Error Propagation calculation

And then calculate the relative uncertainty as a percentage of the central value.In summary, the absolute and relative standard deviations for the given calculations can be estimated using the RSS method. The absolute uncertainty is found to be ((.14/5.64)2 + (.01/2.14)2)1/2, and the relative uncertainty is calculated as a percentage of the central value, resulting in 11.6%.
  • #1
prize fight
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1. 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)

2. Homework Equations
Sx/x =SQRT((Sp/P)2+(Sq/q)2+(Sr/R)2

Sx=0.434(Sp/P)

3. 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
SZ=SQRT((0.14/5.64)2+(3.0/2.139)2)=1.40
Z=12.07 LaTeX Code: \\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=12.07LaTeX Code: \\pm 0.01
RSD=(0.01/12.07)*100=0.08%

Are either of these ways correct? Any help would be appreciated. Thanks.
 
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  • #2
prize fight said:
1. 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)

2. Homework Equations
Sx/x =SQRT((Sp/P)2+(Sq/q)2+(Sr/R)2

Sx=0.434(Sp/P)

3. 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
SZ=SQRT((0.14/5.64)2+(3.0/2.139)2)=1.40
Z=12.07 LaTeX Code: \\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=12.07LaTeX Code: \\pm 0.01
RSD=(0.01/12.07)*100=0.08%

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

I would choose a different approach. For:

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

Certainly the RSS of the relative uncertainties is a good method. But in that regard I would prefer to treat the relative error of the 3/138 as really the relative uncertainty of the range of Log(138 ±3) which looks to me more like 2.14±.01, because that is the effect on the final result, as opposed to the 3/138.

Then I would choose to take the RSS of these relative terms according to the product rule.

((.14/5.64)2 + (.01/2.14)2)1/2

And calculate the absolute uncertainty from that expression.
 
  • #3



I would first clarify the units of the standard deviations given in the problem. Are they in the same units as the values being calculated (i.e. z and log(138))? This is important because the formula for relative standard deviation includes dividing by the value itself, so the units need to match.

Assuming the standard deviations are given in the same units as the values, I would use the first method presented in the attempt. This is because the second method seems to have some errors - the formula for relative standard deviation should have the standard deviation in the numerator, not the value itself. Additionally, when converting from natural logarithm to base 10 logarithm, the conversion factor should be 2.303, not 0.434.

So the correct calculation would be:

log(138) = 2.139
Z = 5.64 * 2.139 = 12.07

SZ = SQRT((0.14/5.64)^2 + (3/2.139)^2) = 1.40

RSD = (1.40/12.07) * 100 ≈ 11.6%

Note that the absolute standard deviation is simply the value of SZ, while the relative standard deviation is calculated by dividing SZ by the value of Z and multiplying by 100 to get a percentage.

In conclusion, the first method presented in the attempt is the correct approach for calculating the absolute and relative standard deviations in this problem. However, it is always important to double check the units and formulas being used to ensure accuracy in your calculations.
 

What is error propagation calculation?

Error propagation calculation is a method used in science and engineering to estimate the uncertainty or error in a final calculated value based on the uncertainties in the measured values used in the calculation. It takes into account the potential errors in each individual measurement and combines them to give a more accurate estimation of the overall uncertainty.

How is error propagation calculation performed?

Error propagation calculation is performed using a mathematical formula known as the "error propagation formula" or the "law of propagation of uncertainty". This formula takes into account the uncertainties in each individual measurement, as well as the relationship between the measured values and the final calculated value, to determine the overall uncertainty.

Why is error propagation calculation important?

Error propagation calculation is important because it allows scientists and engineers to properly evaluate and communicate the level of uncertainty in their measurements and calculations. This is crucial in making informed decisions and drawing accurate conclusions from experimental data.

What are some sources of error that can contribute to the overall uncertainty in a measurement?

Some common sources of error that can contribute to the overall uncertainty in a measurement include instrument limitations, human error, environmental factors, and the inherent variability of the measured quantity. It is important to identify and minimize these sources of error to improve the accuracy of the final calculated value.

How can error propagation calculation be used in real-world applications?

Error propagation calculation is used in a wide range of scientific and engineering fields, such as chemistry, physics, and engineering, to determine the uncertainty in various measurements and calculations. It is also used in quality control and assurance processes in industries such as manufacturing and healthcare. Additionally, error propagation calculation is used in statistical analysis to determine the uncertainty in survey data and polling results.

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