Determine the uncertainties

[lavster]

Homework Statement

what is the uncertainty of $$\frac{a+b}{c+d}$$ if a=b=c=d and $$σ_a = σ_b =σ_c=_d$$

Homework Equations

$$σ_{a+b}=√(σ_a^2+σ_b^2)$$

$$σ_{\frac{a}{b}}=√((\frac{σ_a}{a})^2+(\frac{σ_b}{b}^2))$$

The Attempt at a Solution

since a=b=c=d

$$σ_{a+b}=√2 σ_a$$

$$σ_{\frac{a}{b}}=√2 \frac{σ_a}{a}$$

so $$σ_{\frac{a+b}{c+d}} = \frac{√2 √2 σ _a}{2a} = \frac{σ_a}{a}$$

is this correct?!

Thanks

 

[gneill]

The result looks good to me!

 

[lavster]

Great thanks :) any idea why it is the same as the uncertainty in a single measurement... Just doesn't seem right to me! Xxx

 

[BvU]

It is NOT the same as the uncertainty in a single measurement! That would be $\sigma_a$. Since you are evaluating a ratio, only relative errors matter. The factors $\sqrt 2$ and 2 just happen to cancel.

You can repeat the exercise with ${a+b+c}\over e+f+g$ and see what happens...

 

[HalfLostAndSearching]

for your question. As a scientist, it is important to be precise and clear in our responses, so let me walk you through my thought process in determining the uncertainties in this scenario.

First, let's define our terms. The uncertainty, or error, in a measurement is a measure of how much the measured value may vary from the true value. In this case, we are dealing with the uncertainty in the expression \frac{a+b}{c+d}, where a, b, c, and d are all variables with associated uncertainties (σ_a, σ_b, σ_c, and σ_d).

Next, let's consider the given information. We are told that a=b=c=d, meaning that all the variables have the same value. Additionally, we are given the uncertainties in each variable, σ_a, σ_b, σ_c, and σ_d. From this information, we can see that the uncertainties are all equal, and we can represent this as σ_a = σ_b = σ_c = σ_d = σ.

Now, we can use the uncertainty propagation formula to calculate the uncertainty in our expression. This formula tells us that for a function of multiple variables, f(a,b,c,...), the uncertainty in the output, σ_f, is given by:

σ_f = √((∂f/∂a)^2 * σ_a^2 + (∂f/∂b)^2 * σ_b^2 + (∂f/∂c)^2 * σ_c^2 + ...)

In our case, our function is \frac{a+b}{c+d}, so we can plug in the values and simplify:

σ_{\frac{a+b}{c+d}} = √((∂(\frac{a+b}{c+d})/∂a)^2 * σ_a^2 + (∂(\frac{a+b}{c+d})/∂b)^2 * σ_b^2 + (∂(\frac{a+b}{c+d})/∂c)^2 * σ_c^2 + (∂(\frac{a+b}{c+d})/∂d)^2 * σ_d^2)

= √((\frac{1}{c+d})^2 * σ^2 + (\frac{1}{c+d})^2 * σ^2 + (-\frac{a+b}{(c+d)^2})^2 * σ^2 + (-\frac{a