Gauge Capability: Cg and Cgk
Table of Contents
What a capability study answers
Gauge R and R and gauge capability are often confused because both produce a verdict about a measurement system. They compare against different things, and that difference determines when each is appropriate.
| Study | Compares measurement variation against | Use when |
|---|---|---|
| Gauge R and R | The variation of the parts sampled, or optionally the tolerance | Several operators use the gauge and you need to separate operator effects from gauge effects. |
| Gauge capability | A fixed reference fraction of the tolerance | One operator or an automated gauge, or you want a quick verdict against a specific tolerance without a full study. |
A capability study uses repeated measurements of a single calibrated reference part by one appraiser, conventionally at least 25 readings, though 50 is common. Because there is only one part and one appraiser, it says nothing about reproducibility, and it does not need to.
Cg: is the gauge precise enough?
Cg compares the spread of the gauge against a reference fraction of the tolerance. It considers only scatter, and ignores where the readings are centred.
Cg
Cg = (0.2 × T) ÷ (6 × s)
where T is the total tolerance width and s is the standard deviation of the repeated readings on the reference part.
Read the denominator as the natural spread of the gauge: six standard deviations covers about 99.73 percent of its readings. The numerator is the budget you are allowing the gauge, being 20 percent of the tolerance. Cg is therefore the ratio of the allowance to the requirement, and a Cg of 1.33 means the gauge spread fits into its allowance with 33 percent to spare.
Note the structure is identical to a process capability index. Cp compares the tolerance to the process spread; Cg compares a fraction of the tolerance to the gauge spread. The gauge is being treated as a process whose output is measurements.
Where the 0.2 comes from
The 0.2 factor is a decision, not a derivation. It encodes the judgement that a measurement system should consume no more than 20 percent of the tolerance, which is the same threshold that appears in Gauge R and R as the boundary between acceptable and marginal in many acceptance schemes.
The reasoning behind it is that measurement variation eats into the tolerance available for the process. If the gauge consumes 20 percent, the process still has the large majority of the tolerance to work within, and the risk of misclassifying a part near the specification limit stays modest. Allow the gauge 50 percent and that risk becomes severe.
Cgk: is the gauge also accurate enough?
Cg can look excellent while every reading is offset from the reference value. Cgk adds the bias penalty, in the same way that Cpk adds a centring penalty to Cp.
Cgk
Cgk = (0.1 × T − | x̄ − xref | ) ÷ (3 × s)
where x-bar is the average of the readings and x-ref is the reference value of the master, so the absolute difference is the bias.
The bias is subtracted directly from the half-allowance before dividing by the half-spread. This means Cgk can never exceed Cg, and the gap between them is a direct measure of how much capability the bias is consuming. With zero bias, Cgk equals Cg exactly.
Worked example
Illustrative data. Thirty repeated readings of a 25 mm master on a part with a 0.100 mm tolerance.
| Quantity | Value |
|---|---|
| Total tolerance T | 0.100 mm |
| Reference value of the master | 25.000 mm |
| Average of the readings | 25.004 mm |
| Bias (absolute difference) | 0.004 mm |
| Standard deviation s | 0.0025 mm |
| Cg = (0.2 × 0.100) / (6 × 0.0025) | 1.33 |
| Cgk = (0.010 − 0.004) / (3 × 0.0025) | 0.80 |
Interpreting the pair
| Pattern | What it means | Typical action |
|---|---|---|
| Cg and Cgk both at or above the limit, and close together | The gauge is both precise and well aligned for this tolerance. | Accept and record the study. |
| Cg passes but Cgk fails | Precision is adequate; a bias is consuming the capability. | Adjust or recalibrate. This is usually correctable, because the scatter is already good enough. |
| Cg fails | The gauge scatter alone is too large for this tolerance, regardless of alignment. | Adjustment will not help. Improve the method or fixturing, or use a more precise gauge for this characteristic. |
Resolution and the 10 to 1 rule
Before any capability index is meaningful, the gauge must be able to see the variation you are asking it to quantify. The conventional requirement is that the resolution, meaning the smallest increment the gauge displays, is no more than one tenth of the tolerance.
The 10 to 1 rule
resolution ≤ T ÷ 10
With a 0.100 mm tolerance, this calls for a resolution of 0.010 mm or finer. The reason is that a coarse gauge quantizes the readings: if the resolution is too large relative to the variation, repeated readings return the same displayed value regardless of the real differences between them.
This produces a specific and dangerous failure. The standard deviation of the readings collapses toward zero, Cg becomes very large, and the gauge appears outstanding. It is not outstanding, it is blind. The tell-tale sign is repeated readings showing only two or three distinct values.
Common pitfalls
- Reporting Cg without Cgk. The pair only tells the full story together, and Cg alone conceals bias by construction.
- Comparing a Cg computed with a 0.2 factor against a limit intended for a 0.1 factor. Always state which reference fraction was used.
- Using an uncalibrated part as the reference. Cgk is measured against the reference value, so an unknown master value makes the bias term meaningless.
- Treating a capability study as a complete MSA. It uses one appraiser and one part, so it says nothing about reproducibility and nothing about behaviour across the measuring range.