How Much Impact is Your Gage R&R Having on Process Capability?
Use the calculator below to find out
This calculator takes an observed process capability (as measured) and "subtracts" the gage r&r variation (from your Gage R&R Study) so that you get a view of the underlying process capability  As though you were measuring the process with a perfect gage
1) Enter your Specification Limits into the boxes below
2) Move the slider to select your Process Capability
Minimum Cp  Maximum Cp  

0.50  4.00 
3) Move the slider to select your Precision To Tolerance Ratio (or Gage R&R as % of Process Variation = % or NDC = )
Minimum PTR  Maximum PTR  

0.01  0.25 
4) Get your Results  What's the true Process Capability of your Process?  The green curve will tell you
Lower Specification Limit: 0.002
Upper Specification Limit: 0.002
Observed Process Capability (1):
Precision To Tolerance Ratio (2):
Gage R&R as % of Process Variation (3):
Number of Distinct Categories (NDC) (4):
True Value Process Capability (5):


CALCULATION NOTES & FORMULAE
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 midpoint of the tolerance.
The range of values that the calculator accepts is from 0.5 to 4.0
The general formula is as follows: 
where: 
\scriptsize Cp_{obs}
is the observed process capability
\scriptsize USL is the Upper Specification Limit
\scriptsize LSL is the Lower Specification Limit
\scriptsize \sigma _{obs} is the standard deviation of the observed process
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: 
where: 
\scriptsize PTR
is the Precision To Tolerance Ratio
\sigma _{r\&r} is the standard deviation of the measurement system (usually estimated from a Gage R&R Study)
\scriptsize USL is the Upper Specification Limit
\scriptsize LSL is the Lower Specification Limit
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: 
where: 
\scriptsize GRR\%
is the Gage R&R Percentage
\scriptsize \sigma _{r\&r} is the standard deviation of the measurement system (usually estimated from a Gage R&R Study)
\scriptsize \sigma _{obs} is the standard deviation of the observed process
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: 
where: 
\scriptsize NDC
is the Number of Distinct Categories
\scriptsize \sigma _{r\&r} is the standard deviation of the measurement system (usually estimated from a Gage R&R Study)
\scriptsize \sigma _{obs} is the standard deviation of the observed process
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 midpoint of the tolerance.
The formula is as follows: 
where: 
\scriptsize CP_{tv}
is the True Value Process Capability  the underlying Process Capability, once measurement variation has been removed.
\scriptsize Cp _{obs} is the observed Process Capability (See Note (1))
\scriptsize PTR is the Precision To Tolerance Ratio (see Note (2))
Formula Derivation
Basic Measurement System Analysis Equations  See relevant notes for nomenclature
\scriptsize PTR = \frac{6 \times \sigma _{r\&r}}{USL  LSL} (1)
\scriptsize Cp_{obs} = \frac{USL  LSL}{6 \times \sigma _{obs}} (2)
\scriptsize Cp_{tv} = \frac{USL  LSL}{6 \times \sigma _{tv}} (3)
\scriptsize \sigma_{obs}^2 = \sigma_{tv}^2 + \sigma_{r\&r}^2 (4)
Rearrange (4) :
\scriptsize \sigma_{tv} = \sqrt{\sigma_{obs}^2  \sigma_{r\&r}^2 } (5)
Rearrange (2) :
\scriptsize \sigma_{obs} = \frac{USL  LSL}{6 \times Cp_{obs}} (6)
Substitute (6) into (5):
\scriptsize \sigma_{tv} = \sqrt{{(\frac{USL  LSL}{6 \times Cp_{obs}})}^2  \sigma_{r\&r}^2 } (7)
Substitute (7) into (3):
\scriptsize Cp_{tv} = \frac{USL  LSL}{6 \times \sqrt{{(\frac{USL  LSL}{6 \times Cp_{obs}})}^2  \sigma_{r\&r}^2 }} (8)
Rearrange (1) :
\scriptsize \sigma_{r\&r} = \frac{PTR \times (USL  LSL)}{6} (9)
Substitute (9) into (8):
\scriptsize Cp_{tv} = \frac{USL  LSL}{6 \times \sqrt{{(\frac{USL  LSL}{6 \times Cp_{obs}})}^2  {(\frac{PTR \times (USL  LSL)}{6})}^2 }} (10)
Simplify (10) to get the final formula:
\scriptsize Cp_{tv} = \frac{1}{\sqrt{{(\frac{1}{Cp_{obs}})}^2  {(PTR)}^2}} (11)
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.
This is a simplified model. For a process and gage simulation showing a sample from a bivariate normal distribution, see the Process and Gage Simulator
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 page
Free Giveaway  "What is Measurement System Analysis"
If you would like to learn about where Gage R&R fits into Measurement System Analysis, you might want to take a look at the Free Downloads Page
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If you want to bring anyone up to speed on the importance of MSA and Gage R&R, this package will really help you. It includes:
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