# B Uncertainty calculations?

1. Jan 24, 2017

I know the rules to experimental uncertainties with addition and subtraction, but what about division?

For instance here
The light bulb was measured to have 1.27 ±.05V by the DMM in parallel. Using the DMM in series, its current drawn was .202 ±1A. As a result its resistance was approximately 6.28±.05V/A.

I'm not totally sure if I've done it right...

Is there any large reference book for all of these rules for future insight?

2. Jan 24, 2017

### jambaugh

Division will have an asymmetric effect on uncertainty. You can understand the uncertainty calculations in one of two ways, either as a statistical expression (this is the mean plus or minus some multiple of the standard deviation) or as an interval calculation (the value is in the interval centered at this number plus or minus the interval radius).

In the latter case dividing by an uncertain quantity can yield infinite uncertainty: If b = 0.2 +/- 0.3, and a= 3 then a/b is somewhere below -30 or above +6 since both the positive and negative sides of zero are possible in the denominator you can get numbers ranging to + or - infinity.

But as long as 0 is not in the interval you can apply the arithmetic in a relatively straight forward way. To divide by an interval figure out the minimum and maximum quotient values by respectively dividing by the max and min (note the reversal) values for the denominator. Then re-express in terms of a center plus or minus a radius. There may be discipline specific conventions but that is the general method so far as I know. The subject name is interval arithmetic searching that should give you all sorts of references.

3. Jan 24, 2017

### Staff: Mentor

As long as the relative uncertainty is not too large, you can always estimate the uncertainty contribution of x to f(x,y,z,...) as
$$\Delta_x f(x,y,z,...) = \Delta x |\frac{df(x,y,z,...)}{dx}(x)|$$
where $\Delta x$ is the uncertainty on x, df/dx is the derivative with respect to x (evaluated at your central value for x), and $\Delta_x f$ is the uncertainty on f coming from the uncertainty on x. Multiple independent uncertain input parameters can be evaluated individually and added in quadrature.

This fails if the uncertainty is too large, or if you have correlated uncertainties in the inputs, then you'll need more complex methods.

4. Jan 25, 2017