process variable, such as temperature, in a way that lets the engineer easily determine whether these variations fall within the specified process limits. Control charts are also known as Shewhart charts after Walter Shewhart, who developed them in the early 1900's.
Source: Peter Woolf et al., https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Chemical_Process_Dynamics_and_Controls_(Woolf)/13%3A_Statistics_and_Probability_Background/13.02%3A_SPC_Basic_Control_Charts_Theory_and_Construction%2C_Sample_Size%2C_XBar%2C_R_charts%2C_S_charts
This work is licensed under a Creative Commons Attribution 4.0 License.
Control Chart Background
A process may either be classified as in control or out of control. The boundaries for these classifications are set by calculating the mean, standard deviation, and range of a set of process data collected when the process is under stable operation. Then, subsequent data can be compared to this already calculated mean, standard deviation and range to determine whether the new data fall within acceptable bounds. For good and safe control, subsequent data collected should fall within three standard deviations of the mean. Control charts build on this basic idea of statistical analysis by plotting the mean or range of subsequent data against time. For example, if an engineer knows the mean (grand average) value, standard deviation, and range of a process, this information can be displayed as a bell curve, or population density function (PDF). The image below shows the control chart for a data set with the PDF overlay.
Figure I. Control chart showing PDF for a data set
The centerline is the mean value of the data set and the green, blue and red lines represent one, two, and three standard deviations from the mean value. In generalized terms, if data points fall within three standard deviations of the mean (within the red lines), the process is considered to be in control. These rules are discussed in greater detail later in this section.
Control Charts are commonly used in six sigma control today, as a means of overall process improvement. For more on sixsigma control, see six sigma.
Control Chart Functions
The main purpose of using a control chart is to monitor, control, and improve process performance over time by studying variation and its source. There are several functions of a control chart:
 It centers attention on detecting and monitoring process variation over time.
 It provides a tool for ongoing control of a process.
 It differentiates special from common causes of variation in order to be a guide for local or management action.
 It helps improve a process to perform consistently and predictably to achieve higher quality, lower cost, and higher effective capacity.
 It serves as a common language for discussing process performance.
Sample Size and Subgrouping
There are a few key conditions that must be met when constructing control charts:
 The initial predictions for the process must be made while the process is assumed to be stable. Because future process quality will be compared to these predictions, they must be based off of a data set that is taken while the operation is running properly.
 Multiple subsets of data must be collected, where a subset is simply a set of n measurements taken over a specific time range. The number of subsets is represented as . A subset average, subset standard deviation, and subset range will be computed for each subset.
 From these subsets, a grand average, an average standard deviation, and an average range are calculated. The grand average is the average of all subset averages. The average standard deviation is simply the average of subset standard deviations. The average range is simply the average of subset ranges.
The upper and lower control limits for the process can then be determined from this data.
 Future data taken to determine process stability can be of any size. This is because any point taken should fall within the statistical predictions. It is assumed that the first occurrence of a point not falling within the predicted limits shows that the system must be unstable since it has changed from the predictive model.
 The subsets are defined, based on the data and the process. For example, if you were using a pH sensor, the sensor would most likely output pages of data daily. If you know that your sensor has the tendency to drift every day, you might select a 30 minute subset of data. If it drifts monthly you might set your subset to be 24 hours or 12 hours.
 Finally,
the population size, is assumed to be infinite. Alternatively, if the
population is finite but the sample size is less than 5% of the
population size, we can still approximate the population to be near
infinite. That is, where is the sample size and is
the population size.
XBar, RCharts, and SCharts
There are three types of control charts used determine if data is out of control, bar charts, charts and charts. An bar chart is often paired with either an chart or an chart to give a complete picture of the same set of data.
Pairing Bar with Charts
Bar (average) charts and (range) charts are often paired together. The Bar chart displays the centerline, which is calculated using the grand average, and the upper and lower control limits, which are calculated using the average range. Future experimental subsets are plotted compared to these values. This demonstrates the centering of the subset values. The chart plots the average range and the limits of the range. Again, the future experimental subsets are plotted relative to these values. The chart displays the dispersion of the subsets. Bar/Chart plot a subgroup average. Note that they should only be used when subgroups really make sense. For example, in a Gage R&R study, when operators are testing in duplicates or more, subgrouping really represents the same group.
Pairing Bar with Charts
Alternatively, Bar charts can be paired with charts (standard deviation). This is typically done when the size of the subsets are large. For larger subsets, the range is a poor statistic to estimate the distributions of the subsets, and instead, standard deviation is used. In this case, the Bar chart will display control limits that are calculated using the average standard deviation. The Charts are similar to the charts; however, instead of the range, they track the standard deviation of multiple subsets.
Smoothing Data with a Moving Average
If it is desired to have smooth data, the moving average method is one option. This method involves taking the average of a number of points, and using that average for the middle data point. From this point on, the data is treated the same as any normal group of subsets. Though this method will produce a smoother curve, it has a lag in detecting points, which may be problematic if the points are out of the acceptable range. This time lag would keep the control system from reacting to the problem until after the average is found. For this reason, moving average charts are appropriate mainly for slower processes that can handle the lag.
Reading Control Charts
Control charts can determine whether a process is behaving in an "unusual" way.
Note: The upper and lower control limits are calculated using the grand average and either the average range and average sigma. Example calculations are shown in the Creating Control Charts Section.
The quality of the individual points of a subset is determined unstable if any of the following occurs:
 Rule 1: Any point falls beyond from the centerline(this is represented by the upper and lower control limts).
 Rule 2: Two out of three consecutive points fall beyond on the same side of the centerline.
 Rule 3: Four out of five consecutive points fall beyond on the same side of the centerline.
 Rule 4: Nine or more consecutive points fall on the same side of the centerline.
The quality of a subset is determined unstable according to the following rules:
1. Any subset value is more than three standard deviations from the centerline.
2. Two consecutive subset values are more than two standard deviations from the centerline and are on the same side of the centerline.
3. Three consecutive subset values are more than one standard deviation from the centerline and are on the same side of the centerline.
Creating Control Charts
To establish upper and lower control limits on control charts, there are
a number of methods. We will discuss the method for the number of
components in a subset, , less than . For methods involving
and other techniques, see Process Control and Optimization, Liptak,
2.34. Here, the table of constants for computing limits, and the limit
equations are presented below.
Please note that Table A below does NOT contain data for a sample problem. Any time you make a control chart, you refer to this table. The values in the table are used in the equations for the upper control limit (UCL), lower control limit (LCL), etc. This will be explained in the examples below. If you are interested in how these constants were derived, there is a more detailed explanation in Control Chart Constants.
Subgroup  xbar chart 
Schart  Rchart  
Using Ra  Using Sa  
n  A2 
A3 
B3  B4 
D3  D4 
2  1.886  2.659  0  3.267  0  3.268 
3  1.023  1.954  0  2.568  0  2.574 
4  0.729  1.628  0  2.266  0  2.282 
5  0.577  1.427  0  2.089  0  2.114 
6  0.483  1.287  0.03  1.97  0  2.004 
7  0.419  1.182  0.118  1.882  0.076  1.924 
8  0.373  1.099  0.185  1.815  0.136  1.864 
9  0.337  1.032  0.239  1.761  0.184  1.816 
10 
0.308  0.975  0.284  1.716  0.223  1.777 
11  0.285  0.927  0.322  1.678  0.256  1.744 
12 
0.266  0.886  0.354  1.646  0.283  1.717 
13  0.249  0.85  0.382  1.619  0.307  1.693 
14  0.235  0.817  0.407  1.593  0.328  1.672 
15  0.223  0.789  0.428  1.572  0.347  1.653 
Table A: Table of Constants
To determine the value for , the number of subgroups
In order to determine the upper (UCL) and lower (LCL) limits for the bar charts, you need to know how many subgroups () there are in your data. Once you know the value of , you can obtain the correct constants (A2, A3, etc.) to complete your control chart. This can be confusing when you first attend to create a bar control chart. The value of is the number of subgroups within each data point. For example, if you are taking temperature measurements every min and there are three temperature readings per minute, then the value of would be. And if this same experiment was taking four temperature readings per minute, then the value of would be . Here are some examples with different tables of data to help you further in determining :
Subset#  Values (kg) 
1 (control)  1.02, 1.03, 0.98, 0.99 
2 (control)  0.96, 1.01, 1.02, 1.01 
3 (control)  0.99, 1.02, 1.03, 0.98 
4 (control)  0.96, 0.97, 1.02, 0.98 
5 (control)  1.03, 1.04, 0.95, 1.00 
6 (control)  0.99, 0.99, 1.00, 0.97 
7 (control)  1.02, 0.98, 1.01, 1.02 
8 (experimental)  1.02, 0.99, 1.01, 0.99 
9 (experimental)  1.01, 0.99, 0.97, 1.03 
10 (experimental)  1.02, 0.98, 0.99, 1.00 
11 (experimental)  0.98, 0.97, 1.02, 1.03 
Example 1: since there are four readings of kg.
time (hours) 
pH  
1  7.00  7.30  6.99  7.00 
2  7.12  7.25  7.12  7.20 
3  7.20  7.16  7.20  7.16 
4  6.98  7.00  6.94  7.00 
5  6.99  6.99  6.99  6.98 
6  7.00  6.93  7.02  6.93 
7  6.92 
7.00  6.92  7.02 
8  6.88  6.82  6.94  6.99 
9  7.10  7.00  7.00  7.00 
10 
7.21  7.02  7.21  7.04 
11  7.01  6.86  7.01  6.90 
12  6.86  6.98  6.90  6.98 
13  6.90  7.00  6.87  7.00 
14 
7.01  7.04  7.01  7.05 
15  7.00  6.95  7.00  6.99 
16  7.09  7.20  7.03  7.20 
17  6.89  7.14  6.87  7.15 
18  6.98  6.80  6.98  6.89 
19  7.00  6.90  7.00  6.90 
20  7.20  7.00  7.23  7.00 
21  7.04  7.03  7.08  7.00 
22  6.90  6.92  6.98 
6.92 
23  7.00  7.00  7.00  7.00 
24  7.00  6.97  7.01  6.98 
Example 2: since there are four readings of pH.
time (min) 
T1  T2 
T3 
0 
305.1578 
311.1926 
303.0032 
1 
308.6441 
299.2898 
307.9012 
2 
304.4789 
308.7662 
312.273 
3 
303.2384 
303.7872 
308.4915 
4 
316.6728 
303.9563 
303.3419 
5 
297.3459 
308.0937 
306.353 
6 
310.0358 
304.9309 
304.5568 
7 
302.2579 
304.0973 
317.315 
8 
305.5338 
308.5081 
308.1174 
9 
311.6743 
302.4106 
305.5727 
10 
303.535 
312.9508 
305.1281 
11 
307.5137 
312.0491 
307.6593 
12 
310.6001 
305.5229 
311.1861 
13 
307.6121 
313.0331 
313.4924 
14 
313.2346 
312.1953 
297.2964 
15 
306.0061 
301.9239 
298.6282 
16 
310.8455 
308.7776 
300.404 
17 
306.6952 
299.0904 
304.7548 
18 
305.2398 
307.3239 
297.1759 
19 
303.3781 
305.8241 
306.5276 
20 
309.3113 
316.0451 
309.9065 
Example 3: since there are three readings of temperature.
After creating multiple control charts, determining the value of n will become quite easy.
Calculating UCL and LCL
For the Bar chart the following equations can be used to establish limits, where is the grand average, is the average range, and is the average standard deviation.
Calculating Grand Average, Average Range and Average Standard Deviation
To calculate the grand average, first find the average of the n readings at each time point. The grand average is the average of the averages at each time point.
To calculate the grand range, first determine the range of the n readings at each time point. The grand range is the average of the ranges at each time point.
To calculate the average standard deviation, first determine the standard deviation of the readings at each time point. The average standard deviation is the average of the standard deviations at each time point.
Note: You will need to calculate either the grand range or the average standard deviation, not both.
For bar charts, the UCL and LCL may be determined as follows:
Alternatively, can be used as well to calculate UCL and LCL:
For charts, the and may be determined as follows:
For charts, the and may be determined as follows:
The following flow chart demonstrates the general method for constructing an bar chart, chart, or chart:
Calculating Region Boundaries
To determine if your system is out of control, you will need to section
your data into regions A, B, and C, below and above the grand average.
These regions are shown in Figure III. To calculate the boundaries
between these regions, you must first calculate the UCL and LCL. The
boundaries are evenly spaced between the UCL and LCL. One way to
calculate the boundaries is shown below.
Boundary between A and B above
Boundary Between B and C above
Example 1
Assume that in the manufacture of 1 kg Mischmetal ingots, the product weight varies with the batch. Below are a number of subsets taken at normal operating conditions (subsets 17), with the weight values given in kg. Construct the Bar, charts, and charts for the experimental data (subsets 811). Measurements are taken sequentially in increasing subset number.
Subset #  Values (kg) 
1 (control)  1.02, 1.03, 0.98, 0.99 
2 (control)  0.96, 1.01, 1.02, 1.01 
3 (control)  0.99, 1.02, 1.03, 0.98 
4 (control)  0.96, 0.97, 1.02, 0.98 
5 (control)  1.03, 1.04, 0.95, 1.00 
6 (control)  0.99, 0.99, 1.00, 0.97 
7 (control)  1.02, 0.98, 1.01, 1.02 
8 (experimental)  1.02, 0.99, 1.01, 0.99 
9 (experimental)  1.01, 0.99, 0.97, 1.03 
10 (experimental)  1.02, 0.98, 0.99, 1.00 
11 (experimental)  0.98, 0.97, 1.02, 1.03 
Solution:
First, the average, range, and standard deviation are calculated for each subset.
Subset #  Values (kg)  Average (cc) 
Range (R) 
Standard Deviation(s) 
1 (control)  1.02, 1.03, 0.98, 0.99  1.0050  0.05  0.023805 
2 (control)  0.96, 1.01, 1.02, 1.01  1.0000  0.06  0.027080 
3 (control)  0.99, 1.02, 1.03, 0.98  1.0050  0.05  0.023806 
4 (control)  0.96, 0.97, 1.02, 0.98  0.9825  0.06  0.026300 
5 (control)  1.03, 1.04, 0.95, 1.00  1.0150  0.09  0.040509 
6 (control)  0.99, 0.99, 1.00, 0.97  0.9875  0.03  0.022583 
7 (control)  1.02, 0.98, 1.01, 1.02  1.0075 
0.04  0.028930 
8 (experimental)  1.02, 0.99, 1.01, 0.99  1.0025  0.03  0.025000 
9 (experimental)  1.01, 0.99, 0.97, 1.03  1.0000  0.06  0.025820 
10 (experimental)  1.02, 0.98, 0.99, 1.00  0.9975  0.04  0.027078 
11 (experimental)  0.98, 0.97, 1.02, 1.03  1.0000  0.06  0.029409 
Next, the grand average , average range , and average standard deviation are computed for the subsets taken under normal operating conditions, and thus the centerlines are known. Here .
Bar limits are computed (using ).
Bar limits are computed (using ).
Note: Since (a relatively small subset size), both and can be used to accurately calculate the UCL and LCL.
The individual points in subsets 811 are plotted below to demonstrate how they vary with in comparison with the control limits.
Figure E1: Chart of individual points in subsets 811.
The subgroup averages are shown in the following Bar chart:
Figure E2: Bar chart for subsets 811.
Figure E3: chart for subsets 811.
Figure E4: chart for subsets 811.
The experimental data is shown to be in control, as it obeys all of the rules given above.
Example 2
It’s your first day on the job as a chemical engineer in a plant, and one of your responsibilities is to monitor the pH of a particular process. You are asked by your boss to monitor the stability of the system. She gives you some baseline data for the process, and you collect data for the process during your first day. Construct bar and Charts to report your results.
Table 1: Baseline data
time (hours)  ph  
1  7.00  7.30  6.99  7.00 
2  7.12  7.25  7.12  7.20 
3  7.20  7.16  7.20  7.16 
4  6.98  7.00  6.94  7.00 
5  6.99  6.99  6.99  6.98 
6  7.00  6.93  7.02  6.93 
7  6.92  7.00  6.92  7.02 
8  6.88  6.82  6.94  6.99 
9  7.10  7.00  7.00  7.00 
10  7.21  7.02  7.21  7.04 
11  7.01  6.86  7.01  6.90 
12  6.86  6.98  6.90  6.98 
13  6.90  7.00  6.87  7.00 
14  7.01  7.04  7.01  7.05 
15  7.00  6.95  7.00  6.99 
16  7.09  7.20  7.03  7.20 
17  6.89  7.14  6.87  7.15 
18  6.98  6.80  6.98  6.89 
19  7.00  6.90  7.00  6.90 
20  7.20  7.00  7.23  7.00 
21  7.04  7.03  7.08  7.00 
22  6.90  6.92  6.98  6.92 
23  7.00  7.00  7.00  7.00 
24 
7.00 
6.97 
7.01 
6.98 
To be consistent with the baseline data, each hour you take four pH readings. The data you collect is displayed below.
Table 2: Experimental data
time (hours)  pH  pH  pH  pH 
1  6.99  6.99  7.00  6.89 
2  6.98  7.12  7.05  6.96 
3  7.00  7.18  7.08  7.04 
4  7.01  6.94  6.98  7.00 
5  6.90  6.99  6.93  7.01 
6  6.96  7.01  7.00  7.14 
7  7.04  6.92  6.82  7.01 
8  7.00  6.93  7.00  6.90 
9  7.01  7.00  7.02  6.92 
10  7.04  7.18  6.99  6.93 
11  6.91  7.01  6.90  7.00 
12  7.00  6.97  6.98  7.18 
13  7.00  6.89  7.00  7.03 
14  7.03  7.01  7.05  6.87 
15  6.97  7.00  7.00  6.98 
16  7.03  6.97  7.02  6.98 
17  6.99  6.89  6.87  6.99 
18  6.89  6.98  6.98  6.98 
19  6.98  7.00  7.00  7.02 
20  7.02  7.15  6.97  6.98 
21  7.02  7.08  7.08  7.00 
22  6.97  7.01  6.98  7.05 
23  7.01  7.04  6.99  7.08 
24 
6.97 
7.00 
6.98 
6.98 
Solution
For this situation, there are subsets because there are 24 data sets. For each subset, because there are four pH measurements taken each hour. The first thing you do is calculate the mean and range of each subset. The means are calculated using the AVERAGE() Excel function and the ranges are calculated using MAX()  MIN(). Once these values are calculated, the Grand Average and average range are calculated. These values are simply the means of each subset's mean and range. This data is displayed below.
Table 3: Data used to calculate and grand average and Range.
Control Data
time (hours)  pH  x_ave  range 

1  7.00  7.30  6.99  7.00  7.07  0.31 
2  7.12  7.25  7.12  7.20  7.17  0.13 
3  7.20  7.16  7.20  7.16  7.18  0.04 
4  6.98  7.00  6.94  7.00  6.98  0.06 
5  6.99  6.99  6.99  6.98  6.99  0.01 
6  7.00  6.93  7.02  6.93  6.97  0.09 
7  6.92  7.00  6.92  7.02  6.97  0.10 
8  6.88  6.82  6.94  6.99  6.91  0.17 
9  7.10  7.00  7.00  7.00  7.03  0.10 
10  7.21  7.02  7.21  7.04  7.12  0.19 
11  7.01  6.86  7.01  6.90  6.95  0.15 
12  6.86  6.98  6.90  6.98  6.93  0.12 
13  6.90  7.00  6.87  7.00  6.94  0.13 
14  7.01  7.04  7.01  7.05  7.03  0.04 
15  7.00  6.95  7.00  6.99  6.99  0.05 
16  7.09  7.20  7.03  7.20  7.13  0.17 
17  6.89  7.14  6.87  7.15  7.01  0.28 
18  6.98  6.80  6.98  6.89  6.91  0.18 
19  7.00  6.90  7.00  6.90  6.95  0.10 
20  7.20  7.00  7.23  7.00  7.11  0.23 
21  7.04  7.03  7.08  7.00  7.04  0.08 
22  6.90  6.92  6.98  6.92  6.93  0.08 
23  7.00  7.00  7.00  7.00  7.00  0.00 
24  7.00  6.97  7.01  6.98  6.99  0.04 
Means:  7.01  0.12 
Now that you know and , you can calculate the upper control limit, and lower control limit, , for the bar control chart.
From Table when . Using equations UCL and LCL for bar charts listed above:
Then the and are plotted in Excel along with the average values of each subset from the experimental data to produce the bar control chart.
time (hours)  pH  x_ave  range 

1  6.99  6.99  7.00  6.89  6.97  0.11 
2  6.98  7.12  7.05  6.96  7.03  0.16 
3  7.00  7.18  7.08  7.04  7.08  0.18 
4  7.01  6.94  6.98  7.00  6.98  0.07 
5  6.90  6.99  6.93  7.01  6.96  0.11 
6  7.96  7.01  7.00  7.14  7.03  0.18 
7  7.04  6.92  6.82  7.01  6.95  0.22 
8  7.00  6.93  7.00  6.90  6.96  0.10 
9  7.01  7.18  7.02  6.92  6.99  0.10 
10  7.04  7.01  6.99 
6.93  7.04  0.25 
11  6.91  6.86  6.90  7.00  6.96  0.11 
12  7.00  6.97  6.98  7.18  7.03  0.21 
13  7.00  6.89  7.00  7.03  6.98  0.14 
14  7.03  7.01  7.05  6.87  6.99  0.18 
15  6.97  7.00  7.00  6.98  6.99  0.03 
16  7.03  6.97  7.02  6.98  7.00  0.06 
17  6.99  6.89  6.87  6.99  6.94  0.12 
18  6.89  6.98  6.98  6.98  6.96  0.09 
19  6.98  7.00  7.00  7.02  7.00  0.04 
20  7.02  7.15  6.97  6.98  7.03  0.18 
21  7.02  7.08  7.08  7.00  7.05  0.08 
22  6.97  7.01  6.98  7.05  7.00  0.08 
23  7.01  7.04  6.99  7.08  7.03  0.09 
24  6.97  7.00  6.98  6.98  6.98  0.04 
Table 4: Average subset values and ranges plotted on the bar and chart
Then, to construct the Range charts, the upper and lower control limits were found. For and so then:
From both of these charts, the process is in control because all rules for stabilty are met.
Rule 1: No point falls beyond the UCl and LCL.
Rule 2: Two out of three consecutive points do not fall beyond on the same side of the centerline.
Rule 3: Four out of five consecutive points do not fall beyond on the same side of the centerline.
Rule 4: Nine or more consecutive points do not fall on the same side of the centerline.
It's important that both of these charts be used for a given set of data because it is possible that a point could be beyond the control band in the Range chart while nothing is out of control on the bar chart.
Another issue worth noting is that if the control charts for this pH data did show some points beyond the LCL or UCL, this does not necessarily mean that the process itself is out of control. It probably just means that the pH sensor needs to be recalibrated.
Example 3
A simple outofcontrol example with a sample constructed control chart.
You have been analyzing the odd operation of a temperature sensor in one
of the plant's CSTR reactors. This particular CSTR's temperature sensor
consists of three small thermocouples spaced around the reactor: T1,
T2, and T3. The CSTR is jacketed and cooled with industrial water. The
reaction taking place in the reactor is moderately exothermic. You know
the thermocouples are working fine; you just tested them, but a
technician suggests the CSTR has been operating out of control for the
last 10 days. There have been daily samples taken and there is a control
chart created from the CSTR's grand average and standard deviation from
the year's operation.
You are assigned to see if the CSTR is operating out of control. The
grand average is 307.47 units of temperature and the grand standard
deviation is 4.67 units of temperature. The data is provided for
construction of the control chart in Table 1 and the data from the last
10 troublesome days is shown in Table 2. You decide to plot the
troublesome data onto the control chart to see if it violates any
stability rules.
value  calculate  
Upper Control Limit  316.60  μ + R_{3}σ 
Upper Zone A Threshold  313.56  μ + (2/3)* R_{3}σ 
Upper Zone B Threshold  310.51  μ + (1/3)* R_{3}σ 
Grand Average Value  307.47  μ 
Lower Zone B Threshold  304.42  μ  (1/3)* R_{3}σ 
Lower Zone A Threshold  301.38  μ  (2/3)* R_{3}σ 
Lower Control Limit  298.33  μ  R_{3}σ 
μ  307.47 
σ  4.67 
R_{3}  1.954 
Table 31. Data for Construction of Control Chart
The way I found A_3 or in this case, R_3, I used the control charts
constants table which is found on this wiki page. I decided to use the
xbar using the standard deviations) but you can also use use the
range). I found that the value for n(number of subgroups) is three since
the CSTR's temperature sensor consists of three small thermocouples (T1,T2,T3). Therefore by looking at the constant chart, I get A_3( or R_3 in this case) to be 1.954. Here's the table below:
Subgroup  xbar chart 

Using Ra 
Using Sa 

n 
A2 
A3 
2  1.886  2.659 
3  1.023  1.954 
4  0.729  1.628 
5  0.577  1.427 
6  0.483  1.287 
Also, you will notice if you used the range instead of the standard
deviation to determine the UCL,LCL, etc. that the values will be roughly
the same. Here's the table in comparing the values of UCL and LCL using
either A_2 (range) or A_3(stdev):
A3 (stdev) 
A2 (range) 

1.954  1.023  
Using Sdev 
Using Range  
UCL  316.5997379  316.473784 
LCL  298.3347317  298.460686 
Note: These values were using the same grand average (307.47), the grand standard deviation (4.67) and the grand range (8.80)
Sample Day 
T1  T2  T3 
1  299.82  310.26  306.60 
2  311.68  307.04  300.90 
3  310.94  311.68  306.83 
4  325.00  308.82  304.97 
5  321.43  300.98  311.23 
6  320.74  308.97  305.26 
7  314.77  313.25  303.36 
8  332.75  302.76  306.11 
9  319.87  296.81  305.95 
10  315.21  314.99  309.60 
Table 32. Sample Data from Past 10 Troublesome Days
Solution
When the sample data was graphed onto the control chart, the image below was seen.
Figure 31. 10Day Data Graphed Onto Control Chart
We can see from the control chart that the CSTR system is clearly out of control. Each thermocouple was tested to see which stability rules it violates.
The first thermocouple (T1) violates every stability rule.
 Rule 1  Several points from the T1 data fall above the upper control line.
 Rule 2  There are many instances where at least two out of three consecutive points fall above the zone AB threshold.
 Rule 3  There are eight consecutive points falling above the BC threshold.
 Rule 4  Nine consecutive points fall above the mean value.
Judging on this thermocouple's performance, we can say that the system is out of control, but we will analyze the other thermocouples' performance for good measure.
The second thermocouple (T2) violates stability rule 1, 2, and 3.
 Rule 1  One point falls below the lower control line.
 Rule 2  Two consecutive points (samples 9 and 10) fall beyond the AB threshold.
 Rule 3  Of the last five samples from T2, four are beyond the BC threshold.
The third thermocouple (T3) does not violate any stability rules and the results it displays are within control.
This system is out of control because the data from the thermocouples falls beyond the threshold rules for the unit's control chart. This could be explained with many potential situations. One is explained below.
If the CSTR's agitator is knocked loose, the agitation could become erratic. The erratic agitation could create eddy currents and hot spots in the CSTR.
The entire system is out of control because you know that the thermocouples are operating fine and more than one thermocouple violates the stability rules.