Combining three measurements of same thing to find error

In summary, the three temperatures are averaged to get the temperature of the object. The instrument error is multiplied by the percentage error on each temperature to get the final error.
  • #1
pablo4429
19
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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?
 
  • #5


Hi Paul,

Your question is a common one in error analysis and there are a few different approaches that can be taken to combine the errors from multiple measurements. One method that is often used is called the root sum squared (RSS) method. This method involves taking the square root of the sum of the squares of the individual errors. In your case, this would look something like this:

Error = √(0.75%^2 + 0.75%^2 + 0.75%^2) = 1.3%

This means that the overall error in your temperature measurement would be 1.3%, which is slightly higher than the individual errors of each thermocouple. This is because the RSS method takes into account the fact that there are multiple sources of error and combines them in a way that gives a more accurate representation of the overall error.

Another method that can be used is called the weighted average method. This method involves assigning weights to each measurement based on their individual errors. In your case, since all three thermocouples have the same error, the weights would be equal. The formula for this method would look like this:

Error = (0.75% + 0.75% + 0.75%)/3 = 0.75%

This method would give you a slightly lower overall error compared to the RSS method, but it is important to note that the weights used in this method are subjective and can vary depending on the situation.

I hope this helps and good luck with your error analysis!
 

1. What is the purpose of combining three measurements to find error?

Combining three measurements of the same thing allows scientists to calculate a more accurate and reliable estimate of the true value. It takes into account the variability of the measurements and provides a more precise understanding of the uncertainty associated with the results.

2. How do you combine three measurements to find error?

To combine three measurements, scientists typically use statistical methods such as averaging or weighted averaging. These methods take into account the different magnitudes and potential sources of error for each measurement and produce a single value that represents the best estimate of the true value.

3. What are the benefits of combining three measurements to find error?

Combining three measurements allows scientists to improve the accuracy and precision of their results. It also provides a better understanding of the potential sources of error in the measurements, which can help in the design of future experiments or studies.

4. How does combining three measurements affect the overall error?

Combining three measurements typically reduces the overall error because it takes into account the variability of the measurements and produces a more accurate estimate of the true value. However, this depends on the quality of the measurements and the methods used to combine them.

5. Are there any limitations or assumptions when combining three measurements to find error?

One limitation is that combining measurements assumes that the errors in each measurement are independent and random. If there are systematic errors or correlations between the measurements, the combined error may not accurately represent the true uncertainty. Additionally, the accuracy of the combined result is limited by the accuracy of the individual measurements.

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