How to compare two measurements with uncertainties in terms of sigma?

In summary, the conversation discusses the use of t-values and uncertainties when comparing measurements. The t-value is calculated using the difference between two measurements and their standard errors. It is used to determine if the measured value is within a certain range of the accepted value. There may be some variation in how people discuss this concept depending on their country or background. Finding references for more complex error propagation techniques can be challenging.
  • #1
CCofADoa
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Homework Statement
For example, we measured some quantity in our lab 1.0 +/- 0.1 GeV, but another lab measured 0.98 +/- 0.01 GeV. How can we say something like, "Our measurement is within 2 sigma of the other measurement or reference value"? Does that kind statement event make sense?
Relevant Equations
There was something called t-value, which is defined by:

t = (measured value - accepted value) / standard error of measured value
Perhaps that statement is just saying how big the t-value is.

Like, in this case:

t = (1.0 - 0.98) / 0.1 = 0.1

So we can say that our measured value is within 1 sigma from the other measured value.
In this case, do we just ignore the uncertainties of the other measured/reference value?

It's possible this is a language problem, and you just don't say this in English. Because I was not able to find anything from googling about it.
 
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  • #2
It's a bit more complicated when the measurements have different standard errors. Suppose independent measurements x and y are Normally distributed, with means μx and μy, and variances σx2 and σy2. (We use variance rather than standard deviation because variances are additive). The difference x-y is Normal with mean μx - μy, and variance σx2 + σy2. The t-value is then
t = (x - y)/√(σx2 + σy2)
 
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  • #3
Thanks a lot.

So, do people even say, "Our result is within 1 sigma from the accepted value" when they are talking about the t-value? Perhaps it's just the people in my country.

Personally I always find it difficult to find the references for this stuff, beyond basic error propagation. For example, how error propagation works in a curve fitting, or in a matrix inversion, etc.
 

FAQ: How to compare two measurements with uncertainties in terms of sigma?

1. What is the purpose of comparing two measurements with uncertainties in terms of sigma?

The purpose of comparing two measurements with uncertainties in terms of sigma is to determine the level of agreement between the two measurements. This allows scientists to assess the reliability and accuracy of their experimental data.

2. How do you calculate the uncertainty in terms of sigma?

The uncertainty in terms of sigma can be calculated by finding the difference between the two measurements and dividing it by the average of the two measurements. This value is then multiplied by the standard deviation of the two measurements.

3. Can the uncertainty in terms of sigma be negative?

No, the uncertainty in terms of sigma cannot be negative. It is always expressed as a positive value, representing the distance from the mean or central value.

4. How is the level of agreement determined when comparing two measurements with uncertainties in terms of sigma?

The level of agreement is determined by calculating the ratio of the uncertainty in terms of sigma to the difference between the two measurements. This ratio is then compared to a reference value, typically set at 1 or 2. If the ratio is less than the reference value, the two measurements are considered to have good agreement.

5. What is the significance of using sigma in comparing two measurements with uncertainties?

Sigma is a unit of measurement that represents the standard deviation of a set of data. Using sigma in comparing two measurements with uncertainties allows for a standardized and consistent way of evaluating the agreement between the two measurements. It also provides a quantitative measure of the uncertainty, allowing for a more accurate analysis of the data.

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