SPC Indicators Formulas#
Point Statistics#
Point statistics are calculated based on the single data point data and are mostly applicable to variable charts. The point statistics that are displayed in the GUI are described in the table below:
| Indicator | Formula | Display For |
|---|---|---|
| Count | Count | All Variable Charts |
| Max | Max | All Variable Charts |
| Min | Min | All Variable Charts |
| Mean | \(\overline{x}=\frac{x_1+x_2+x_3+\cdots+x_n}{n}\) | All Variable Charts |
| Standard Deviation | \(s=\sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}}\) | All Variable Charts when the number of readings is greater than one |
| Median | \(\widetilde{x}\)= The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is the average of the two middle values. | All Variable Charts |
| Range | \(Range=Max(x_{1\cdots n})-Min(x_{1\cdots n})\) | All Variable Charts |
| Cp - Process Capability | \(C_p=\frac{USL-LSL}{6s}\) Note: this value is only calculated in both the USL and the LSL are present for the data point. | All Variable Charts when the number of readings is greater than one |
| Cpk - Process Capability Index | \(Min(\frac{USL-\overline{x}}{3s},\frac{\overline{x}-LSL}{3s})\) | All Variable Charts when the number of readings is greater than one |
| Cir | \(Cir=\frac{3\cdot \sqrt{(\overline{x}-TargetSpecValue)^2+s^2}}{Min(USL\;-\;TargetSpecValue,\;TargetSpecValue-LSL)}\) | All Variable Charts when the number of readings is greater than one |
Table: Point Statistics
Chart Statistics#
The Chart statistics described below are calculated based on all the data points that are visible in the GUI for the Logical Chart and the calculations exclude Deleted and Excluded data points. All formulas apply to the indicator one only.
Some Chart statistics use a single value for specification limits and control limits. Because the specification limits (Lower, Target, Upper) as well as Centerlines and Control Limits (Lower Control Limit and Upper Control Limit) may change from data point to data point, changes in these values will be handled based on the value defined in the configuration entry Cmf/System/Configuration/SPC/MultipleDataPointsStatistics/CalculationMode/ that can have two values:
- Average (default and used if no value is present) - uses the average of the values for the considered data points (default). The considered data points are only the ones that contains the required values (Lower Specification Limit, Target Specification Value, Upper Specification, Lower Control Limit, Centerline and Upper Control Limit).
- Last - uses the values (Lower Specification Limit, Target Specification Value, Upper Specification, Lower Control Limit, Centerline and Upper Control Limit) from the last data point, if the values are defined for the data point.
| Indicator | Formula | Data Points | Readings | Display For | Notes |
|---|---|---|---|---|---|
| Count | Count | x | All Charts | All data points | |
| Max | Max | x | All Charts | ||
| Min | Min | x | All Charts | ||
| Mean | \(\overline{x}=\frac{x_1+x_2+x_3+\cdots+x_n}{n}\) | x | All Charts | ||
| Overall Mean | \(\mu=\frac{x_1+x_2+x_3+\cdots+x_n}{n}\) | x | Not Displayed | This indicator considers all individual readings and it excludes data points which are aggregated | |
| Standard Deviation | \(s=\sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}}\) | x | All Charts (when Data Points > 1) | ||
| Overall Standard Deviation | \(\delta=\sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}}\) | x | Not Displayed | This indicator considers all individual readings and it excludes data points which are aggregated | |
| Median | \(\widetilde{x}\) | x | All Charts | The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is the average of the two middle values. | |
| Range | \(Range=Max(x_{1\cdots n})-Min(x_{1\cdots n})\) | x | All Charts | ||
| Within Standard Deviation | For Individuals and Moving Range Charts: \(\hat{\delta}_{Within}=\frac{\overline{MR}}{d2(w)}\) where \(d2(w)\) with w=2 is 1,1284.For the other types of Variable Charts, the Pooled Standard Deviation (Sp) is calculated according to the formula: \(S_p=\sqrt{\frac{\left [ \sum_{i}\sum_{j}{(X_{ij} - \overline{X_i})^2} \right ]}{\left [ \sum_{i}{(n_{i} - 1} \right ]}}\) Then, the degrees of freedom (d) is calculated as: \(\sum(n_i-1)\) And finally, the Within Standard Deviation is calculated using the formula: \(\hat{\delta}_{Within}=\frac{Sp}{c4(d+1)}\) | x | x | Not Displayed | |
| Cp - Process Capability | \(C_p=\frac{USL-LSL}{6\hat{\delta}_{Within}}\) | x | x | All Variable Charts (when Data Points > 1) | About the data points to be considered - in case of changing or missing USL/LSL, please refer to the Calculation Mode described earlier in this document. |
| Cpu - Upper Process Capability | \(C_{p,upper}=\frac{USL-\mu}{3\hat{\delta}_{Within}}\) \(\mu\) to be used is the overall mean for the all the individual readings of the data points considered - same as Pp | x | x | Not Displayed | About the data points to be considered - in case of changing or missing USL/LSL, please refer to the Calculation Mode described earlier in this document. This indicator considers all individual readings and it excludes data points which are aggregated. |
| Cpl - Lower Process Capability | \(C_{p,lower}=\frac{\mu-LSL}{3\hat{\delta}_{Within}}\) \(\mu\) to be used is the overall mean for the all the individual readings of the data points considered - same as Pp | x | x | Not Displayed | Never displayed - used for calculations later; About the data points to be considered - in case of changing or missing USL/LSL, please refer to the Calculation Mode described earlier in this document. This indicator considers all individual readings and it excludes data points which are aggregated. |
| Cpk - Process Capability Index | \(C_{pk}=min(C_{p,lower},C_{p,upper})\) | x | x | All Variable Charts (when Data Points > 1) | If single sided, the system considers only the non-nullable part of the equation |
| Z-Score - Standard Score | \(Z=3C_{pk}\) | x | x | All Variable Charts (when Data Points > 1) | Only display if configuration entry /Cmf/System/Configuration/SPC/ChartStatistics/IncludeZScore is set to true |
| Pp - Process Performance | \(P_p=\frac{USL-LSL}{6\delta}\) | x | x | All Variable Charts (when Data Points > 1) | About the data points to be considered - in case of changing or missing USL/LSL, please refer to the Calculation Mode described earlier in this document. This indicator excludes all the data points which are aggregated. |
| Ppu - Upper Process Performance | \(P_{p,upper}=\frac{USL-\mu}{3\delta}\) | x | x | Not Displayed | About the data points to be considered - in case of changing or missing USL/LSL, please refer to the Calculation Mode described earlier in this document. This indicator excludes all the data points which are aggregated. |
| Ppl - Lower Process Performance | \(P_{p,lower}=\frac{\mu-LSL}{3\delta}\) | x | x | Not Displayed | |
| Ppk - Process Performance Index | \(P_{pk}=min(P_{p,lower},P_{p,upper})\) | x | x | All Variable Charts (when Data Points > 1) | Note: If single sided, the system considers only the non-nullable part of the equation |
| OOS - Out of Spec | Counts the number of all readings of all data points which are either above the upper spec limit or below the lower spec limit. | x | All Charts | ||
| OOC - Out of Control | Counts the number of data points which are either above the upper control limit or below the lower control limit. | x | All Charts | For the Variable Charts, this indicator only considers the Indicator 1 | |
| OOC% - Out of Control Rate | \(OOC(\%)=100\%\frac{OOC}{number\;of\;data\;points}\) | x | All Charts | ||
| OTI - On Target Indicator | \(OTI=\frac{\overline{x}-Target\;Spec\;Value}{s}\) | x | All Variable Charts (when Data Points > 1) | About the data points to be considered - in case of changing or missing Target Spec Values, please refer to the Calculation Mode described earlier in this document. Only display if configuration entry /Cmf/System/Configuration/SPC/ChartStatistics/IncludeOTISection is set to true | |
| OCI - On Center Indicator | \(OCI=\frac{\overline{x}-Centerline}{s}\) | x | All Variable Charts (when Data Points > 1) | About the data points to be considered - in case of changing Centerline Values, please refer to the Calculation Mode described earlier in this document. Only display if configuration entry /Cmf/System/Configuration/SPC/ChartStatistics/IncludeOTISection is set to true | |
| LCLCR - Lower Control Limit Change Ratio | \(LCLCR=\frac{LCL-(\overline{x}-3s)}{s}\) | x | All Variable Charts (when Data Points > 1) | About the data points to be considered - in case of changing LCLs, please refer to the Calculation Mode described earlier in this document. Only display if configuration entry /Cmf/System/Configuration/SPC/ChartStatistics/IncludeCLRSection is set to true | |
| UCLCR - Upper Control Limit Change Ratio | \(UCLCR=\frac{(\overline{x}+3s)-UCL}{s}\) | x | All Variable Charts (when Data Points > 1) | About the data points to be considered - in case of changing UCLs, please refer to the Calculation Mode described earlier in this document. Only display if configuration entry /Cmf/System/Configuration/SPC/ChartStatistics/IncludeCLRSection is set to true | |
| CLCR - Control Limits Change Ratio | \(CLCR=If\;(Abs(LCLR)\:\gt\:Abs(UCLCR))\;then\;LCLR\;;\;else\;UCLCR\) | x | All Variable Charts (when Data Points > 1) | Only display if configuration entry /Cmf/System/Configuration/SPC/ChartStatistics/IncludeOTISection is set to true |
Table: Chart Statistics