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I got into a discussion with a colleague today about some of our data sensors. Our sensors weigh objects anywhere from 10 to 100 tonnes. currently we calculate the error as:

$$ error = \frac{\left (x_{field} - x_{true} \right )}{x_{true}} $$

The problem is that if the error is consistent in terms of value, the percentages aren't. For example (fictitious data):

True_wt Field_wt    error       diff_wt
12000   13500       -0.125      -1500
34000   32500       0.0441      1500
65000   63000       0.0307      2000
24000   26000       -0.083      -2000
18000   19500       -0.083      -1500

My point was that estimating the error in such a way for our sensors is incorrect since the real issue is that they're off by x amount and not necessary off by a percentage.

Question: In situations such as the example above, what would be a, or some of the, way(s) that the error in measurement could be measured in?

Edit 1: To reflect on @Whuber's comment

The context of the error is to solve the calibration of the sensor itself. The idea is to calibrate the field sensor to return a value that is very close to the true weight, so that over the year or so the sensor is in the ground, it stores as accurate and precise of weights as possible

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    $\begingroup$ Please tell us this: why is the error important? If, for instance, you are weighing trucks and collecting a fee for weight, then the error translates directly into a fee and you might be very interested in the total error during a fiscal year. If you are determining whether the truck is light enough to cross a limited-load bridge, all you care about is the chance that the truck is too heavy: now even a huge error with a small truck is of no consequence, but a small error at certain other weights would be alarming. In short, the context of the problem determines how to measure error. $\endgroup$
    – whuber
    Jun 27 '13 at 16:40
  • $\begingroup$ @whuber I edited the question to reflect that. Does that clarify the problem? $\endgroup$
    – dassouki
    Jun 27 '13 at 16:44
  • $\begingroup$ Have you considered building a model to predict absolute error as a function of total weight? Given the percentage error is not constant, probably this model would be non linear. Then, you can use the model to produce a calibration curve. $\endgroup$ Jun 27 '13 at 23:01
  • $\begingroup$ @AndreSilva No I haven't, do you mind posting that as an answer perhaps with some explanation as well? $\endgroup$
    – dassouki
    Jun 28 '13 at 14:03
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    $\begingroup$ In addition to @whuber's point: different types of sensors can have different "behaviour" in terms of error. Some measurement techniques may have constant error, for others the error may increase with increased weight. In my field (chemometrics) it is quite common to have both types of error, e.g. a behaviour that has constant variance for low weights, but turns into a proportional standard deviation for higher weights. $\endgroup$ Jun 28 '13 at 15:41
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I am suggesting the following:
Try to find the pattern of the error produced by your sensor ranging from 10 tons to 100 tons objects.

You would need as many samples as it was possible to capture the variation of the error in all classes of weight.

Let's suppose you could find a pattern of error (see figure below). Just as an example, look to the series "d". As you said, the absolute error is constant (% errors are not).

enter image description here

If you find a pattern in the error you can build a model from regression techniques to describe the error as a function of true weight. Then, somehow (don't ask me why) you would adapt the calibration curve to the sensor. If it reads "x' value of weight, the sensor offsets the true value to "y", according to the calibration curve produced.

Off course, the scatter plot is just an very simplified example to illustrate the patterns, but to build the model you would need lots of observations (do not know how many) to sample all classes of weight (e.g. 18 classes with 5 tons of interval).

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