**What is Gage Repeatability & Reproducibility (GR&R)?**

Gage Repeatability & Reproducibility, commonly known as a GR&R, is a statistical method used in process control (SPC) to measure the precision and variation present in a measuring device and the subsequent effectiveness of the instrument to be used as a measuring tool. There are, as the name suggests, two components that comprise the GR&R, repeatability, related to the ability of the gage itself to give consistent results (precision)under repeated test of the same manner, and reproducibility, or the ability of the gage to provide repeated results regardless of the operator performing the test (variation among operators).

In Rockwell Hardness Testing a GR&R study can be a valuable measure of how accurate the tester is performing and how much variation the tester may be contributing to the overall process. However, GR&R evaluation in Rockwell are not achieved without some complications, but once these are understood and accounted for, GR&R assessment provides useful process data. While many types of gages, such as micrometers or calipers, can be GR&R performance tested relatively easy and without sample variation (using a gage block of a known value to make repeated measurements on the same block and calculating the repeatability), a GR&R evaluation on a Rockwell tester poses inherent difficulties. As a Rockwell test in never performed in the same spot, a primary obstacle is the test samples variation itself. As a result, as no material is completely uniform in hardness, a variation is introduced. This phenomenon is typically accounted for through introducing process tolerances to the mathematical analysis when performing the GR&R calculations.

**The Purpose**

The purpose of performing a Gage Repeatability " Reproducibility (GR&R) study is to determine how much of the process tolerance is being used up by variation in a hardness testing instrument (also referred to as equipment variation or repeatability) as well as between operators (also referred to as appraiser variation or reproducibility). When the combination of these sources (repeatability and reproducibility or R"R) becomes a significant portion of the process tolerance, an operator cannot be sure whether they are measuring the hardness of a part or simply generating random numbers with the Rockwell tester machine. For statistical process control (SPC) to work effectively, the combined variation should be less than 10 percent of the process tolerance (< 10% GR&R). Rockwell hardness testing machines with a GR&R between 10% and 30% may be acceptable on an interim basis, and machines with a GR&R of greater than 30% should not be used for SPC.

**Why Perform a GR&R?**

A survey done several years ago for ASTM on 30 testers in daily use, showed that 90% of the instruments checked failed a direct verification even though they passed a normal indirect verification using test blocks. There is no doubt that these testers were using up most if not all of the allowable tolerance. This will significantly increase the error calculation when used to determine the uncertainty of the machine. Performing a GR&R can reveal a lot about how well your system is reading Rockwell hardness and if you need additional verification such as direct, performed. Nearly all of the testers in use today have not had their basic functions (force, depth measuring, time cycle) verified since they were manufactured. Many times the shipping process alone can cause problems with these delicate instruments. ASTM does not require direct verifications of the test forces and displacement measuring system after testers are shipped to the user. It's not uncommon to see testers in use that are over 40 years old. That means the forces they apply during the test and the devices they use to measure the depth have not been checked for accuracy for 40 years. Springs are frequently used to apply the test forces. Springs are known to lose their force in time and after 40 years, no one knows what the actual force really is since only indirect verifications using standardized test blocks have to be done periodically to monitor the instruments. .Clever servicemen have spent a lifetime tweaking testers to “read the blocks” using up the full ASTM allowed tolerances. With the unit of measurement for one regular Rockwell point equal to only 2 µ (approximately 0.002 mm or 0.000080 inch depth) it becomes obvious that such exact measurement requires a very precise measuring system and as importantly controlled process. Failure to properly prepare and execute a Rockwell hardness test and ensure a frequently calibrated and maintained tester can result in compromised test data or false readings, potentially contributing to the production and delivery of substandard product that could have detrimental and catastrophic effects in the performance and the integrity of the goods they are used in. When it is easy to see that errors in the machine performance can quickly translate to poor product, a GR&R can provide a measurable and revealing assessment of how well your machine is performing.

**Understanding your Testers Repeatability and Reproducibility**

Most users don't have quantitative knowledge of how well their testers perform. The repeatability and reproducibility of a tester are frequently determined by performing a GR&R study. By doing GR&R studies periodically it is easy to establish and monitor the performance of an instrument. A typical GR'R study can quickly establish the short and long-term performance of your tester including operator influence.

A full or long method GR&R study involves 10 different test blocks and three operators each making 3 tests on each block. The total of 90 tests will tell you what part of your samples tolerance is going to be used up by the inaccuracy of the tester. A short or mini GR&R involves 10 different test blocks with 3 tests on each by a single operator for a total of 30 tests. Comparisons on Rockwell testers between the two methods show negligible differences in the final result.

**The Process**

Due to the previously mentioned variation in materials, particularly on production parts, GR&R testing should be performed on standardized test blocks to reduce the material variation influence as much as possible. Test blocks, by design, are manufactured to be as uniform as possible thereby making them the best material to perform the GR&R test. In a full or long method GR&R, each of three operator should perform a single hardness test on each of the 10 test block sequentially, beginning with block #1 and through block #10 to complete the first run. To compensate for the random non-uniformity inherent in a test block, the operator must insure that each set of indents on each block is repeated in close proximity as possible to each other (grouped or radially). The process is repeated for 3 full run for a total of 90 tests, 9 on each test block.

**The Calculation**

The GR&R calculation for Rockwell testers is essentially the comparison of the combination of machine and operator variation with the process tolerance. If the variation is low or the process tolerance wide by comparison, then the % GR&R will also be low. Conversely, if the variation is high or the process tolerance narrow by comparison, the % GR&R will be high. In the analysis, the ranges of readings for each operator are calculated and then the average range is generated for each. In addition the average test value is also determined for each operator. This data is utilized to generate the full GR&R result.

**Tolerance**

The process tolerance aspect of the calculation is relatively simple: it is plugged in directly from the engineering specification for part's hardness (for example, a part calling for a hardness of 42 to 48 HRC would have a total tolerance of 6 points). Note that the calculation of GR&R is only relevant in the context of process tolerance, comparing machine and operator variation to a test block tolerance for example is not meaningful, as it says nothing about the machine’s suitability to measure real parts. Test block tolerances are for insuring the accuracy of a machine, not its repeatability.

**Variation**

The calculations for variation can appear somewhat vague, but all they are doing is converting average range values and operator differences into an approximation for six sigma (six times the standard deviation for all the data). Six sigma is the statistical description for a machine's total variation. Assuming the machine is varying in a normal manner, six sigma says that over 99&337; of all tests done on a given block (or set of ten blocks in the case of GR&R) will fall within this region. It is also, in a sense, the uncertainty of the machine at that hardness level, meaning that for a given reading; the actual hardness value could be up to plus or minus three sigma away.

**The Result**

Given that well trained operators and/or automatic machines yield negligible operator (appraiser) variation, it is easiest to understand the final calculation for % GR&R in the context of only the machine (equipment) variation. This calculation simply divides the six sigma approximation (average range for all tests multiplied by a statistical constant) by the process tolerance and multiplies by 100. A 10% or less GR&R then requires that six sigma (total machine variation) be 10% or less of the process tolerance. The graph in figure 2 demonstrates typical GR&R results from various types of Rockwell testers, including analog and digital deadweight as well as closed loop. As expected, the analog instruments yield the poorest GR&R performance while a closed loop tester with highly regulated force application yields the best performance.

Some Rockwell testers have features that contribute to high performing gage repeatability and reproducibility and some manufacturers ensure that every tester produced will undergo a GR&R evaluation and will not be shipped unless it completes this test with a GR&R performance equal to or better than a stated percentage. Proof of the study is typically provided in a GR&R certificate issued with each tester. Some contributing factors to high GR&R performance include:

- Closed-loop load control: load cell provides means of feedback through the load cell and indenter (wear and friction from mechanical parts are compensated for within the loop). Closed loop control provides the most accurate form of force application
- Attaching the penetrator to the load cell so that errors from friction are eliminated
- Designing so that the force measuring device and the depth measuring scale are directly in line with the indenter in a single axis arrangement
- Elimination of elevating screws (source of non-recoverable deflection)
- Utilizing digital readout as opposed to analog and increasing resolution to .01

Gage Repeatability and reproducibly is undoubtedly a useful and informative tool in assessing the performance level of a Rockwell hardness tester. With hardness testing utilized as an important and informative process in materials testing, quality control and acceptance and performance of materials we depend on the data produced to verify heat treatment, structural integrity, and quality of components to determine if a material has the properties necessary to ensuring that the materials utilized in the things we use every day contribute to a well-engineered, efficient, and safe world. Proper technique, procedure and strict adherence to standards in addition to ensuring a well performing instrument will greatly contribute to the accuracy and usefulness of Rockwell testing.

Buehler is a manufacturer of scientific instruments and supplies for cross-sectional analysis. Buehler sectioning, mounting, grinding and polishing, imaging and analysis and hardness testing equipment along with consumables are used by metallurgical, metallography, petrography, research & development, university laboratories, quality control departments and failure analysis facilities for the analysis of all types of materials, including ceramics, composites, semiconductors, metals, rocks and minerals, and plastics. Buehler is well established as the world's leading supplier of materials preparation and analysis instruments, equipment, consumable supplies and application solutions.

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