## Interactive Calculator – Impact of Gage R&R on Process Capability

### How much Impact is your Gage R&R having on Process Capability

Use the calculator below to find out

### Calculation Notes and Formulas

#### (1) Observed Process Capability

The calculator is written to accept the Cp or Pp index, as calculated from a statistical package like Minitab, or a spreadsheet. It is the centered Process Capability of the process.
The blue curve, on the chart above, is drawn to +/- 3 standard deviations from the mid-point of the tolerance.
The range of values that the calculator accepts is from 0.5 to 4.0

The general formula is as follows: –

$Cp_{obs} = \dfrac{USL - LSL}{6 \times \sigma _{obs}}$

$where \qquad{\text Cp}_{\text obs} {\text is \ the \ observed \ process \ capability} \\\\where \qquad{\text USL} {\text \ is \ the\ Upper \ Specification \ Limit} \\\\where \qquad{\text LSL} {\text \ is \ the\ Lower \ Specification \ Limit} \\\\and \qquad \sigma_{obs} {\text \ is \ the \ standard \ deviation \ of \ the \ observed \ process} \\$

#### (2) Precision To Tolerance Ratio – PTR – (or Gage R&R as a percentage of Tolerance)

The Precision To Tolerance Ratio is a metric used to assess a Measurement System.
It is the ratio between the measurement variation (as a spread of 6 standard deviations), and the tolerance band.
The minimum value that the calculator accepts is 0.01
The maximum value that the calculator accepts is limited by the observed variation. To explain this, the observed variation comprises 2 sources of variation, the underlying (true value) variation, and the measurement variation. These variations are “added” together to create the observed process variation. Therefore, the measurement variation cannot be larger than the observed process variation (otherwise the underlying process would have to have negative variation – which is impossible). This is why, when the Observed Process Capability value is changed, the Precision To Tolerance Ratio may adjust to compensate.

The general formula is as follows: –

$PTR = \dfrac{6 \times \sigma _{r\&r}}{USL - LSL}$

$where \qquad{\text PTR \ is \ the \ Precision \ To \ Tolerance \ Ratio} \\\\where \qquad \sigma_{r\&r} {\text is \ the\ standard \ deviation \ of \ the \ measurement \ system} \\{\text estimated \ from \ a \ Gage \ R\&R \ Study}\\\\where \qquad{\text USL} {\text \ is \ the\ Upper \ Specification \ Limit} \\\\where \qquad{\text LSL} {\text \ is \ the\ Lower \ Specification \ Limit} \\\\$

#### (3) Gage R&R as % of Process Variation

The Gage R&R as a % of Process Variation is a metric used to assess a Measurement System.
It is the ratio between the standard deviation of the measurement variation, and the standard deviation of the observed process, expressed as a percentage
The minimum value that the calculator accepts is 1%
The maximum value that the calculator accepts is 95%.

Side Note: A Gage R&R Study may produce a result that shows the measurement variation as a percentage of the STUDY VARIATION (ie the variation of the parts used in the study). Obviously, the study variation may not be as wide as the process variation – depending on the parts you used in the study. This naturally creates a higher Gage R&R Percentage. For this reason, many experts recommend either calculating this % using a historical process standard deviation, or, when choosing parts for the Gage R&R Study, choose a range that represents the full process spread.
I would suggest using the historical process standard deviation, where possible.

The formula is as follows: –

$GRR \ \%= \dfrac{\sigma _{r\&r}}{\sigma_{proc}} \times 100\%$

$where \qquad{\text GRR \ \% \ is \ the \ Gage \ R\&R \ Percentage} \\{\text \ based \ on \ the \ process \ variation}\\\\where \qquad \sigma_{r\&r} {\text is \ the\ standard \ deviation \ of \ the \ measurement \ system} \\{\text estimated \ from \ a \ Gage \ R\&R \ Study}\\\\where \qquad \sigma_{proc} {\text \ is \ the \ standard \ deviation\ of \ the \ observed \ process} \\\\$

#### (4) Number of Distinct Categories – NDC

The Number of Distinct Categories is a metric used to assess a Measurement System.
If we calculated a confidence interval on the measurement error, this would give us a measurement error “width”. The ndc is the number of times that a 97% Confidence Interval on the measurement error will fit into the width of the variation due to the observed process.

Side Note: A Gage R&R Study may produce a result that calculates NDC based on the STUDY VARIATION (ie the variation of the parts used in the study). Obviously, the study variation may not be as wide as the process variation – depending on the parts you used in the study. This naturally creates a smaller NDC. However, you are using gages to measure the whole process, not just the parts used in your Gage R&R Study, therefore this calculator uses the standard deviation of the observed process to calculate NDC

The formula is as follows: –

$NDC = {1.41 \times \dfrac{\sigma _{proc}}{\sigma _{r\&r}} }$

$where \qquad{\text NDC \ is \ the \ Number \ of \ Distinct \ Categories } \\\\where \qquad \sigma_{r\&r} {\text is \ the\ standard \ deviation \ of \ the \ measurement \ system} \\{\text estimated \ from \ a \ Gage \ R\&R \ Study}\\\\where \qquad \sigma_{proc} {\text \ is \ the \ standard \ deviation\ of \ the \ observed \ process} \\\\$

#### (5) True Value Process Capability

The True Value Process Capability is an estimate of the Process Capability you would see if you were to measure the process using an absolutely perfect gage
In effect, the measurement variation is “subtracted” from the observed variation, so that we can see the underlying process.
The green curve, on the chart above, is drawn to +/- 3 standard deviations from the mid-point of the tolerance.

The formula is as follows: –

${\text Cp}_{\text Act} = \dfrac{{\text Cp}_{\text Obs}}{\sqrt{1 - ( {\text Cp}_{\text Obs} \times {\text PTR})^2}}$

$where \qquad{\text Cp}_{\text act} {\text is \ the \ actual \ or \ true \ value \ process \ capability} \\\\where \qquad{\text Cp}_{\text obs} {\text is \ the \ observed \ Process \ Capability} \\\\where \qquad{\text PTR \ is \ the \ Precision \ To \ Tolerance \ Ratio} \\\\$

#### Things To Consider when using this calculator

###### Everything is an estimate

Your estimate of Cp – Observed Process Capability is calculated based on a confidence interval
The results of a Gage R&R Study – PTR, %GRR, or NDC are based on a sampling estimate and therefore have an upper and lower confidence bound. – If you know the confidence limits on these metrics, you may want to take them into account.

###### Where would you spend your money ?

Set the Cp at 1.33 and the PTR at 0.31 – This gives a Gage R&R % as 41% and NDC as 3.
According to most sources, this makes the gage unsuitable to be used on this process.
But, now scroll up and look at the chart – and tell me –
Where would you spend your money ?

• Improving the Gaging to better than 10% ?
• Improving the process?
• Both ?
• Neither ?

If you think this is an interesting question, you might want to take a look at this discussion of the AIAG Guidelines for % Gage R&R

#### Alternative Online Cp Calculators

If you have just want some simple data entry calculators with no charts, there a two calculators available on this page – Impact of Gage R&R on the Process Capability Index Cp

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