Gage Repeatability and Reproducibility (R&R)

Photo by Iain
Photo by Iain

Photo by Iain


  • getting consistent results repeatedly.
  • the repeatability of the gage.


  • unbiased true values are obtained.
  • must be assured before an R&R can be performed.
  • This is why we calibrate.


  • ability to detect differences in measurement.
  • must be assured before an R&R can be performed.


  • checked by comparing the results of different operators taken at different times.
  • Examining the variation between the average of the individual inspectors for all parts measured.


  • determined by examining the variation between the individual inspectors and within their measurement readings.


  • the reproducibility of the gage

Process Variation:

  • Examine the variation between part averages that are averaged among inspectors.


  • reduces the number of characteristics of a system.


  • long-term performance level after a process has attained statistical control.


  • when a product or service has met requirements.


  • the accuracy of a measuring instrument mapped to US national standards.

Reproducibility: Variation is average of measurement made by different operators using the same gage measuring the same object. For example three operators are tasked to measure the same object by using same gage then all three operators given different readings.

Repeatability: The random variations in measurement when one operator uses the same gage to measure the same part at several times. When repeatability suffers due to the gage some common factors might be at play (temperature, position etc.)

Measurement Wariability

Gage R&R studies establish repeatability and reproducibility components of measurement Variability

Measurement variability can be broken down as;

Variance (measured error) = Variance (repeatability error) + Variance (reproducibility error)

(Also see Measurement Systems Analysis)

Gage Variability

Gage R&R studies involve data collection schemes that break the total measurement variability or “gage variability” into two portion. There are several factors that impact on the true measured value i.e.

Measured Value = f (TV + AC + Rep + Rpr)


TV: True value
AC: gage accuracy ——– established by calibration
Rep: gage repeatability
Rpr: gage reproducibility

We will look at “repeatability” and “reproducibility” as primary contributors to measurement error.

How to find Gage Repeatability?

The term “Gage Error” is used interchangeably with “repeatability” or “Equipment Gage Variation (EV)”. The question is how do we actually find this?

It is required to measure path of an object using same gage and let say “p” operators are involved in doing measurements.

Every one using the same gage to measure same paths ‘r-times’.

The standard deviation due to repeatability (equipment or gage variation) found by measuring each path r-times and formula to find Standard deviation of repeatability is as follows;

gage r and r

d2: is based on r-times the number of observations or trial per part made by each operator.

σ: is standard deviation in gage to gage measurement.
R : An average of range by different measurements using same gage.

Let’s look at an example that we have different readings of same gage by different measurements. These reading have slide variation due to repeatability. By applying the same gage on same object we made some measurement using the same operators and readings. Let’s we have 5 different readings;

x1 x2 x3 x4 x5

  • Now find the range of the measured values by using this formula,

Range = “R” = Max – Min = X4 – X2

  • Find Average range R of many such measurements.
  • Let we have five measurements so there are 5 number of observation. Which is n = 5
  • This ‘n’ is very useful for finding d2.
  • d2: it is constant and depend upon ‘n’ and its value is calculated from Table directly.
  • n: it is numbers of observations of n times that you made measurements.


Table for finding d2 in Gage R&R Studies

table to find d2 in gage r and r

Example of finding d2 in Gage R&R Studies

Here, the Number of Observations in Samples is 5. For the value n = 5, the chart states the value of d2 = 2.326. So put this value in a formula of repeatability

repeatability d2  is 5

This repeatability is due to gage and it gives an idea of amount of variation actually contributed by the gage by repeatability measurement of same object.

How to find reproducibility?

Remember that, reproducibility is dependent upon the operator – the person who is doing an experiment.

Simple Method for finding σ (reproducibility)

The standard deviation for reproducibility or Appraisal variability (AV). By using this formula we will calculate the reproducibility.

Simple manner for reproducibility


Let’s take an example to find reproducibility and repeatability by given data using the gage capability study logic method. This methods allows you to calculate the two components of measurements system variation.

  • Repeatability
  • Reproducibility

However we must have several operators measure several parts by several times.

  • We selected 5 parts that represent the variation inherent in the process for a given characteristics.
  • We select 2 operators A&B who will be involve in the study.
  • We record the results of the measurements made by the operators on all 5 parts over 3 trials using calipers.

operator a and b

Now find the average value of all parts for each operator.


Now,use the formula calculate repeatability and reproducibility. For repeatability number of repeated measurements are n = 3 so in this case d2 will be 1.693, again by using the table above.


And for reproducibility there are two operators so in this case d2 will be 1.128. So by using formula of reproducibility;


(As we know Variance is a square of Standard Deviation)

3 comments… add one
  • Nguyen

    You you help to explain how to calculate Gage R & R for true position (with x and y values), please? I dont see any article discuss about this

    Thank you very much!

    • Hi Nguyen, would you be able to post an example problem or question set that would help me provide what you are looking for?

  • Jovon

    How did you get the X-bar min and max figures?


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