# Combining three measurements of same thing to find error

#### pablo4429

Hi All,

So here is an error analysis question for you all:

I am measuring the temperature of an object at three positions simultaneously. They are type K thermocouples so their individual error is 0.75% of their reading.

However, I can also take long term data which shows that their individual temperatures wander much less than that.

The question is, how do I combine the instrument error with the statistical error to get the actual temperature and error on that temperature of the object?

I am hoping since there are three thermocouples, I can leverage them off of each other to get a more accurate and precise value. I am going to check the Squires book soon but I am unsure of what this process would be called to start the lookup process.

This page looks promising, I think it is similar to what I want except that I would just extend the calculation to three measurements from two, would you agree?

http://isi.ssl.berkeley.edu/~tatebe/whitepapers/Combining Errors.pdf

Thanks a ton!
Paul

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#### Simon Bridge

Science Advisor
Homework Helper
Each temperature reading should be recorded to the accuracy of the instrument (unless the operator is less accurate). The extra DP are surplus to requirements and should be discarded as providing no further information.

Consider - if the temperature of the plate is prett constant with time, then wouldn't the thermocouples indicate a fairly constant temperature too? It may be the wrong constant temperature. To see how constant it is, you'd have to restart the measurement from scratch. In which case you would be taking repeated measurements.

#### pablo4429

Ah ok. So would the best way to claim the temperature to be take an average of the three instruments measuring the same object and just claim the instrument error as the measurement error even though the final measurement is an average?

Thanks again.

#### Simon Bridge

Science Advisor
Homework Helper
You should check the device documentation to see what the quoted error value means.
In general - if you estimate a value of something by the average of N independent measurements, each with standard error s, then the std error on that average is s/√N

I your case the standard errors may not be the same - but you do have the same percentage error on each reading. So you need to propagate the percentage error through the calculation for the average temperature.

$\bar T = \frac{1}{3}(T_1+T_2+T_3)$

... do you know how to propagate errors?

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