ARTICLES
# Gauge Bias Linearity

Bias in the measurement system is the difference between the average of measured values and the actual value of a part. The actual value of the part is called the reference value. Ideally the bias should be zero.

The above image shows the variation in the different measurements on the same part about the average and the deviation of the average from the reference value. In this instance the bias is negative.

It cannot be assumed that bias is constant over all ranges of measurements. It is possible that bias increases, or reduces, as the magnitude of measurements increases or decreases, such as shown in the image below. This can be due to several reasons, such as poor calibration, poor maintenance. For example weighing scales used to weigh light weights, medium and heavy weights may provide different biases at each level, if the scales have not been maintained.

A linearity study is performed to establish if there is a linear relationship with the magnitude of the measured itemâ€™s characteristic. For such a study reference parts are chosen varying between the normal operating range. Usually, for a simple analysis several measurements are taken for each reference part by the same appraiser. The bias is then plotted on a scatter chart, as shown below, and a statistical analysis performed to determine if the observed linearity is likely to be real, or due to chance. Sometimes due to the way the numbers fall it may appear that there is a trend, but if another set of measurements is taken the trend is no longer apparent.

The scatter diagram visualizes the relationship. It is possible that even though there is a relationship between bias and magnitude of the measurements, the relationship is non-linear. Typically, as per example, a line of best fit is drawn with confidence bands as shown. If the green line (zero bias) falls within the confidence bands then there is no evidence for the bias to be dependent on magnitude, i.e. bias is constant at zero.

BIS.Net MSA provides the following analytical output to further place statistical significance on the relationship

The contents have the following meaning.

Repeatability is the combined standard deviation of measurement error for the same part obtained over all parts. You can expect an error of plus or minus 3 times this value for each part, irrespective of the magnitude of the part. However, it is possible that the error standard deviation varies with magnitude of the measurements, just as the bias can. A more in-depth analysis is required to establish this.

%EV (Equipment Variation) is the percent of the repeatability relative to the total (process standard deviation of a stable process). E.g. if the process standard deviation equals 1 and repeatability is .1 then %EV =100*.1/1=10%. One would like the percentage to be small.

The t values for the slope and intercept of the straight line is compared to the critical t value to establish significance. For linearity one is normally only concerned with the slope. If the t-value is greater than the critical value the appraiser can be reasonably (never 100%) certain that there is a linear effect at the chosen level of significance.

Alternatively, and preferably use the p value for the slope to see if there is a relationship between magnitude of measurements. Typically, the value would be less than .05

R-Squared is a measure of the variation explained by a linear relationship. This value should be over 90% if there is true linear relationship. For the above example the value is only 75% indicated that the relationship is non-linear, however because the green line does not fall within the red confidence bands one can conclude that the bias is dependent on magnitude of measurements, which needs to be investigated.

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