# How To Incorporate Sensor Uncertainties With Multiple Measurements?

1. Sep 15, 2013

### 012anonymousx

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.

Last edited: Sep 15, 2013
2. Sep 15, 2013

### Staff: Mentor

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. Sep 15, 2013

### 012anonymousx

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. Sep 15, 2013

### cypherpunks

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. Sep 15, 2013

### Staff: Mentor

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. Sep 15, 2013

### 012anonymousx

Ah, so a systematic error cannot be reduced. I appreciate it a lot.

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

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