# Percent difference: What to place on the denominator?

I'm genuinely confused on what to put on the denominator for the formula $$\%~\mathrm{difference} = \mathrm{\left|theoretical~-~experimental\right|\over theoretical}$$ or some other similar formula with different variable names that mean the same thing.

Specifically, I have a reading on the iron content per pill written on the bottle, and I have an amount that I calculated from doing some math with the experimental data. The problem is, I know that the calculated value is experimental, but the iron content per pill isn't really theoretical because it's known. Anyway, which should I put in the denominator?

• Also, welcome to Chem.SE! :-) – hBy2Py Jun 16 '16 at 0:58
• If it's %difference, shouldn't you multiply the RHS of your equation by 100? – Carlos Gouveia Jun 16 '16 at 1:54

The experimental value is the one you measured/calculated yourself, from your own experiments. The theoretical value is the one that you obtained from some third party, to which you are comparing your experimental value.

So, the number on the bottle is the theoretical, and the number from your data and calculations is the experimental.

It seems that you are really asking about the errors in the measurement otherwise its trivial and you plug in the numbers and multiply by 100 to get percentage. You should then place the decimal so that your result is to the same number of decimal places as the smallest number of places you have in either number used in the calculation.
If you are asking about errors you can use the method below.
You have two readings from different sources with different error values. Before you can decide what the overall error is you must have some estimate of the respective errors, even if one is very small or zero. Next you should use an error propagating formula to estimate the error in % difference. Let your equation be y = f( x, z ) where f is the function in x and z then the variance (std deviation squared) is

thus in your example x could be theoretical and z experimental values thus you have
y=(x-z)/x = 1-z/x. The sigma's are the respective measured standard deviations. Once you have the total error you can decide how to write %difference +- error. (The absolute value does not matter here as the formula squares values)

• While this type of uncertainty propagation calculation is important to know how to carry out, I don't think it answers the OP's question. – hBy2Py Jun 17 '16 at 23:24
• I added a comment that should answer this at the end of the first paragraph. If the OP is doing the calculation and is worried about precision then its not much more effort to calculate the errors properly. – porphyrin Jun 20 '16 at 6:50