You can find the necessary formulae following the link below. The covariance should be zero unless you have any systematic errors in the adsorption experiment.

https://en.wikipedia.org/wiki/Propagation_of_uncertainty#Example_formulas

Dear Betty,

From your question I understand that you are transforming the raw data (adsorption) into concentration. I assume that the uncertainty is associated with both adsorption estimates (test(s) and initial) and that no uncertainty (or negligible) is associated with the concentration.

To ilustrate I will use an example:

Ainitial=80 (units are not critical for the final result)

sinitial=14

Atest=40

stest=10

Conc=120 (unknown units)

In this case the "critical operation" will be the ratio of Atest±stest/Ainitial±sinitial.

The first step will be to convert the standard deviations to RSD (relative std dev) also known as CV (coeficient of variation). This is done by RSDtest=stest/Atest and RSDinitial=sinitial/Ainitial, (divide the std deviation by the average adsorption value).

One obtains: RSD initial=14/80=0.175=17.5% and RSDtest=10/40=0.25=25%

The RSD of the ratio Atest/Ainitial will be the square root of the sum of the squared RSD's:

RSD(t/i)=SQRT(0.175^2+0.25^2)=0.30516=30.5%

The average ratio is Atest/Ainitial=40/80=0.50

Now we need to back transform the RSD into std deviation:

std dev=average * RSD = 0.50 * .30516 = 0.15258

The ratio Atest/Ainitial±stdev is equal to 0.50±0.15

Now we need to multiply both estimates by the concentration:

Atest/Ainitial*concentr = 0.50 * 120 = 60 (concentration units)

The final stdev of the Atest/Ainitial*concentr is = 0.15258 * 120 = 18.31

This should be the value to use as error bars.

I hope it was clear

Best

Luis

Betty -

I am not a chemist, but could you possibly being trying to predict y (here a value of some "final" condition) from a single regressor x (here a value of an "initial" condition), and be looking for prediction intervals?  ("Prediction" just means use of regression, not forecasting, though there is a time element in this case.) 

Here is an illustration of this for regression through the origin:

 https://www.researchgate.net/publication/263265199_CRE_Prediction_%27Bounds%27_and_Graphs_Example_for_Section_4_of_Properties_of_WLS_article

Or maybe this is a better example:

 https://www.researchgate.net/publication/262143393_A_Note_on_Estimating_Impact_of_Data_Collection_Mode_Change_for_a_Given_Key_Response_Item

Perhaps this is not what you are doing at all, but I thought I'd mention it, in case this might be what you are doing.

Cheers - Jim

Data CRE Prediction 'Bounds' and Graphs Example for Section 4 of ...

Technical Report A Note on Estimating Impact of Data Collection Mode Change f...