Error Propagation in Calculations: Estimate Absolute & Relative SDs

[prize fight]

Homework Statement

Estimate the absolute and relative standard deviations of the following calculations. The number in parentheses is the standard deviation of the preceding value.

a) z=5.64(s=0.14)*log(138)(s=3)

Homework Equations

S_x/x =SQRT((S_p/P)²+(S_q/q)²+(S_r/R)²

S_x=0.434(S_p/P)

The Attempt at a Solution

I thought of two ways to go about the problem I am not sure which way is correct here are both attempts:

log(138)= 2.139
Z=12.069
S_Z=SQRT((0.14/5.64)²+(3.0/2.139)²)=1.40
Z=12.07 $$\pm$$1.40

RSD=(1.4/12.07)*100 =11.6%

Or my other attempt:

Z=12.07
(0.14/5.64)²+(0.434*(3.0/138))=0.01
Z=12.07$$\pm$$0.01
RSD=(0.01/12.07)*100=0.08%

Are either of these ways correct? Any help would be appreciated. Thanks.

 

[nate808]

Thank you for your question. Both of your attempts are on the right track, but there are a few corrections that need to be made.

Firstly, when calculating the standard deviation of a product or quotient, we use the formula you provided: Sx/x =SQRT((Sp/P)2+(Sq/q)2+(Sr/R)2. However, in this case, we are not dealing with a product or quotient - we are dealing with a logarithm, which is a function. Therefore, we need to use a different formula, which is: S(log(x))=Sx/x*1/ln(10). This formula takes into account the fact that the standard deviation of a logarithm is proportional to the standard deviation of the original value.

So, for your first attempt, the correct formula to use would be: S(log(138))= 0.14/138*1/ln(10)=0.00033. This is the standard deviation of the logarithm, not the value of the logarithm itself.

Then, to calculate the standard deviation of the final value, we use the formula: Sz=Z*S(log(x)). Plugging in the values, we get: Sz=5.64*0.00033=0.00186. This is the absolute standard deviation of the final value, which is z=12.07.

To calculate the relative standard deviation (RSD), we use the formula: RSD=Sz/z*100%. Plugging in the values, we get: RSD= 0.00186/12.07*100%=0.0154%. This is the correct RSD for this calculation.

For your second attempt, the formula you used for calculating the standard deviation of a value (0.434*(3.0/138)) is incorrect. The correct formula to use is: Sx=0.434*S(log(x)). Plugging in the values, we get: Sx=0.434*0.00033=0.00014. This is the absolute standard deviation of the final value, which is z=12.07.

To calculate the relative standard deviation (RSD), we use the formula: RSD=Sx/z*100%. Plugging in the values, we get: RSD= 0.00014/12.07*100%=0.0012%. This is the correct RSD for this

 

[Blueshift5]

Both of your attempts are correct, but they are slightly different approaches to calculating the absolute and relative standard deviations. In the first attempt, you used the formula for calculating the overall standard deviation of a product of values, which is appropriate for this problem. In the second attempt, you calculated the individual standard deviations and then added them together, which is also a valid approach. However, the second attempt would be more appropriate if you were given the individual standard deviations for each value, rather than just the overall standard deviation for the preceding value.

In terms of which approach to use, it depends on the information given and the specific context of the problem. In this case, both approaches give similar results, with the first approach giving a slightly larger standard deviation due to the use of the overall standard deviation for the preceding value. However, both approaches give a relatively small standard deviation, indicating a high level of precision in the calculation.

Overall, your understanding and application of error propagation in calculations is correct, and your solutions are valid. Just make sure to carefully consider the given information and context of the problem when deciding which approach to use.