(easy) Uncertainty Analysis question

[ausdreamer]

Homework Statement

If I measure A=50, with a minimum value of 48, and a maximum value of 51,
and measure B=100, with a minimum value of 92, and a maximum value of 115,

and I add the two (C=A+B) together, what is the resulting minimum and maximum value of C?

Homework Equations

If A and B had uncertainties A=(50+/-2) and B=(100+/-10), rather than the lower and upper uncertainties differing, then the uncertainty in C would be:

SQRT((2^2)+(10^2))

The Attempt at a Solution

Can I use the above expression for the lower uncertainties, and then again on the upper uncertainties, to individually calculate the upper and lower C values? So we'd get:

C_minimum = SQRT((2^2)+(8^2)) = SQRT(4+64)=SQRT(68)=8.246 =:= 8
and
C_maximum = SQRT((1^2)+(15^2))=SQRT(1+225)=SQRT(226)=15.03 =:= 15

so C=150 with a minimum value of 142 and a maximum value of 165. Does this seem correct?

 

[DiracRules]

That's not correct, at least according to the standard theory of propagation of error.

The correct formula is, as you stated
$u_C^2=u_A^2+u_B^2$ where u is the uncertainty.

Actually, the most general formula for $y=f(x_1,x_2,\ldots,x_i,\ldots,x_N)$ is

$u_y^2=\sum_{i=1}^N c_i^2u_{x_i}^2+2\sum_{i=1}^{N-1}\sum_{j=i+1}^N c_ic_j r_{ij}u_{x_i}u_{x_j}$ where $r_{ij}$ is the correlation coefficient ($r_{ij}=\frac{ u_{(x_i,x_j)} }{ u_{x_i}u_{x_j} },\,\,u_{(x_i,x_j)}$ is the covariance)

The expression above can be calculated, and http://mathworld.wolfram.com/ErrorPropagation.html" is a nice explanation.

 

[ausdreamer]

Thanks for your reply DiracRules, but I don't think it has answered my question. I just want to know how to calculate the uncertainty given my uncertainty is not equal above or below the measured value. (Ie. not A=50+/-2, but a measured value of A=50 with a lower value of 48, and maximum value of 51)

If your answer does answer this question, then can you please be more descriptive in your answer?

 

[DiracRules]

Oh, sorry I misread.

Now I can't say if your solution is correct or not - I will check my textbook, but I don't understand one thing: what does actually mean that you measured A=50 with a lower measure of 48 and an higher of 51?

I can only guess that either you did 3 measurement (48, 50, 51) or you did more than three, so that you get A in [48,51] and average(A)=50.
Both the way, usually $u_B=\sigma(B)$ or $u_B=2\sigma(B)$ or $u_B=3\sigma(B)$, that is the uncertainty is a multiple of the standard deviation (this way you have a symmetric interval for the measure).

 

[ausdreamer]

I mean I used a program to find the values of A and B, and it also gave me the lowest possible value and largest possible value for each. But it turns out the lower and upper values aren't equally far away from the actual result it gave for the quantity.

So if you pictures the measurement of A plotted on the y-axis, the error bar in the vertical direction (for the measurement of A) would be 2 values down from 50, and 1 value above 50. So not equal on either side.

 

[HallsofIvy]

The lowest A can be is 48 and the lowest B can be is 92 so the lowest possible value for A+ B= 48+ 92= 140. The highest A can be is 51 and the highest B can be is 115 so the highest possible value for A+ B= 51+ 115= 166.

IF the question was "what is the resulting minimum and maximum value of C?" as you say in your original post, nothing more is necessary.

 

[ausdreamer]

The lowest A can be is 48 and the lowest B can be is 92 so the lowest possible value for A+ B= 48+ 92= 140. The highest A can be is 51 and the highest B can be is 115 so the highest possible value for A+ B= 51+ 115= 166.

IF the question was "what is the resulting minimum and maximum value of C?" as you say in your original post, nothing more is necessary.

Yes, but I'm required to write an uncertainty analysis and that reasoning isn't detailed enough :(

I'm just looking for confirmation of my idea to carry out the standard uncertainty formula (sqrt(...)) but substitute in the lower uncertainty in A and B to find the lower uncertainty in C, and then substitute the upper uncertainty in A and B to find the upper uncertainty in C. This makes sense to me but I'm just looking for confirmation that this'll work.

 

[DiracRules]

As HallsofIvy said before, IF you need to say the maximum and minimum values you just add the maximum and minimum values of A and B.

IF you need to do an analysis of uncertainty, then maximum and minimum values of A and B are worthless.

To do a proper analysis, you should find the standard deviation of your measures and perform your calculations on that.

for example, you find:
$\bar{A}=50, u(A)=\sigma(A),\bar{B}=100, u(B)=\sigma(B)$ then $\bar{C}=150, u(C)=\sqrt{u(A)^2+u(B)^2}$
(or, if you wish, you may choose $u=2\sigma\,\,or\,\, u=3\sigma$).