# Uncertainty in the standard deviation

1. May 6, 2009

### Shukie

1. The problem statement, all variables and given/known data
A health physicist is testing a new detector and places it near a weak radioactive sample. In five separate 10-second intervals, the detector counts the following numbers of radioactive emissions:

16, 21, 13, 12, 15.

a) Find the mean and standard deviation (SD) of these five numbers.

b) Compare the standard deviation with its expected value, the square root of the average number.

c) Naturally, the two numbers in part b) do not agree exactly, and we would like to have some way to assess their disagreement. This problem is, in fact, one of error propagation. We have measured the number v. The expected standard deviation in this number is just $$\sqrt{v}$$, a simple function of v. Thus, the uncertainty in the standard deviation can be found by error propagation. Show, in this way, that the uncertainty in the SD is 0.5. Do the numbers in part b) agree within this uncertainty?

3. The attempt at a solution

a) Mean = 15.4
Standard deviation = 3.5

b) Expected SD = $$\sqrt{15.4} = 3.9$$

c) I know that the following is true: $${\sigma}(\sqrt{v})= {\sigma}_v*\frac{d(\sqrt{v})}{dv} = 0.5$$ where $${\sigma}_v = \sqrt{v}$$. How do I show by error propagation that the uncertainty in the SD is 0.5?