# Online Gage R&R Calculators

 Gage R&R Study Design Tool Gage R&R Calculator Gage R&R Process Capability Calculator - See below

# 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
Lower Specification Limit
Upper Specification Limit

###### 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
(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: -

\scriptsize Cp_{obs} = \frac{USL - LSL}{6 \times \sigma _{obs}}

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

(2) Precision To Tolerance Ratio

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: -

\scriptsize PTR = \frac{6 \times \sigma _{r\&r}}{USL - LSL}

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

(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: -

\scriptsize GRR\% = {\frac{\sigma _{r\&r}}{\sigma _{obs} } \times 100\%

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

(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: -

\scriptsize NDC = {1.41 \times \frac{\sigma _{obs}}{\sigma _{r\&r}} }

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

(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: -

\scriptsize Cp_{tv} = \frac{1}{\sqrt{{(\frac{1}{Cp_{obs}})}^2 - {(PTR)}^2}}

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)

Re-arrange (4) :

\scriptsize \sigma_{tv} = \sqrt{\sigma_{obs}^2 - \sigma_{r\&r}^2 }        (5)

Re-arrange (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)

Re-arrange (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)

(6) 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.

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

- 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 page

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