# Statistical error and error propagation

I have a quantity defined as: $P_{frac} = \frac{F_{max}-F_{min}}{F_{max}+F_{min}}$

I also have the value for $F_{max}$, $F_{min}$, and their statistical errors.

How can I calculate the error for $P_{frac}$?

• What, precisely, do you mean by "their statistical errors"? Are $F_{max}$ and $F_{min}$ dependent? – Glen_b -Reinstate Monica Mar 11 '14 at 8:54
• They are results from a measurement, and then they also are reported with their error. They are not dependent. – Py-ser Mar 11 '14 at 9:04
• They are not dependent. -- out of curiosity, how do you know this? – Glen_b -Reinstate Monica Mar 11 '14 at 9:49
• They are the maximum and the minimum value of a series of data. – Py-ser Mar 11 '14 at 11:46
• In which case they're very likely not independent (though if the samples are large, it may be a reasonable approximation). – Glen_b -Reinstate Monica Mar 11 '14 at 19:34

If you can work it out for $R=\frac{F_{max}}{F_{min}}$, and you can work it out for a reciprocal (multiplicative inverse), then rewrite:

$$P_{frac}=\frac{1}{1+\frac{2}{R-1}}$$

Which involves only shifts, inverses and a scalar multiplication.

• Thanks Glen, but... what about then? – Py-ser Mar 11 '14 at 11:46
• I don't think I should say more because at present: (i) my very first question in comments remains unanswered; and (ii) you've made some statements that don't seem consistent with each other (and which I don't presently have enough information to choose between). So I don't know for certain what I am supposed to be calculating nor the circumstances under which I am doing it. If you clarify your question sufficiently, I might be able to say more. – Glen_b -Reinstate Monica Mar 11 '14 at 19:36

so, $\delta_{frac}=\frac{\Delta_{max}+\Delta_{min}}{F_{max}-F_{min}}+\frac{\Delta_{max}+\Delta_{min}}{F_{max}+F_{min}}=(\Delta_{max}+\Delta_{min})\frac{2F_{max}}{F_{max}^2-F_{min}^2}$
absolute error $\Delta_{frac}=(\Delta_{max}+\Delta_{min})\frac{2F_{max}}{(F_{max}+F_{min})^2}$