# Error Propagation

I hope I am right in this section.

I am unsure with error propagation. When calculation the error in a titration, many errors has to be taken into account:

Error in Glassware/ Error in Balance/ Error in Burette etc.

I learned that the absolute and relative error have only $1$ significant figure and that the total amount is rounded to the decimal place of the error.

Therefore $5.34532g ± 0.001428g$ would be $5.345g ± 0.001g$

The relative error is $0.001g/5.345g = 0.00018709 = 0.0002$.

If there is an experiment with a lot of steps and error propagation wouldn't the rounding of all the errors in every single step change the result a lot? Wouldn't rounding the error just in the end make more sense?

I know of several conventions in active use. The most precise that I have seen will quote two digits of uncertainty and the two relevant uncertain digits. For example, http://physics.nist.gov/cgi-bin/cuu/Value?bg|search_for=universal_in! quotes $G$ as $(6.67408 \pm 0.00031) \cdot 10^{-11} \operatorname{m}^3 \operatorname{kg}^{-1} \operatorname{s}^{-2}$. I have mostly seen this used for things like universal constants, which get used over and over again, so it is important that we use as much precision as we have.
Getting slightly less precise, one can include one digit of uncertainty, the digit where the uncertainty kicks in, and one more digit of the original quantity. In the $G$ example this would be $(6.67408 \pm 0.0003) \cdot 10^{-11}$. Often the digit which is completely uncertain will get subscripted in this convention, as in $(6.6740_8 \pm 0.0003) \cdot 10^{-11}$. We used this in my quantitative analysis class in undergrad (which worked out of Quantitative Chemical Analysis by Harris).
The next one in precision, which is the most common in my experience, includes one digit of uncertainty and the digit where the uncertainty kicks in. So in the $G$ example this would be $(6.6741 \pm 0.0003) \cdot 10^{-11}$.