2.65.5.2 Algorithm for Control ChartsAlgorithmControlCharts
Tests for Special Causes
 Test 1: One Point More Than 3 From Center Line
 Test if the point (subgroup) is out of the center line more than 3 .
 Test 2: Nine Points in a Row on The Same Side of The Center Line
 Test if there are nine consecutive points (subgroups) on the same side (all above or all below) the center line.
 Test 3: Six Points in a Row, All Increasing or All Decreasing
 Test if there are six consecutive points (subgroups) strictly monotonous.
 Test 4: Fourteen Points in a Row, Alternating Up and Down
 Test if there are fourteen consecutive points (subgroups) alternating up and down, that is one point is bigger than the previous point, and then the next point is smaller this one, alternately.
 Test 5: Two Out of Three Points More Than 2 From The Center Line (Same Side)
 Test if in 3 consecutive points (subgroups), there are 2 points out of the center line more than 2 on the same side, that is all points are above or below the center line.
 Test 6: Four Out of Five Points More Than 1 From Center Line (Same Side)
 Test if in 5 consecutive points (subgroups), there are 4 points out of the center line more than 1 on the same side, that is all points are above or below the center line.
 Test 7: Fifteen Points in a Row Within 1 of Center Line (Either Side)
 Test if there are 15 consecutive points (subgroups) within 1 of the center line, that is, the ranges of all points to the center line are less than 1 .
 Test 8: Eight Points in a Row More Than 1 From Center Line (Either Side)
 Test if there are 8 consecutive points (subgroups) out of the center line more than 1 , that is, the ranges of all points to the center line are more than 1 .
Variables Charts for Subgroups
Charts include XbarR, XbarS, IMRR/S (Between/Within), Xbar, R, S, and Zone charts.
XbarR
 Sigma Estimation: If the historical value is specified, this historical value is used, otherwise, estimated from data.
 Xbar Chart
 Plotted Points: The mean of the observations for each subgroup.

 where is the observation in the subgroup, and is the number of observations in subgroup .
 Center Line: Represents the process mean, the historical value is used if specified, otherwise, uses the mean of data calculated as follows:
 , where is the total number of observations.
 Control Limits
 For each subgroup , lower control limit (LCL) is calculated by
 For each subgroup , upper control limit (UCL) is calculated by
 where is the process mean, is the parameter for Test 1, is the process standard deviation, and is the number of observations in subgroup .
 R Chart
 Plotted Points: The range for each subgroup.

 Center Line
 , where is the number of observations in subgroup , is the value of unbiasing constant , and is the process standard deviation.
 Control Limits
 For each subgroup , lower control limit (LCL) is calculated by
 For each subgroup , upper control limit (UCL) is calculated by
 where is the parameter for Test 1, is the process standard deviation, is the number of observations in subgroup , is the value of unbiasing constant , and is the value of unbiasing constant .
XbarS
 Sigma Estimation: If the historical value is specified, this historical value is used, otherwise, estimated from data.
 Sbar: Please refer to Average of Subgroup Standard Deviations (Sbar) in Standard Deviation Estimation section for more details about the formula.
 Pooled Standard Deviation: Please refer to Pooled Standard Deviation in Standard Deviation Estimation section for more details about the formula.
 Xbar Chart: Please refere to Xbar Chart in XbarR section above.
 S Chart
 Plotted Points: The standard deviation for each subgroup, .
 Center Line
 Not use unabiasing constant:
 Use unabiasing constant:
 where is the number of observations in subgroup , is the value of unbiasing constant , and is the process standard deviation.
 Control Limits
 For each subgroup , lower control limit (LCL) is calculated by:
 Not use unabiasing constant:
 Use unabiasing constant:
 For each subgroup , upper control limit (UCL) is calculated by:
 Not use unabiasing constant:
 Use unabiasing constant:
 where is the parameter for Test 1, is the process standard deviation, is the number of observations in subgroup , is the value of unbiasing constant , and is the value of unbiasing constant .
IMRR/S (Between/Within)
 Sigma Estimation: Please refer to Standard Deviation Estimation for the details. And please note, if the historical between standard deviation is specified, is calculated by:

 I Chart
 Plotted Points: For each data point, plot the mean of each subgroup.
 Center Line: The process mean, . If a historical value is specified, use this historical value, otherwise, estimate the mean of the data.
 Control Limits
 Lower control limit (LCL) is calculated by
 Upper control limit (UCL) is calculated by
 where is the parameter for Test 1, is the process standard deviation, and is the process mean.
 MR Chart
 Plotted Points: For each data point, plot the moving range () of the means of subgroups.
 Center Line: Estimate the unbiased average of the moving range by:

 where is moving range of the means of subgroups, is the unbiasing constant, and is the number of points in the moving range.
 Control Limits
 Lower control limit (LCL) is calculated by
 Upper control limit (UCL) is calculated by
 where is the parameter for Test 1, is the process standard deviation, is the number of points in the moving range, is the value of unbiasing constant , and is the value of unbiasing constant .
 R Chart: Please refer to R Chart in XbarR section above.
 S Chart: Please refer to S Chart in XbarS section above.
Xbar
Please refer to Xbar Chart in XbarR section above.
R
Please refer to R Chart in XbarR section above.
S
Please refer to S Chart in XbarS section above.
Zone
 Sigma Estimation: Please refer to Standard Deviation Estimation section for more details about the formula.
 Plotted Points: Cumulative scores based on zones at 1, 2, and 3 standard deviations from center line. For the first point, it is plotted zone score or weight of , and then the subsequent plotted point is sum of sequential weights. If the point crosses the center line, the sum is reset to 0.
 Center Line: Overall average of the individual observations or subgroup means.
 Zone Score: There are 4 zones, and different zone has different weight.
 Zone 1: Between center line and , weight of 0
 Zone 2: Between and , weight of 2
 Zone 3: Between and , weight of 4
 Zone 4: Beyond , weight of 8
Variables Charts for Individuals
Charts include IMR, ZMR, Individuals, and Moving Range charts.
IMR
 Please refer to I Chart and MR Chart in IMRR/S(Between/Within) section above.
ZMR
 Sigma Estimation: Please refer to Standard Deviation Estimation section for more details about the formula. And there are 4 methods for estimating :
 By Runs: is estimated for each run independently.
 By Parts (Combine All Observations for Same Part): All runs data of the same part are used to estimate .
 Constant (Combine All Observations): All the data across runs and parts are used for estimation.
 Relative to Size (Combine All Observations, Use ln): First transform the data by natural log, and then use the transformed data across all runs and all parts for estimation.
 Process Mean: For different part, process mean is calculated separately. Historical values can be specified as process means too.
 Z Chart
 Plotted Points: Plot Z Chart by the data point calculated as follows:

 where is observation, is mean of group, is the standard deviation of group, and is the width of moving range.
 Center Line: It is always 0 because the data are standardized already.
 Control Limits: Because of the standarization of data, lower and upper control limits are always 3 and 3 respectively.
 MR Chart
 Plotted Points: Plot the moving range of the values in each group.
 Center Line: It is always 1.128 because the data are standardized already.
 Control Limits: Because of the standarization of data, lower control limit is always 0. And upper control limit is different for different estimation method. For average moving range, upper control limit is always 3.686, and for median moving range, it is 3.12.
Individuals
 Please refer to I Chart in IMRR/S(Between/Within) section above.
Moving Range
 Please refer to MR Chart in IMRR/S(Between/Within) section above.
Attributes Charts
Charts include P Chart Diagnostic, P, Laney P', NP, U Chart Diagnostic, U, Laney U', and C charts.
P Chart Diagnostic
 Plotted Points
 X Data
 Adjusted Counts: First of all, compute the adjusted defective counts () as follows:
 , where is the count of defectives for subgroup , is the size of subgroup , and is the mean of subgroup size.
 Transformed Counts: Then transform the adjusted counts using the formula below to get the X data:

 Y Data
 Four methods aer provided for Y data calculation, including Median Rank (Benard), Mean Rank (HerdJohnson), Modified KaplanMeier, and KaplanMeier. And formulas for these for methods are:

 where , and is the number of data points.
 Y Data Types: There are three data types for Y data available, including Percent, Probability, and Normal Score. The function calculation above for Y data is the Probability, and Percent and Normal Score are computed as:

 , where is the inverse standard normal distitribution function.
 Ratio of Observed Variation to Expected Variation
 Expected Variation
 , where is the mean of subgroup size.
 Observed Variation
 First of all, calcuate the normal scores of transformed counts (see above). Note, this normal score is different from the one for Y data above. Here is the procedure:
 From the first point of transformed counts to the last point, find out each subsequence points, which are all the same value. For each subsequence, compute normal scores by:
 , where is the data point, is the mean of the corresponding subsequence, and is the total number of data points.
 Then get the middle 50% (excluding those less than the 25th percentile or greater than the 75th percentile) of the data for use, along with the corresponding , and then perform the linear fit by the following equation:
 , then get observed variation:

 Ratio

 95% Confidence Limits for Ratio

 where is the number of subgroups, is the mean of subgroup size, , is the count of defectives for subgroup , is the size of subgroup .
 , that is to fix the lower confidence limit for the ratio to 60%.
 Decision
 Compare the ratio to the 95% upper/lower confidence limit.
 Ratio > Upper Confidence Limit: Traditional P chart may result in an elevated false alarm rate, and Laney P' chart is recommended.
 Ratio < Lower Confidence Limit: Traditional P chart may result in control limits that are too wide and Laney P' chart is recommended.
P
 Plotted Points
 , where is the number of defectives for subgroup , and is the size of subgroup .
 Center Line
 If a historical value is specified, use this historical value, otherwise, use the mean proportion of defectives from data, calculated by:
 , where is the number of defectives for subgroup , and is the size of subgroup .
 Control Limits


 where is the process proportion, is the parameter for Test 1, and is the size of subgroup .
Laney P'
 Plotted Points: The proportion of defectives for each subgroup:
 , where is the number of defectives for subgroup , and is the size of subgroup .
 Center Line
 If a historical value is specified, use this historical value, otherwise, use the mean proportion of defectives from data, calculated by:
 , where is the number of defectives for subgroup , and is the size of subgroup .
 Sigma Z
 Firstly, convert each subgroup proportion to zscore:
 Then, apply moving range of length 2 to zscore, and get sigma Z as:

 where is proportion of defectives for subgroup , is the process proportion, is the size subgroup , and is the moving range of length 2.
 Control Limits


 where is the process proportion, is the parameter for Test 1, is the size of subgroup , and is the Sigma Z calculated above.
NP
 Plotted Points: The number of defectives in each subgroup () is plotted.
 Center Line
 If a historical value is specified, use this historical value, otherwise, use the mean proportion of defectives from data, calculated by:
 , where is the number of defectives for subgroup , and is the size of subgroup .
 Then center line for each subgroup is computed as follows:

 Control Limits


 where is the process proportion, is the parameter for Test 1, and is the size of subgroup .
U Chart Diagnostic
 Please refer to P Chart Diagnostic section above for the similar procedure, but with the different calculations summaried below:
 Plotted Points
 X Data
 Transformed Counts

 Ratio of Observed Variation to Expected Variation
 Expected Variation

 95% Confidence Limits for Ratio

 where is the number of subgroups, is the mean of subgroup size, , is the defect count for subgroup , is the size of subgroup .
 Decision
 Compare the ratio to the 95% upper/lower confidence limit.
 Ratio > Upper Confidence Limit: Traditional U chart may result in an elevated false alarm rate, and Laney U' chart is recommended.
 Ratio < Lower Confidence Limit: Traditional U chart may result in control limits that are too wide and Laney U' chart is recommended.
U
 Plotted Points: The defect rate for each subgroup:
 , where is the number of defects for subgroup , and is the size of subgroup .
 Center Line
 If a historical value is specified, use this historical value, otherwise, use the mean of the data, calculated by:
 , where is the number of defectives for subgroup , and is the size of subgroup .
 Control Limits


 where is the process mean, is the parameter for Test 1, and is the size of subgroup .
Laney U'
 Plotted Points: The defect rate for each subgroup:
 , where is the number of defects for subgroup , and is the size of subgroup .
 Center Line
 If a historical value is specified, use this historical value, otherwise, use the mean of the data, calculated by:
 , where is the number of defectives for subgroup , and is the size of subgroup .
 Sigma Z
 Firstly, convert each subgroup rate to zscore:
 Then, apply moving range of length 2 to zscore, and get sigma Z as:

 where is defect rate for subgroup , is the process mean, is the size subgroup , and is the moving range of length 2.
 Control Limits


 where is the process mean, is the parameter for Test 1, is the size of subgroup , and is calculated above.
C
 Plotted Points: The number of defects in each subgroup () is plotted.
 Center Line
 If a historical value is specified, use this historical value, otherwise, use the process mean is estimated by data:
 , where is the number of defects in subgroup , and is the number of subgroups.
 Control Limits


 where is the process mean, and is the parameter for Test 1.
TimeWeighted Charts
Charts include Moving Average, EWMA, and CUSUM charts.
Moving Average
 Plotted Points

 where is the mean of the subgroup, and is the moving number for average.
 Center Line
 If a historical value is specified, use this historical value, otherwise, use the process mean is estimated by data:
 , where is the observation, and is the number of observations.
 Control Limits


 where is the process mean, is the parameter for Test 1, is the standard deviation, is the moving number for average, and is the subgroup size.
EWMA
 Plotted Points

 where is the process mean, is the mean of the subgroup, and is the weight.
 Center Line
 If a historical mean is specified, use this historical mean, otherwise, use the process mean is estimated by data:
 , where is the observation, and is the number of observations.
 Control Limits
 The standard deviation of the plotted points is calcuated by:

 And then control limits are computed by:


 where is the process mean, is the parameter for Test 1, is the standard deviation, can be the specified historical value, or calculated from data, is the weight, and is the subgroup size.
CUSUM
Tabular CUSUM
 Plotted Points
 The data plotted in a tabular CUSUM chart are and . Normally, they are initialized at 0, but if the process is out of control at startup, FIR (fast initial response) method can be used for initialization, that is


 Then the lower and upper tabular CUSUM plotted points are:


 where is the number of standard deviation for FIR, is the process standard deviation, is the subgroup size, is the mean of the subgroup, is the target, and is the size of the shift to detect.
 If the previous lower point is smaller than the lower control limit, or the previous upper point is larger than the upper control limit, and you want to reset the signal, then the calculation for and will use and instead respectively.
 Center Line
 The center line is 0.
 Control Limits


 where is the decision interval, is the process standard deviation, and is the subgroup size.
Vmask CUSUM
 Plotted Points
 The data plotted in a Vmask CUSUM chart are :

 where is the mean of the subgroup, is the target, and .
 Vmask Slope

 where is the slope of the Vmask arm, is the process standard deviation, and is the subgroup size.
 Vmask Width at Origin

 where is the decision interval, is the process standard deviation, and is the subgroup size.
 Vmask Origin
 By default, origin is estimated by number of subgroups.
