Propagation of error: exponents

[AndrewBworth]

Hi all. I have been trying to understand propagation of error of exponents. Most an. Chem textbooks I see say y = a^x, sy/y = (sa/a)*x. But say y = a*b, then sy/y = ((sa/a)^2 + (sb/b)^2)^.5 . if a = b then sy/y= (2*(sa/a)^2)^.5 = 2^.5*abs(sa/a). This shows the rule y=a^x, sy/y= x^.5*abs(sa/a).

 

[brainpushups]

I think I understand your question. You are asking why the standard way of representing uncertainty in a product (the so-called quadrature method of addition) does not apply to powers.

The most straightforward way to explain this that I've seen may satisfy you. Recall that the quadrature method only applies to uncertainties that are independent of each other. Take a simple example of measuring two sides of a rectangle - the uncertainty in your first measurement is independent of the uncertainty of the second measurement. It is therefore preferable to state the uncertainty in the area of the rectangle using the quadrature method.

Contrast this with measuring only one side of a square and calculating its area. In this circumstance the calculation involves multiplying the same value together and the uncertainties are clearly no longer independent; the quadrature method is not justifiable in this circumstance.

 

[AndrewBworth]

Thank you! I guess I have a thing or two to learn about statistics.

 

[brainpushups]

You're welcome. One book that I found extremely useful for the basics of dealing with experimental uncertainty is John R. Taylor's An Introduction to Error Analysis. I would definitely recommend checking it out. I ended up buying a copy.

 

[DrDu]

We also have a statistics subforum for this kind of questions.