2.65.5.2 Algorithm for Control Charts

Tests for Special Causes

  • Test 1: One Point More Than 3 \sigma From Center Line
    Test if the point (subgroup) is out of the center line more than 3 \sigma.
  • 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 \sigma 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 \sigma 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 \sigma From Center Line (Same Side)
    Test if in 5 consecutive points (subgroups), there are 4 points out of the center line more than 1 \sigma on the same side, that is all points are above or below the center line.
  • Test 7: Fifteen Points in a Row Within 1 \sigma of Center Line (Either Side)
    Test if there are 15 consecutive points (subgroups) within 1 \sigma of the center line, that is, the ranges of all points to the center line are less than 1 \sigma.
  • Test 8: Eight Points in a Row More Than 1 \sigma From Center Line (Either Side)
    Test if there are 8 consecutive points (subgroups) out of the center line more than 1 \sigma, that is, the ranges of all points to the center line are more than 1 \sigma.

Variables Charts for Subgroups

Charts include Xbar-R, Xbar-S, I-MR-R/S (Between/Within), Xbar, R, S, and Zone charts.

Xbar-R

  • 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.
      \bar{X_i}=\frac{\sum_{j=1}^{n_i}X_{ij}}{n_i}
      where X_{ij} is the j^{th} observation in the i^{th} subgroup, and n_i is the number of observations in subgroup i.
    • Center Line: Represents the process mean, the historical value is used if specified, otherwise, uses the mean of data calculated as follows:
      \bar{X}=\frac{\sum_{i=1}^NX_i}{N}, where N is the total number of observations.
    • Control Limits
      For each subgroup i, lower control limit (LCL) is calculated by LCL_i=\mu-\frac{k\sigma}{\sqrt{n_i}}
      For each subgroup i, upper control limit (UCL) is calculated by UCL_i=\mu+\frac{k\sigma}{\sqrt{n_i}}
      where \mu is the process mean, k is the parameter for Test 1, \sigma is the process standard deviation, and n_i is the number of observations in subgroup i.
  • R Chart
    • Plotted Points: The range for each subgroup.
      r_i=\max(Subgroup_i)-\min(Subgroup_i)
    • Center Line
      \bar{R_i}=d_2(n_i)*\sigma, where n_i is the number of observations in subgroup i, d_2(\cdot) is the value of unbiasing constant d_2(), and \sigma is the process standard deviation.
    • Control Limits
      For each subgroup i, lower control limit (LCL) is calculated by LCL_i=\max(0, d_2(n_i)*\sigma-k*\sigma*d_3(n_i))
      For each subgroup i, upper control limit (UCL) is calculated by UCL_i=d_2(n_i)*\sigma+k*\sigma*d_3(n_i)
      where k is the parameter for Test 1, \sigma is the process standard deviation, n_i is the number of observations in subgroup i, d_2(\cdot) is the value of unbiasing constant d_2(), and d_3(\cdot) is the value of unbiasing constant d_3().

Xbar-S

  • 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 Xbar-R section above.
  • S Chart
    • Plotted Points: The standard deviation for each subgroup, s_i.
    • Center Line
      Not use unabiasing constant: \bar{S_i}=\sigma
      Use unabiasing constant: \bar{S_i}=c_4(n_i)*\sigma
      where n_i is the number of observations in subgroup i, c_4(\cdot) is the value of unbiasing constant c_4(), and \sigma is the process standard deviation.
    • Control Limits
      For each subgroup i, lower control limit (LCL) is calculated by:
      Not use unabiasing constant: LCL_i=\sigma-\frac{c_5(n_i)}{c_4(n_i)}k\sigma
      Use unabiasing constant: LCL_i=c_4(n_i)\sigma-c_5(n_i)k\sigma
      For each subgroup i, upper control limit (UCL) is calculated by:
      Not use unabiasing constant: LCL_i=\sigma+\frac{c_5(n_i)}{c_4(n_i)}k\sigma
      Use unabiasing constant: LCL_i=c_4(n_i)\sigma+c_5(n_i)k\sigma
      where k is the parameter for Test 1, \sigma is the process standard deviation, n_i is the number of observations in subgroup i, c_4(\cdot) is the value of unbiasing constant c_4(n_i)=\frac{\Gamma(\frac{n_i}{2})}{\Gamma(\frac{n_i-1}{2})}\sqrt{\frac{2}{n_i-1}}, and c_5(\cdot) is the value of unbiasing constant c_5(n_i)=\sqrt{1-c_4(n_i)^2}.

I-MR-R/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, \sigma_{xbar} is calculated by:
    \sigma_{xbar}=\sqrt{\sigma_{between}^2+\frac{\sigma_{within}^2}{SubgroupSize}}
  • I Chart
    • Plotted Points: For each data point, plot the mean of each subgroup.
    • Center Line: The process mean, \mu. 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 LCL=\mu-k\sigma
      Upper control limit (UCL) is calculated by UCL=\mu+k\sigma
      where k is the parameter for Test 1, \sigma is the process standard deviation, and \mu is the process mean.
  • MR Chart
    • Plotted Points: For each data point, plot the moving range (MR) of the means of subgroups.
    • Center Line: Estimate the unbiased average of the moving range by:
      CenterLine=MR*d_2(w)
      where MR is moving range of the means of subgroups, d_2(w) is the unbiasing constant, and w is the number of points in the moving range.
    • Control Limits
      Lower control limit (LCL) is calculated by LCL=\max(0, d_2(w)*\sigma-k*\sigma*d_3(w))
      Upper control limit (UCL) is calculated by UCL=d_2(w)*\sigma+k*\sigma*d_3(w)
      where k is the parameter for Test 1, \sigma is the process standard deviation, w is the number of points in the moving range, d_2(\cdot) is the value of unbiasing constant d_2(), and d_3(\cdot) is the value of unbiasing constant d_3().
  • R Chart: Please refer to R Chart in Xbar-R section above.
  • S Chart: Please refer to S Chart in Xbar-S section above.

Xbar

Please refer to Xbar Chart in Xbar-R section above.

R

Please refer to R Chart in Xbar-R section above.

S

Please refer to S Chart in Xbar-S 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 \bar{X_i}, 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 1\sigma, weight of 0
    Zone 2: Between 1\sigma and 2\sigma, weight of 2
    Zone 3: Between 2\sigma and 3\sigma, weight of 4
    Zone 4: Beyond 3\sigma, weight of 8

Variables Charts for Individuals

Charts include I-MR, Z-MR, Individuals, and Moving Range charts.

I-MR

Please refer to I Chart and MR Chart in I-MR-R/S(Between/Within) section above.

Z-MR

  • Sigma Estimation: Please refer to Standard Deviation Estimation section for more details about the formula. And there are 4 methods for estimating \sigma:
    • By Runs: \sigma 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 \sigma.
    • Constant (Combine All Observations): All the data across runs and parts are used for \sigma 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 \sigma 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:
      z_{i}=\frac{X_i-\mu_i}{\sigma}
      where X_i is observation, \mu is mean of group, \sigma is the standard deviation of group, and w 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 z 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 I-MR-R/S(Between/Within) section above.

Moving Range

Please refer to MR Chart in I-MR-R/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 (a_i) as follows:
      a_i=\frac{d_i}{n_i}\bar{n}, where d_i is the count of defectives for subgroup i, n_i is the size of subgroup i, and \bar{n} is the mean of subgroup size.
      Transformed Counts: Then transform the adjusted counts using the formula below to get the X data:
      X_i=\arcsin\left(\sqrt{\frac{a_i+0.375}{\bar{n}+0.75}}\right)
    • Y Data
      Four methods aer provided for Y data calculation, including Median Rank (Benard), Mean Rank (Herd-Johnson), Modified Kaplan-Meier, and Kaplan-Meier. And formulas for these for methods are:
      Y_i=\left\{\begin{array}{ll}\frac{i-0.3}{N+0.4}&Median\;Rank\;(Benard)\cr\frac{i}{N+1}&Mean\;Rank\;(Herd-Johnson)\cr\frac{i-0.5}{N}&Modified\;Kaplan-Meier\cr\frac{i}{N}&Kaplan-Meier\end{array}\right.
      where i=1, 2, 3,...,N, and N 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:
      Y_i(Percent)=Y_i*100
      Y_i(NormalScore)=\Phi^{-1}(Y_i), where \Phi^{-1} is the inverse standard normal distitribution function.
  • Ratio of Observed Variation to Expected Variation
    • Expected Variation
      ExpectedVariation=\frac{1}{\sqrt{4\bar{n}}}, where \bar{n} is the mean of subgroup size.
    • Observed Variation
      First of all, calcuate the normal scores of transformed counts (see X_i 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:
      NormalScores_i=\Phi^{-1}(\frac{Mean-0.375}{N+0.25}), where i is the i^{th} data point, Mean is the mean of the corresponding subsequence, and N 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 X_i data for use, along with the corresponding NormalScores, and then perform the linear fit by the following equation:
      NormalScores=\beta_0+\beta_1X, then get observed variation:
      ObservedVariation=\frac{1}{\beta_1}
    • Ratio
      Ratio=\frac{ObservedVariation}{ExpectedVariation}*100
  • 95% Confidence Limits for Ratio
    UpperLimit=\exp(0.185+\frac{5.62}{m}+\frac{0.274}{\bar{n}*\bar{p}})*100
    where m is the number of subgroups, \bar{n} is the mean of subgroup size, \bar{p}=\frac{\sum d_i}{\sum n_i}, d_i is the count of defectives for subgroup i, n_i is the size of subgroup i.
    LowerLimit=60, 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
    p_i=\frac{x_i}{n_i}, where x_i is the number of defectives for subgroup i, and n_i is the size of subgroup i.
  • Center Line
    If a historical value is specified, use this historical value, otherwise, use the mean proportion of defectives from data, calculated by:
    \bar{p}=\frac{\sum x_i}{\sum n_i}, where x_i is the number of defectives for subgroup i, and n_i is the size of subgroup i.
  • Control Limits
    LCL=\max(0, p-k\sqrt{\frac{p(1-p)}{n_i}})
    UCL=\min(1, p+k\sqrt{\frac{p(1-p)}{n_i}})
    where p is the process proportion, k is the parameter for Test 1, and n_i is the size of subgroup i.

Laney P'

  • Plotted Points: The proportion of defectives for each subgroup:
    p_i=\frac{x_i}{n_i}, where x_i is the number of defectives for subgroup i, and n_i is the size of subgroup i.
  • Center Line
    If a historical value is specified, use this historical value, otherwise, use the mean proportion of defectives from data, calculated by:
    \bar{p}=\frac{\sum x_i}{\sum n_i}, where x_i is the number of defectives for subgroup i, and n_i is the size of subgroup i.
  • Sigma Z
    Firstly, convert each subgroup proportion p_i to z-score: z_i=\frac{p_i-p}{\sqrt{\frac{p(1-p)}{n_i}}}
    Then, apply moving range of length 2 to z-score, and get sigma Z as:
    \sigma_z=\overline{MR}/1.128
    where p_i is proportion of defectives for subgroup i, p is the process proportion, n_i is the size subgroup i, and \overline{MR} is the moving range of length 2.
  • Control Limits
    LCL=\max(0, p-k\sqrt{\frac{p(1-p)}{n_i}}*\sigma_z)
    UCL=\min(1, p+k\sqrt{\frac{p(1-p)}{n_i}}*\sigma_z)
    where p is the process proportion, k is the parameter for Test 1, n_i is the size of subgroup i, and \sigma_z is the Sigma Z calculated above.

NP

  • Plotted Points: The number of defectives in each subgroup (x_i) 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:
    p=\bar{p}=\frac{\sum x_i}{\sum n_i}, where x_i is the number of defectives for subgroup i, and n_i is the size of subgroup i.
    Then center line for each subgroup is computed as follows:
    CenterLine_i=n_ip
  • Control Limits
    LCL=\max(0, n_ip-k\sqrt{n_ip(1-p)})
    UCL=\min(n_i, n_ip+k\sqrt{n_ip(1-p)})
    where p is the process proportion, k is the parameter for Test 1, and n_i is the size of subgroup i.

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
      X_i=\sqrt{a_i}+\sqrt{a_i+1}
  • Ratio of Observed Variation to Expected Variation
    • Expected Variation
      ExpectedVariation=1
  • 95% Confidence Limits for Ratio
    UpperLimit=\exp(0.182+\frac{5.75}{m}+\frac{0.195}{\bar{n}*\bar{u}})*100
    where m is the number of subgroups, \bar{n} is the mean of subgroup size, \bar{u}=\frac{\sum d_i}{\sum n_i}, d_i is the defect count for subgroup i, n_i is the size of subgroup i.
  • 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:
    u_i=\frac{x_i}{n_i}, where x_i is the number of defects for subgroup i, and n_i is the size of subgroup i.
  • Center Line
    If a historical value is specified, use this historical value, otherwise, use the mean of the data, calculated by:
    \bar{u}=\frac{\sum x_i}{\sum n_i}, where x_i is the number of defectives for subgroup i, and n_i is the size of subgroup i.
  • Control Limits
    LCL=\max(0, u-k\sqrt{\frac{u}{n_i}})
    UCL=u+k\sqrt{\frac{u}{n_i}}
    where u is the process mean, k is the parameter for Test 1, and n_i is the size of subgroup i.

Laney U'

  • Plotted Points: The defect rate for each subgroup:
    u_i=\frac{x_i}{n_i}, where x_i is the number of defects for subgroup i, and n_i is the size of subgroup i.
  • Center Line
    If a historical value is specified, use this historical value, otherwise, use the mean of the data, calculated by:
    \bar{u}=\frac{\sum x_i}{\sum n_i}, where x_i is the number of defectives for subgroup i, and n_i is the size of subgroup i.
  • Sigma Z
    Firstly, convert each subgroup rate u_i to z-score: z_i=\frac{u_i-u}{\sqrt{\frac{u}{n_i}}}
    Then, apply moving range of length 2 to z-score, and get sigma Z as:
    \sigma_z=\overline{MR}/1.128
    where u_i is defect rate for subgroup i, u is the process mean, n_i is the size subgroup i, and \overline{MR} is the moving range of length 2.
  • Control Limits
    LCL=\max(0, u-k\sqrt{\frac{u}{n_i}}*\sigma_z)
    UCL=u+k\sqrt{\frac{u}{n_i}}*\sigma_z
    where u is the process mean, k is the parameter for Test 1, n_i is the size of subgroup i, and \sigma_z is calculated above.

C

  • Plotted Points: The number of defects in each subgroup (x_i) is plotted.
  • Center Line
    If a historical value is specified, use this historical value, otherwise, use the process mean is estimated by data:
    \bar{c}=\frac{\sum x_i}{m}, where x_i is the number of defects in subgroup i, and m is the number of subgroups.
  • Control Limits
    LCL=\max(0, c-k\sqrt{c})
    UCL=c+k\sqrt{c}
    where c is the process mean, and k is the parameter for Test 1.

Time-Weighted Charts

Charts include Moving Average, EWMA, and CUSUM charts.

Moving Average

  • Plotted Points
    MA_i=\left\{\begin{array}{ll}\frac{\overline{X}_1+\overline{X}_2+\cdots+\overline{X}_i}{i}&i\le v\cr \frac{\overline{X}_{i-v+1}+\overline{X}_{i-v+2}+\cdots+\overline{X}_i}{v}&i>v\end{array}\right.
    where \overline{X}_i is the mean of the i^{th} subgroup, and v 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:
    \mu=\frac{\sum x_i}{m}, where x_i is the observation, and m is the number of observations.
  • Control Limits
    LCL_i=\left\{\begin{array}{ll}\mu-k\sigma\frac{\sqrt{\frac{1}{n_1}+\frac{1}{n_2}+\cdots+\frac{1}{n_i}}}{i}&i\le v\cr \mu-k\sigma\frac{\sqrt{\frac{1}{n_{i-v+1}}+\frac{1}{n_{i-v+2}}+\cdots+\frac{1}{n_i}}}{v}&i>v\end{array}\right.
    UCL_i=\left\{\begin{array}{ll}\mu+k\sigma\frac{\sqrt{\frac{1}{n_1}+\frac{1}{n_2}+\cdots+\frac{1}{n_i}}}{i}&i\le v\cr \mu+k\sigma\frac{\sqrt{\frac{1}{n_{i-v+1}}+\frac{1}{n_{i-v+2}}+\cdots+\frac{1}{n_i}}}{v}&i>v\end{array}\right.
    where \mu is the process mean, k is the parameter for Test 1, \sigma is the standard deviation, v is the moving number for average, and n_i is the i^{th} subgroup size.

EWMA

  • Plotted Points
    z_i=\left\{\begin{array}{ll}w\overline{X}_i+(1-w)\mu&i=1\cr w\overline{X}_i+(1-w)z_{i-1}&i>1\end{array}\right.
    where \mu is the process mean, \overline{X}_i is the mean of the i^{th} subgroup, and w is the weight.
  • Center Line
    If a historical mean is specified, use this historical mean, otherwise, use the process mean is estimated by data:
    \mu=\frac{\sum x_i}{m}, where x_i is the observation, and m is the number of observations.
  • Control Limits
    The standard deviation of the plotted points is calcuated by:
    \sigma_z(i)=\left\{\begin{array}{ll}\frac{w\sigma}{\sqrt{n_i}}&i=1\cr w\sigma\sqrt{\sum_{j=1}^{i}{\frac{(1-w)^{2*(i-j)}}{n_j}}}&i>1\end{array}\right.
    And then control limits are computed by:
    LCL_i=\mu-k\sigma_z(i)
    UCL_i=\mu+k\sigma_z(i)
    where \mu is the process mean, k is the parameter for Test 1, \sigma is the standard deviation, can be the specified historical value, or calculated from data, w is the weight, and n_i is the i^{th} subgroup size.

CUSUM

Tabular CUSUM

  • Plotted Points
    The data plotted in a tabular CUSUM chart are CL_i and CU_i. 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
    CL_0=\left\{\begin{array}{ll}0&No\;FIR\cr -f\frac{\sigma}{\sqrt{n_1}}&Use\;FIR\end{array}\right.
    CU_0=\left\{\begin{array}{ll}0&No\;FIR\cr f\frac{\sigma}{\sqrt{n_1}}&Use\;FIR\end{array}\right.
    Then the lower and upper tabular CUSUM plotted points are:
    CL_i=\min(0, CL_{i-1}+\overline{X}_i-(T-k\frac{\sigma}{\sqrt{n_i}}))
    CU_i=\max(0, CU_{i-1}+\overline{X}_i-(T+k\frac{\sigma}{\sqrt{n_i}}))
    where f is the number of standard deviation for FIR, \sigma is the process standard deviation,n_i is the i^{th} subgroup size, \overline{X}_i is the mean of the i^{th} subgroup, T is the target, and k 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 CL_{i-1} and CU_{i-1} will use CL_0 and CU_0 instead respectively.
  • Center Line
    The center line is 0.
  • Control Limits
    LCL_i=-h\frac{\sigma}{\sqrt{n_i}}
    UCL_i=h\frac{\sigma}{\sqrt{n_i}}
    where h is the decision interval, \sigma is the process standard deviation, and n_i is the i^{th} subgroup size.

V-mask CUSUM

  • Plotted Points
    The data plotted in a V-mask CUSUM chart are C_i:
    C_i=C_{i-1}+\overline{X}_i-T
    where \overline{X}_i is the mean of the i^{th} subgroup, T is the target, and C_0=0.
  • V-mask Slope
    Slope_i = k\frac{\sigma}{\sqrt{n_i}}
    where k is the slope of the V-mask arm, \sigma is the process standard deviation, and n_i is the i^{th} subgroup size.
  • V-mask Width at Origin
    Width_i=2h\frac{\sigma}{\sqrt{n_i}}
    where h is the decision interval, \sigma is the process standard deviation, and n_i is the i^{th} subgroup size.
  • V-mask Origin
    By default, origin is estimated by number of subgroups.