Gauge Capability Study

Gauge Capability Study [Introduction to SQC – Chapter 7] Introduction [p.352] In any situation involving measurement, some of the observed variability will be due to product itself and some will be due to measurement error or gauge variability. 2 2 σ Total = σ Pr2 oduct + σ Gauge 2 Where σ Total is the total observed variance. σ Pr2 oduct is the variance that is due to the 2 product and σ Gauge is the variance that is due to the measurement error.

A control chart can be used to separate the components of variance identified above and provide an assessment of how capable the gauge is for the intended purpose.

Example: [p.353] An instrument is planned for use in a proposed SPC implementation. An assessment of gauge capability is required. Twenty units of product are obtained and numbered. The same operator measures each unit twice with the gauge. A X and R control chart is then developed.

The X chart may exhibit many parts outside the limits as many of the units may have had vastly different dimensions. The R chart directly shows the magnitude of the measurement error. The R value shows the difference in measurements made on the same units using the same instrument. Out-of-control points on the Range chart would indicate that the operator is having difficulty using the instrument.

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Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

x1 21 24 20 27 19 23 22 19 24 25 21 18 23 24 29 26 20 19 25 19

x2 20 23 21 27 18 21 21 17 23 23 20 19 25 24 30 26 20 21 26 19

R 1 1 1 0 1 2 1 2 1 2 1 1 2 0 1 0 0 2 1 0

X 20.5 23.5 20.5 27.0 18.5 22.0 21.5 18.0 23.5 24.0 20.5 18.5 24.0 24.0 29.5 26.0 20.0 20.0 25.5 19.0

X = 22.3 R = 1.0

Xbar/R Chart 1

30

Means

1

1

25 20 1 Subgroup 0

Ranges

3

1

1 10

1

1 1

3.0SL=24.18

1

X=22.30 -3.0SL=20.42

20

3.0SL=3.267

2 1

R=1.000

0

-3.0SL=0.000

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The standard deviation of the measurement error, σ Gauge can be estimated from the Range chart.

σ Gauge =

R 1.0 = = 0.887 d 2 1.128

The distribution of measurement error follows the normal distribution. 6σ Gauge is a good estimate of gauge capability.

6σ Gauge = 6(0.887) = 5.32 This implies that individual measurements can vary by as much as ± 3σ Gauge (i.e. ± 2.66) due to gauge error or measurement error.

Precision To Tolerance Ratio [p.354] It is useful to compare the gauge capability (of the instrument) to the total specification (of the product). This is called the Precision to Tolerance Ratio (or P/T

Ratio) and is essentially the reciprocal of the process capability ratio.

6σ Gauge P = T USL − LSL

If the part being inspected has an USL and LSL of 60 and 5 respectively, then the P/T ratio is calculated as follows:

P 5.32 = = 0.097 (9.7%) T 60 − 5 A general rule of thumb is that the P/T ratio should be ≤ 0.10 (i.e. 10%). P/T ratios above this indicate that there is inadequate gauge capability. Montgomery cautions against too much reliance on this rule of thumb.

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Others measures of gauge capability have been proposed such as the ratio of product variability to total variability and also the ratio of measurement system variability to total variability.

Gauge Variability as a Percentage of Total Variability [p.355] The total variability for the data presented earlier (i.e. 20 units, each measured twice) can be determined by calculating the sample standard deviation S, of all the measurements (i.e. n = 40). This is determined as S = 3.172 and represents an estimate of the total variability and includes both the product variability and gauge variability. 2 σ Total = S 2 = (3.172) 2 = 10.06 2 σ Gauge = (0.887) 2 = 0.79 2 2 = σ Pr2 oduct + σ Gauge As σ Total , it is now possible to calculate the variability due to the

product.

σ Pr2 oduct = 10.06 − 0.79 = 9.27

σ Pr oduct = 9.27 = 3.04 The ratio of product variability to total variability can be expressed as: 2 σ product 9.26 ρp = 2 = = 0.92.1 10.05 σ Total

Similarly we can calculate the ratio of measurement system variability to total variability.

ρm =

2 σ Gauge 0.79 = = 0.079 2 σ Total 10.05

Note ρ p = 1 − ρ m

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ρ p and ρ m are often expressed as percentages. This means that the measuring instrument contributes 7.9% of the total observe variance in the measurements.

Expressing the gauge variability as a percentage of the product variability is often a more meaningful expression of gauge capability than the P/T ratio, as it does not depend on the specification.

Components of Measurement Error [p.357] In the previous example we have considered the situation of a single operator. In most cases, more than one operator will be involves in using the gauge. It is therefore important to also consider the variability between operators. We can therefore consider the components of measurement error as: 2 2 2 2 σ Measuremen t Error = σ Gauge = σ Reproducability + σ Repeatability

Reproducability: Measurement error due to different operators using the gauge. Repeatability: Measurement error due to the inherent precision of the gauge (same operator).

Example: In the earlier example we consider only one operator. We now consider two additional operators, (i.e. Operator 2 and Operator 3). Each of these operators also measure each of the units twice. The results are shown as follows:

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Unit

x11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 24 20 27 19 23 22 19 24 25 21 18 23 24 29 26 20 19 25 19

Operator 1 x12 X1 20 20.5 23 23.5 21 20.5 27 27.0 18 18.5 21 22.0 21 21.5 17 18.0 23 23.5 23 24.0 20 20.5 19 18.5 25 24.0 24 24.0 30 29.5 26 26.0 20 20.0 21 20.0 26 25.5 19 19.0

X 1 = 22.3

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R1

1 1 1 0 1 2 1 2 1 2 1 1 2 0 1 0 0 2 1 0 R1 = 1.00

x 21

20 24 19 28 19 24 22 18 25 26 20 17 25 23 30 25 19 19 25 18

Operator 2 x 22 20 24 21 26 18 21 24 20 23 25 20 19 25 25 28 26 20 19 24 17

X2 20.0 24.0 20.0 27.0 18.5 22.5 23.0 19.0 24.0 25.5 20.0 18.0 25.0 24.0 29.0 25.5 19.5 19.0 24.5 17.5

X 2 = 22.28

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R2

x31

Operator 3 x32

0 0 2 2 1 3 2 2 2 1 0 2 0 2 2 1 1 0 1 1 R2 = 1.25

19 23 20 27 18 23 22 19 24 24 21 18 25 24 21 25 20 21 25 19

21 24 22 28 21 22 20 18 24 25 20 19 25 25 20 27 20 23 25 17

X3 20.0 23.5 21.0 27.5 19.5 22.5 21.0 18.5 24.0 24.5 20.5 18.5 25.0 24.5 20.5 26.0 20.0 22.0 25.0 18.0

X 3 = 22.10

R3

2 1 2 1 3 1 2 1 0 1 1 1 0 1 1 2 0 2 0 2 R3 = 1.20

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The gauge repeatability is obtained from the average of the three ranges.

R =

R1 + R2 + R3 = 1.15 3 R 1.15 = = 1.02 d 2 1.128

σ Repeatability =

d 2 is based on a sample size of n = 2, as each range value was based on the range between two measurements.

The gauge reproducability is based on the variability between the three operators. If the X i values differ it is due to the difference between the operators since all measure the same parts.

(

)

(

)

X Max = Max X 1 , X 2 , X 3 = 22.3 X Min = Min X 1 , X 2 , X 3 = 22.10

R X = X Max − X Min = 22.3 − 22.10 = 0.20

σ Reproducability =

RX d2

=

0.20 = 0.12 1.693

d 2 is chosen as 1.693 as R X is based on three operators (i.e. n = 3) We have now estimated both components of the measurement error. 2 2 2 σ Gauge = σ Reproducab ility + σ Repeatability 2 2 2 σ Gauge = (0.12 ) + (1.02 ) = 1.0548

σ Gauge = 1.0548 = 1.03 Calculate the P/T ratio

6σ Gauge P 6(1.03) = = = 0.11 T USL − LSL 60 − 5 Page 7 of Gauge Capability

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The conclusion drawn is that the gauge capability is not adequate. As σ Repeatability >> σ Reproducability , training the operators will not solve the problem. The problem lies with the gauge itself and we should look to another inspection device.

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