My question is on how the input/output equations for sensors are found, for example a pressure sensor which relates force [N] to resistance [ohms]. Would this equation be found by measuring resistance of the sensor at different force values, multiple times, then plotting the average as a curve? Where an equation for the curve is then found to be either linear, power function or polynomial? How are the sensor uncertainties then calculated?
Textbooks and online sources often say the industry standard for uncertainty is given at a confidence interval of 95 %. This means out of 100 measurements, 95 would achieve the specified value and deviate around this value in range of the +/- uncertainty given. For example temperature sensor: states uncertainty of 1 deg C within a certain range. So 95% of the time the uncertainty is within this range.
However i am not sure how this uncertainty is calculated. Is the uncertainty 2 times standard deviation?
If for example a temperature sensor says it measures temperature with an uncertainty of +/- 1 deg C, then did the manufacturer test this sensor, say a thousand times at a controlled temperature, for example 25 C, the mean temperature measured was then 25 C of those 1000 measurements, and then the uncertainty is +/- 2 times standard deviation of the 1000 measured data points, to get the final uncertainty of +/- 1 deg C at a confidence of 95%?
Uncertainty can be expressed in terms of (1) standard uncertainty (SU), and (2) expanded uncertainty (EU) at a specified confidence level or coverage probability, say 95%. Under the assumption of normality, EU=2xSU. SU is the standard error of the sample mean for a series of observations, which is a Type A SU. A manufacturer's specification may give a SU for its sensor. This is a Type B SU and is for a single measurement (observation). However, a manufacturer often gives a spec like +/- 1 deg C in your example. This number is often known as the Maximum Permissible Error (MPE) It means that the error of measured temperature using this sensor would not exceed the MPE (at 100% of certainty). You need to check with the manufacturer to find out what they mean by +/- 1 deg C: SU or MPE.
Dear Fritz,
Please see:
https://www.bipm.org/en/publications/guides/gum.html
For uncertainty in measurement.
Hello Fritz
The manufacturer should report the sensor uncertainty in the data sheet in some way. The exact metric that they use to quantify it depends upon the construction of the sensor and their personal opinion.
One simple solution would be to hold the measurand (the true value of the signal you want to measure) constant at some known value and then sample a distribution of values from the sensor (sample it many times with constant measurand). This would give you the distribution of the sensor noise at one point. You could repeat this for as many points as you wish and fit a curve to the means of the distributions to get an equation for the input / output characteristics.
The equations for propagating the uncertainty through the equations can be found in the GUM linked by Ciro above.
It is important to look at your data and convince yourself that it is acceptable to assume that the noise is normally distributed before making this assumption. After that the number of standard deviations you report your noise to is a subjective choice.
Dear Fritz,
When talking about the uncertainty in the manufacturer's specification, for instance, for a temperature sensor, presented as +/- value without a probability level, it is most likely that it is the maximum permissible error what is meant, corresponding to 100 % probability. Such an error limit that usually characterizes the accuracy of a measuring instrument is commonly not derived from a statistical analysis of repeated measurements, as you would expect. Moreover, this is usually not based on propagating the uncertainties through the measurement model, as the GUM methodology generally suggests.
The performance characteristics such as the (linear) range, the limit of detection and of quantification, the precision and trueness may be of practical use in description of a newly developed sensor. A detailed discussion of these issues as applied to electrochemical sensors can be found in the paper of Desimoni and Brunetti (2013)
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