# Need clarification on sig-figs and propagation of error.

• Ian Mount
In summary: The second part of this statement is not true. -- You can use the error propagation law to be more precise: If X = A*B then relative uncertainty in X is: (error in A/A) + (error in B/B) and the uncertainty in X is: (error in A)*(B) + (A)*(error in B).
Ian Mount
Hello,

I have a question asking me to find the volume of a rectangular prism. The dimensions are as follows:

x = 20 ± 0.2 cm, y = 30 ± 0.2 cm, z = 70 ± 0.4 cm

I am asked to report the answer with the correct number of significant figures and include the error.

What I have so far:

V = xyz = 42000 ± 558.93 cm3

So I'm stuck on determining how many significant figures the answer should have and how to round the error.

I'm leaning towards 3 significant figures because 20 ± 0.2 is anything between 19.8 and 20.2, which have 3 significant figures. The same can be said for the other given dimensions. Therefore the volume would be V = 4.20*104 cm3.

Also, I'm not sure how to round errors. Do I round to the same number of significant figures or do I round to the same precision as the answer? If it's the same precision, assuming three significant figures, the answer would be 4.20*104 ± 6*102 cm3. This feels correct, if you know what I mean.

Any help would be greatly appreciated. Thanks!

I moved the thread to our homework section.

With a proper uncertainty estimate the concept of significant figures is not that relevant any more. Typically two digits of the uncertainty are reported, but as your input uncertainties have only one significant figure rounding to one digit of the uncertainty should be fine as well.

mfb said:
I moved the thread to our homework section.

With a proper uncertainty estimate the concept of significant figures is not that relevant any more. Typically two digits of the uncertainty are reported, but as your input uncertainties have only one significant figure rounding to one digit of the uncertainty should be fine as well.

So would the answer, 42000 cm3, be reported with only one significant figure as well? Leading to a final answer of 40000 ± 600 cm3? What you're saying makes sense to me it just feels so wrong, as 20*30*70 is clearly 42000, or were you only referring to the uncertainties, with regards to the number of significant figures?

No, the central value always keeps the same absolute precision as the uncertainty: (420 +- 6) * 10^2

mfb said:
No, the central value always keeps the same absolute precision as the uncertainty: (420 +- 6) * 10^2

Oooh okay, that makes sense. Thank you so much! I was having a ridiculously hard time finding a definitive answer on google.

There is an engineer's "rule of thumb" that "when quantities add, their errors add, when quantities multiply their relative errors add".

## 1. What are significant figures and why are they important in scientific calculations?

Significant figures represent the precision of a measurement and indicate the number of digits that are known with confidence. They are important in scientific calculations because they help to ensure the accuracy and reliability of the results.

## 2. How do you determine the number of significant figures in a measurement?

The general rule is that all non-zero digits are significant, zeros between non-zero digits are significant, and zeros at the end or beginning of a number are not significant. However, there are some exceptions and rules for rounding when dealing with certain operations and scientific notation.

## 3. What is the purpose of propagation of error in scientific experiments?

Propagation of error is a method used to estimate the uncertainty in the final result of a calculation based on the uncertainties in the measured values. It helps to determine the range of possible values for the final result and evaluate the reliability of the data and experimental procedures.

## 4. How do you perform propagation of error calculations?

To perform propagation of error, you need to first identify the sources of uncertainty in your measurements and determine the appropriate mathematical formula to calculate the overall uncertainty. This can involve adding, subtracting, multiplying, or dividing the uncertainties, depending on the specific operation being performed.

## 5. Can you provide an example of how to use sig-figs and propagation of error in a real-life scientific experiment?

Sure, let's say you are measuring the volume of a liquid using a graduated cylinder with a precision of 0.1 mL. You record the initial volume as 10.0 mL and the final volume as 27.3 mL. The change in volume would be 17.3 mL, but to account for the uncertainty in the measurements, you would report the result as 17.3 ± 0.2 mL, where the uncertainty is calculated by adding the uncertainties of the initial and final volumes. This result would then be used in subsequent calculations to determine the density or other properties of the liquid.

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