Statistical Process Control
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.
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.