Working a Gage R&R Study by Hand

Software will run a gage study for you. It will also hand you a clean-looking report built on a bad study. An inspector who can do the math by hand can check that report, see which number is driving the result, and tell a real gauge problem from a badly run study. This walks through one full study using the average and range method from the auto industry's Measurement Systems Analysis (MSA) manual. Your customer or your site may use a different method, and their rules win. Reminder: repeatability is one person's scatter on the same part, and reproducibility is the difference between people.

What You Will Be Able to Do

• Check a study's ranges before you trust its results.
• Calculate EV, AV, GRR, PV and TV from the summary numbers.
• Report percent of tolerance, percent of study variation, and ndc, and say which one matters.
• Read a failed study and decide where to start.
• Work out a guard band for readings near a limit.

The Study

A shaft has a diameter of 0.5000 +/- 0.0050, so the tolerance band is 0.0100. Three appraisers (A, B and C) each measure the same 10 parts, three times each, in random order, without seeing each other's numbers. That makes 90 readings.

From the raw data you build a few summary numbers.

Average range is the mean of the 10 ranges for that appraiser. A range is the largest minus the smallest of the three trials on one part. So a large average range means that appraiser's own readings scatter.

You also need three more numbers:

• R-double-bar, the average of the three average ranges: (0.00042 + 0.00050 + 0.00046) / 3 = 0.00046.
• X-diff, the highest appraiser average minus the lowest: 0.50031 - 0.50006 = 0.00025.
• Rp, the range of the part averages. The largest part average minus the smallest here is 0.0060.

Step 1: Check the Ranges First

Before you calculate anything, look for readings that do not belong. A single bad trial can inflate the whole result.

Upper limit for a range = D4 x R-double-bar

For 3 trials, D4 is about 2.57, and some tables round it to 2.58. For 2 trials, it is 3.27.

Here the limit is 2.57 x 0.00046 = 0.00118. Scan all 30 ranges. Any range above 0.00118 means something odd happened on that part in that series, such as a misread digit, a dirty part, or a different spot on the shaft. Find out what happened. Have that appraiser repeat that part, or remove the problem and rerun it. Do not delete a reading only because it looks bad.

Also check for resolution. If most of the ranges are zero, or the readings take only two or three values, the gauge's steps are too coarse. The math will still run, and it will mislead you.

Step 2: The Constants

The method uses constants based on how many trials, appraisers and parts you used. They turn ranges into standard deviations, which are the common measure of spread.

Older editions of the manual built a multiplier into these constants, such as 4.56 and 3.05. Use one set throughout. Mixing them is a common mistake in old spreadsheets.

Step 3: Calculate the Five Numbers

Equipment variation (EV) is repeatability.

EV = R-double-bar x K1 = 0.00046 x 0.5908 = 0.000272

Appraiser variation (AV) is reproducibility. It starts from the difference between people, then takes out the part of that difference caused by repeatability. Here n is the number of parts and r is the number of trials.

AV = square root of [ (X-diff x K2) squared - (EV squared / (n x r)) ]

• X-diff x K2 = 0.00025 x 0.5231 = 0.0001308, and its square is 0.0000000171.
• EV squared / 30 = 0.0000000739 / 30 = 0.0000000025.
• Subtract: 0.0000000146. The square root is 0.000121.

If the number inside the square root comes out negative, AV is taken as zero.

Gage R&R (GRR) combines them. Standard deviations do not add directly. Their squares add.

GRR = square root of (EV squared + AV squared) = square root of (0.0000000739 + 0.0000000146) = 0.000298

Part variation (PV) is the spread of the parts themselves.

PV = Rp x K3 = 0.0060 x 0.3146 = 0.00189

Total variation (TV) combines the gauge and the parts.

TV = square root of (GRR squared + PV squared) = 0.00191

Step 4: Turn Them Into Decisions

Percent of tolerance. Take the measurement spread as 6 standard deviations, and compare it with the tolerance band.

Percent of tolerance = 6 x GRR / tolerance = 6 x 0.000298 / 0.0100 = 17.9 percent

Percent of study variation. Compare GRR with total variation.

Percent of study variation = GRR / TV = 0.000298 / 0.00191 = 15.6 percent

Number of distinct categories (ndc).

ndc = 1.41 x PV / GRR = 1.41 x 0.00189 / 0.000298 = 8.9, which is 8 after dropping the decimal. The manual truncates, so 8.9 does not round up to 9.

So this gauge is usable for many jobs, with a decision needed on risk. A quality engineer, or the customer's rule, makes that final call.

Which Number Matters

Use percent of tolerance when the gauge is used to pass or fail parts. Use percent of study variation and ndc when the gauge feeds a control chart, because then it has to see small changes in the process. A gauge can be fine for one job and poor for the other.

A wide tolerance and a tight process can show 8 percent of tolerance and 35 percent of study variation. That gauge sorts good from bad, but it cannot see what the process is doing.

Two Traps in the Percentages

The percentages do not add to 100. EV, AV and PV are ratios of standard deviations, and only their squares add. In this study, 15.6 squared plus 98.8 squared (PV's share of TV) comes to about 10,000, which is 100 squared.

The spread factor also changes percent of tolerance. Older reports use 5.15 standard deviations in place of 6. This same study at 5.15 gives 15.3 percent, not 17.9. If two reports disagree, check the factor before you blame the gauge.

When the Study Fails

Compare EV and AV. The larger one tells you where to start.

Worked example. A second study on the same shaft, with a new group of appraisers, gives R-double-bar of 0.00025, so EV is 0.00015. But X-diff is 0.0012.

• X-diff x K2 = 0.000628, and its square is 0.000000394.
• AV is about 0.000627. GRR is about 0.000645.
• Percent of tolerance = 6 x 0.000645 / 0.0100 = 38.7 percent. That fails.

EV is small, so each person repeats well and the gauge itself is fine. AV is four times larger. The people disagree with each other. You watch them, and you find that appraiser C measures near the shoulder, and A and B measure near the middle of the shaft. That shaft has a slight taper.

The fix is a written method with a marked spot. After that, X-diff falls to 0.00020. EV stays at 0.00015, the new AV is 0.00010, GRR is 0.00018, and percent of tolerance is 10.8. Nobody bought a new gauge.

ANOVA (analysis of variance) is a more detailed calculation. Software runs it, and it can separate a true appraiser-by-part interaction. For example, people may agree on round parts and disagree on out-of-round parts. The average and range method cannot show that.

When Good Gauges Look Bad

PV comes from the parts you chose. If you pick 10 parts that are nearly the same, PV is small, and percent of study variation and ndc look terrible, even with a sound gauge.

Worked example. A study on 10 parts pulled from one hour of production gives Rp = 0.0008, so PV = 0.00025. With GRR at 0.000298, TV is 0.00039, percent of study variation is 76 percent, and ndc is 1. But the percent of tolerance is still 17.9, so the gauge did not change. The parts did.

The fix is to pick parts that span the process, over several days or shifts. If your site allows it, TV can come from a known process standard deviation instead of the sample of parts. Read your procedure first.

Guard Bands Near a Limit

Even a good system gives uncertain answers for parts near a limit. A guard band is a tighter internal limit that accounts for this.

Worked example. The upper limit is 0.5050, and GRR is 0.000298. About 95 percent of readings fall within 2 standard deviations of the true size, so plus or minus 0.0006.

• A reading of 0.5049 could be a true size up to about 0.5055, which is out of tolerance.
• A guard band of 0.0006 sets the accept limit at 0.5044.
• Parts reading from 0.5044 to 0.5050 get a closer look with a better gauge, or are treated as rejects.

A guard band protects the customer and costs you some good parts. Do not set one yourself. Quality or the customer decides whether to use one. If a part keeps passing and failing on repeat readings, tell someone. Measuring it until it passes falsifies the record.

Common Mistakes

Check Your Understanding

1. A study has 3 trials and R-double-bar of 0.0010. One range reads 0.0031. What do you do?

The limit is 2.57 x 0.0010 = 0.00257, so 0.0031 is over it. Find out what happened on that part. Repeat or correct it for a real reason before you finish the math.

2. EV is 0.0003, and AV is 0.0004. What is GRR?

Square each and add them: 0.00000009 + 0.00000016 = 0.00000025. The square root is 0.0005. You do not add 0.0003 and 0.0004.

3. A tolerance band is 0.020, and GRR is 0.0005. What is the percent of tolerance?

6 x 0.0005 / 0.020 = 15 percent, which falls in the 10 to 30 band.

4. A gauge shows 9 percent of tolerance, 41 percent of study variation and ndc of 3. It is used for pass or fail only. Is it good enough?

For pass or fail, it looks acceptable under the common guideline, subject to your customer's rule. It would not be good enough to run a control chart, because it cannot see small process changes.

5. Appraiser averages differ by a lot, but each person's ranges are tight. What is the first thing to do?

Watch how each person measures. Tight ranges mean the gauge repeats, so the difference comes from the method: the spot, the force, the setup. Write the method down and rerun.

Key Takeaways

• Check ranges and resolution before you believe any result.
• EV is repeatability, AV is reproducibility, and their squares add to the square of GRR.
• Use percent of tolerance for pass or fail, and percent of study variation and ndc for control charts.
• The larger of EV and AV tells you where to start.
• Parts that are too much alike make a good gauge look bad.
• A guard band handles uncertainty near a limit, and only quality or the customer sets one.