This pages summarizes the calculations used for the control charts including plotted values, process sigma, process average and control limits. The calculations are divided up as follows:

- Subgroup Charts (X-R, X-s, X, R, s, Median-R, Median)
- Individuals Charts (X-mR, X, mR, z-mR, Levey_Jennings, Run)
- Between/Within Charts (X-mR-R, X-mR-s)
- Time Weighted Charts (MA/MR, EWMA, CUSUM)
- Multivariate Charts (T2)

**Subgroup Charts (X-R, X-s, X, R, s, Median-R, Median)**

**Plotted Values**

**Subgroup Average:** the average of the values in the subgroup; the equation below shows the average for the i^{th} subgroup containing n_{i} values; plotted on the X chart.

Note: the software handles varying subgroup sizes.

**Subgroup Range:** the range of the values in a subgroup; the equation below shows the range of the i^{th} subgroup where X_{max} is the maximum value in the subgroup and X_{min} is the minimum value in the subgroup; plotted on the range (R) chart.

R_{i} = X_{max} – X_{min}

**Subgroup Standard Deviation:** the standard deviation of the values in a subgroup; the equation below shows the standard deviation of the ith subgroup where X_{ij} is the jth sample in subgroup i, X_{i} is the subgroup average, and n_{i} is the number of values in the subgroup; plotted on the s (standard deviation) chart.

**Subgroup Median:** the median of the values in a subgroup (the “middle” value if the subgroup size is odd; the average of the two middle values if the subgroup size is even); plotted on the median chart.

**Process Sigma**

**Sigma:** sigma (s) can be estimated from the average subgroup range, from the average subgroup standard deviation, or from the pooled standard deviation.

**Sigma from Average Range (default option on all subgroups charts except X-s control chart):** the following equation is used to estimate sigma:

where

and r_{i} is the range for the subgroup i, d_{2} and d_{3} are control chart constants that depend on subgroup size (n_{i}).

**Sigma from Average Subgroup Standard Deviation (default for X-s control chart):** the following equation is used to estimate sigma:

where

and where s_{i} is the standard deviation of the ith subgroup and c_{4} is a control chart constant that depends on subgroup size (n_{i}).

**Sigma from Pooled Standard Deviation:** the following equation is used to estimate sigma:

s = s_{p}/c_{4}

where

and where s_{p} is the pooled standard deviation, c_{4} is a control chart constant that depends on subgroup size, x_{ij} is the jth sample of the ith subgroup, x_{i} is the average of the ith subgroup and n_{i} is the subgroup size for the i^{th} subgroup.

**Process Averages**

**Overall Average:** The overall average is the average of all the individual sample results. This will equal the average of the subgroup averages if the subgroup size does not vary. However, if the subgroup size varies, the average of the subgroup averages does not equal the average of the individual sample results. The software uses the individual sample results to calculate the overall average.

**Average Range:** the average range for a subgroup depends on the subgroup size using the following equation:

R = d_{2}s

where d_{2} is the control chart constant based on the subgroup size (n_{i}) and s is the estimate of the sigma.

**Overall Average Median (X _{m}):** the overall average median is the average of the subgroup medians.

**Control Limits**

**X Chart Control Limits**

where X is the overall average, n_{sl} is the number of sigma limits (default is 3), n is the subgroup size, and s is the estimate of sigma.

**R Chart Control Limits**

where n_{sl} is the number of sigma limits (default is 3), d_{2} and d_{3} are the control chart constants based on the subgroup size (n), and s is the estimate of sigma.

**s Chart Control Limits:** the upper control limit (UCLi) and the lower control limit (LCLi) for subgroup i are given by the following equations:

where n_{sl} is the number of sigma limits (default is 3), c_{4} and c_{5} are the control chart constants based on the subgroup size (n_{i}), and s is the estimate of sigma.

**Median Chart Control Limits: ** the upper control limit (UCLi) and the lower control limit (LCLi) for subgroup i are given by the following equations:

where X_{m} is the average subgroup median, n_{sl} is the number of sigma limits (default is 3), e_{1} is a control chart constant to adjust sigma for using the median instead of the average for the subgroup size (n), and s is the estimate of sigma.

**Individual Charts (X-mR, X, mR, z-mR, Levey-Jennings, Run)**

**Plotted Values**

**X Values**: the individual values; plotted on the X chart, Levey-Jennings chart, and run chart.

**z Values:** the z values for a given product; the following equation is the i^{th} z value for product j, X_{i} the i^{th} value for product j, ?_{j} is the estimated sigma for the product (based on the moving range of 2), Nominal_{j} is the nominal value for product j (can also use the average of product j); plotted on the z chart.

z_{i} = (X_{i}-Nominal_{j})/?_{j}

**mR Values:** the moving range between consecutive points, the following equation is the i^{th} moving range, X_{i} and X_{i-1} are two consecutive points; plotted on the moving range (mR) chart.

R_{i} – |X_{i} – X_{i-1}|

**Process Sigma**

**Sigma:** sigma (s) is estimated from the moving range for the X-mR, X, mR, and z-mR chart; the Levey-Jennings uses the calculated standard deviation; the run chart does not require an estimate of sigma.

**Sigma from the Average Moving Range:** the following equation is used to determine sigma, where mR is the average moving range and d_{2} is a control chart constant that depends on subgroup size (SPC for Excel uses n = 2 for the moving range).

**Sigma from the Calculated Standard Deviation:** the following equation is used to determine sigma for the Levey-Jennings chart, where X_{i} is the i^{th} value, X is the overall average and N is the total number of values.

**Process Averages**

The overall process average for individuals charts is calculated from the individual samples except for the z-mR chart.

**Overall Average:** the average of the individual samples, except for z chart: process average is always 0.

**Average Moving Range:** the average moving range is given by:

mR = d_{2}s

where d_{2} is the control chart constant based for n = 2 and s is the estimate of sigma from the average moving range.

**Control Limits**

**X Chart Control Limits**

where n_{sl} is the number of sigma limits (default is 3), and s is the estimate of the sigma from the average moving range.

**mR Chart Control Limits**

where n_{sl} = number of sigma limits, s = estimate of sigma from the average moving range, and d_{2} and d_{3} are control chart constants set for a subgroup size of 2.

**z Chart Control Limits**

UCL = 3

LCL = -3

**Levey-Jennings Chart Control Limits**

where n_{sl} is the number of sigma limits (default is 3), and s is the estimate of the sigma from the calculated standard deviation.

**Between/Within Charts (X-mR-R, X-mR-s)**

The calculations for these charts are the same as those given above for the X, R, s, and mR charts. Please refer to those calculations.

**Time Weighted Charts (MA/MR, EWMA, CUSUM)**

Note: the moving average/moving range (MA/MR) chart calculations are the same as given for the subgroup averages charts above. Please refer to those calculations.

**CUSUM Chart: One-Sided**

**Plotted Values**

**One-Sided Upper CUSUM:** the one-sided cumulative sum on the “high” side (above the average).

where SH_{i} = upper CUSUM for ith sample, SH_{i-1} = upper CUSUM of (i-1)^{th} sample, X_{i} = i^{th} sample result, T = target, K = difference to detect, ? = estimated standard deviation, n_{i} = size of ith sample (note: X represents an individual sample results or a subgroup average).

** One-Sided Lower CUSUM:** the one-sided cumulative sum on the “low” side (below the average).

Note: if the fast initial response option is not selected, SH_{0} = SL_{0} = 0. If the fast initial response option is selected (F = value of fast initial response) then:

**Process Average:** the average for the CUSUM is 0.

**Process Sigma:**

If the subgroup size is 1, sigma is estimated from the moving range for n = 2 (see above); if the subgroup size is greater than 1, sigma is estimated as the pooled variance (see above).

**Control Limits**

where H = action limits.

**CUSUM Chart: V-Mask**

**Plotted Values**

**Cumulative Sum Statistic:** the cumulative sum of the difference from target.

S_{i} = S_{i-1} + X_{i} – T

where S_{0} = 0, X_{i} = the i^{th} sample, and T = target.

**Process Average:** the average for the CUSUM is 0.

**Process Sigma**

If the subgroup size is 1, sigma is estimated from the moving range for n = 2 (see above); if the subgroup size is greater than 1, sigma is estimated as the pooled variance (see above).

**Control Limits**

For the last point (H = action limits):

For remaining points (K = difference to detect):

**EWMA Chart**

**Plotted Values:** the exponentially weighted moving average determined by:

EWMA_{i} = EWMA_{i-1} + w(X_{i} – EWMA_{i-1})

where w = the weighting factor, X_{i} = i^{th} sample result (or subgroup average).

**Process Sigma:** if the subgroup size is 1, sigma is estimated from the moving range for n = 2 (see above); if the subgroup size is greater than 1, sigma is estimated as the pooled variance (see above).

**Process Average:** this is the target value or the overall average.

**Control Limits**

where n_{sl} is number of sigma limits (default is 3), s = estimate of sigma, CL is the center line (either the average or the target), w = weight, and n= subgroup size.

**Multivariate Charts (T ^{2})**

**Plotted Values**

**Subgroups**

where n = subgroup size, x = subgroup average, x = overall average, S = S matrix

**Individuals**

where x = individual sample, x = overall average and S = S matrix.

**Control Limits**

**Subgroups**

where n = subgroup size, m = number of subgroups, p = number of variables, ? = confidence level, F is the F distribution.

**Individuals**

where B is the beta distribution.

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