# Intro to Measurement System Analysis (MSA) of Continuous Data – Part 3: Linearity

#### Miner

##### Forum Moderator
Staff member
This is the third in a series of articles about MSA. The focus of this article will be on measurement linearity.

Linearity is simply measurement bias throughout the entire range of the measurement device. For this reason bias and linearity are often combined into a single study.

A good calibration system will check the calibration of a measurement device at a minimum of three locations (both extremes and the middle of the measurement range). A thorough linearity study will check at least five locations (e.g., both extremes, and at 25%, 50% and 75% of the measurement range). Standards such as used in calibration should be used instead of actual parts unless the parts can be measured with less measurement variation with a different measurement device. Each standard is measured repeatedly at least ten times. All measurements must be made randomly to minimize the appraiser recalling previous results. The results can be analyzed using statistical software such as Minitab, or with the attached file as shown.
• Calculate the bias for each individual measurement and the average bias for each reference standard (or part).
• Plot the individual and average biases on the y-axis of a scatter plot versus the values of the standards on the x-axis.
• Perform a regression analysis using the individual biases as the response and the reference values as the predictor variable.
• Plot the regression line and the 95% confidence limits for the regression line on the scatter plot.
• Plot the bias = 0 line for all reference values.
• Verify that the Bias = 0 line lies within the +/- 95% confidence limits of the regression line.
• The y – intercept and slope of the regression equation should each be approximately equal to zero. These values may be statistically evaluated if desired per the formulae in the AIAG MSA 3rd edition manual. For practical purposes the graphical analysis is sufficient. The statistical analysis is only necessary for borderline cases.
Ideally, linearity will be statistically equal to zero. However, this will not always be the case. There are several possible scenarios:
• [Constant bias – All measurements are offset by the same amount regardless of size. This is essentially a calibration issue. Calibrate the measurement device and repeat the linearity study.
• ]Non-constant, linearly increasing/decreasing bias –The bias either increases or decreases as the location within the measuring range increases. Possible causes: measurement scale is proportionally small/large (gage issue), thermal bias (thermal expansion/contraction is proportional to the size of the dimension), pressure bias (deflection under pressure is proportional to size), etc.
[Non-constant, nonlinear bias – The bias changes in a nonlinear fashion throughout the measurement range. Possible cause: measurement scale is nonlinear (gage issue, particularly with electronics), gage wear in one section of the measurement range, worn standards.

If the linearity is acceptable within the range actually used for measurement, the gage may be accepted for a specified range of measurement. This must be clearly noted on the gage and the practice documented in the appropriate quality procedures.

The next article will be:
Intro to Measurement System Analysis (MSA) of Continuous Data – Part 4: Stability

Last edited by a moderator:

#### Ron Rompen

Trusted Information Resource
Thanks for the update, Marc, and of course to Miner for posting the article in the first place.

G

#### Guest

I got a situation where the No of Parts are 5, No appraisers 3 and No of trials 3, Can I conduct MSA for this situation ? if so what are Values of Constants Like D2,D3,D4? and K1, K2, K3 for this situation as per MSA edition 3? can some one Help on this Urgently Please?

#### Miner

##### Forum Moderator
Staff member
dsvkannagi;bt514 said:
I got a situation where the No of Parts are 5, No appraisers 3 and No of trials 3, Can I conduct MSA for this situation ? if so what are Values of Constants Like D2,D3,D4? and K1, K2, K3 for this situation as per MSA edition 3? can some one Help on this Urgently Please?
Yes
K1 = 0.5908
K2 = 0.5231
K3 = 0.4030
d2 = 1.693
d3 = 0.888
D4 = 2.58

G

#### Guest

hello Miner,

I've been using MSA since 2nd Ed., in which the linearity test were far more simple... the linear regression were based on mean bias values, not on individual values (which is the case since 3rd Ed)...

in fact, my concern is about the R-sq value, which is (by consequence) derived from individual values.. the way I see it, the estimation of residuals based on individuals will be testing not only the bias variation (over op. range) itself, but also repeatability of the MS... in other words, based on individuals, this index will not be usefull to caracterize true linearity error.. by 'true linearity error' I mean the variation of the gain (slope) error over the op. range, not the slope of the mean line itself (MSA mixes these two different errors in only one).. with residuals based on means, it was possible to state if there is a true linearity error on the MS (by observing how good the mean bias values fit in a line)...

Last edited:
G

#### Guest

Hello Guys,
I was wondering if someone can help me out with my deliemma. I am trying to create an Uncertainty Budget for the optical microscope that we have in our lab. I am following the VDA German Automotive standards which make alot of refrences to AIAG MSA. I cannot figure out how to derive the linearity uncertainty. Does anybody have an suggestions? Formulas/ Explaination would help alot.
Thanks

#### Miner

##### Forum Moderator
Staff member
Hello Guys,
I was wondering if someone can help me out with my deliemma. I am trying to create an Uncertainty Budget for the optical microscope that we have in our lab. I am following the VDA German Automotive standards which make alot of refrences to AIAG MSA. I cannot figure out how to derive the linearity uncertainty. Does anybody have an suggestions? Formulas/ Explaination would help alot.
Thanks
I am not an expert in Uncertainty. I recommend that you post your question in the ISO 17025 forum, but expert responses.

#### Miner

##### Forum Moderator
Staff member
Francisco Augusto;bt556 said:
hello Miner,

I've been using MSA since 2nd Ed., in which the linearity test were far more simple... the linear regression were based on mean bias values, not on individual values (which is the case since 3rd Ed)...

in fact, my concern is about the R-sq value, which is (by consequence) derived from individual values.. the way I see it, the estimation of residuals based on individuals will be testing not only the bias variation (over op. range) itself, but also repeatability of the MS... in other words, based on individuals, this index will not be usefull to caracterize true linearity error.. by 'true linearity error' I mean the variation of the gain (slope) error over the op. range, not the slope of the mean line itself (MSA mixes these two different errors in only one).. with residuals based on means, it was possible to state if there is a true linearity error on the MS (by observing how good the mean bias values fit in a line)...