How To Incorporate Sensor Uncertainties With Multiple Measurements?

In summary, when taking multiple measurements to calculate an average, the standard deviation of the mean is equal to 1/sqrt(N) times the standard deviation of the measurements. However, this does not take into account the imprecision of the individual data points, which can be attributed to both random and systematic errors. While random errors decrease with the square root of the number of measurements, systematic errors cannot be reduced. In order to fully understand and account for uncertainty in measurements, it is important to consider both the variance of the data and the possible sources of error.
  • #1
012anonymousx
47
0
Lets say a sensor measures within accuracy of +/- 0.05N

And you take multiple measurements and graph it out. (I.e., 5.12N, 5.15N, 5.05N...).

What is the uncertainty of the final average?

One way I read is:

Sx = s/√N where s = std. dev.
N = number of data points

But this doesn't incorporate the sensor's uncertainty.
 
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  • #2
The standard deviation of the mean (aka the standard error) is equal to ##1/\sqrt{N}## times the standard deviation of the measurement. You can derive this from the equation in the paper I posted to your previous thread.
 
  • #3
That uncertainty comes from variance of the data.

But it does not incorporate the fact that the data points themselves have imprecision.

Does it?
 
  • #4
RANDOM errors decrease with the square root of the number of measurements. (If the measurements are roughly normally distributed.) SYSTEMATIC errors do not.

In order to see if multiple leasurements help at all, you have to break the stated measurement uncertainty into those two components. Simple instruments only give an overall uncertainty, which makes it almost impossible.

Random errors can be estimated by taking the standard deviation (the SAMPLE standard deviation should be used) of a large number of measurements.

Systematic errors are much harder to estimate. They impose a lower bound on the quality that can be attained with even an infinite number of re-measurements.

If you look at metrology papers, you will see that they generally re-measure until the random error is somewhat lower than the systematic error; additional effort gains too little to bother.
 
  • #5
012anonymousx said:
That uncertainty comes from variance of the data.

But it does not incorporate the fact that the data points themselves have imprecision.

Does it?
In the end it doesn't matter. You have a single random variable, the measurement. Without additional information there is no way to separately model the variance due to the measurement and the variance due to the thing being measured.

Perhaps you are leaving out some information in the description.
 
  • #6
Ah, so a systematic error cannot be reduced. I appreciate it a lot.

I left out information of the experiment but it was irrelevant.
 

1. How do I take sensor uncertainties into account when I have multiple measurements?

The best way to incorporate sensor uncertainties with multiple measurements is to use statistical methods such as error propagation or weighted least squares. These methods take into account the uncertainties of each individual measurement and provide a more accurate estimate of the true value.

2. Can I simply average multiple measurements to account for sensor uncertainties?

No, averaging multiple measurements does not take into account the uncertainties of each measurement and can actually result in a less accurate estimate. It is important to use statistical methods that consider the uncertainties of each measurement.

3. How do I determine the uncertainties of my sensors?

The uncertainties of sensors can be determined through calibration and validation processes. This involves comparing the sensor readings to a known reference value and calculating the difference between the two. The standard deviation of this difference can then be used as the uncertainty for that particular sensor.

4. Are there any software or tools available to help incorporate sensor uncertainties with multiple measurements?

Yes, there are various software and tools available that can help with incorporating sensor uncertainties. Some examples include MATLAB, Python, and R, which have packages specifically designed for error propagation and weighted least squares calculations.

5. Can sensor uncertainties be completely eliminated?

No, it is not possible to completely eliminate sensor uncertainties. However, by using appropriate statistical methods and ensuring proper calibration and validation processes, the uncertainties can be minimized and the accuracy of the measurements can be improved.

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