Measurement System Linearity (a Type A Uncertainty)
Linearity looks at the accuracy of the measurements over the full
range of the device.

In order to measure the linearity of a device, we must take
repeated measurements of parts or samples that cover its entire
range.

So that we don't introduce reproducibility error into the picture,
the same operator must make all the measurements.
To check linearity, measure at least 5 samples that cover the full
the range of the instrument.

Reference measurements for each of the samples (made by your
quality group or by an outside laboratory) will be needed to
determine linearity.

The reference measurements will be compared to the results from
the instrument whose linearity is being studied.

Measure each of the samples randomly at least 10 times.

For each of the parts, calculate the average and the range of the
measurements made. The sample averages and ranges will be used with
the reference values to determine linearity either graphically or by
calculations.
Graphical Method:

Plot the average measured values (on the yaxis) for each sample
against the reference value (on the xaxis). If the resulting line
is approximates a straight line with a 45degree slope, the
measurement device is linear.

If the measured values do not form a straight line, or the line
diverges from the optimal 45degree slope, you may have a problem
with linearity.
Calculating Linearity:

A technique does exist that provides a precise mathematical
evaluation of the linearity.

The evaluation is based on the equation of a line that defines the
relationship between the bias and the reference values of the parts
or samples.
 The bias is the value of the sample measurement minus the
reference measurement.
 To calculate the line of best fit, use the equation:
y = ax + b
where: y = bias value a = slope of the line x = reference value b = the yintercept
 To calculate the slope, a:
where: n = total number of measurements made
 To calculate the yintercept, b:

With values for a and b, we can
complete the regression equation (y = ax
+ b); it gives us the line of best fit.

Using the results of the regression
equation, we can determine the “goodness
of fit” by calculating the Coefficient
of Determination, R^{2}.
 R^{2} lets us know what amount of
the variation in the bias values the
regression line explains.
 If R^{2} is 0.6 (60%) or more, the
regression line is an adequate
representation of the line of best
fit.
Calculating the linearity and bias.

Linearity = a, slope of the line of
best fit

Bias = b, yintercept of the line of
best fit
Test the linearity
H_{0}:
a = 0
Test the bias
H_{0}:
b = 0
If linearity or bias fails the ttest, the device is unacceptable.
The options include:

Recalibrate the device and recheck the linearity and bias.

Replace the device with one that is linear.

Use the device over only the linear portion of the range.

