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An example is this pressure transmitter which has a BSL accuracy of 0.08%. I know the term "Best fit straight line" - BFSL. Is BSL just a short hand for this ("Best straight line")?

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BFSL (Best Fit Straight Line) is the same as BSL (Best Straight Line). It is calculated from the calibration's extracted data, where the manufacturer test the sensor for several known input values (in this case, pressures). Once these calibration points are measured, they are approximated with a straight line (least-square method) and the BSL value is an error value that indicates how close the measured points are to this straight line.

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