Question about “uncertainty”

Question

I'm a bit confused about all the different "definitions" of uncertainty?
One can calculate "uncertainty" of a single measurement by simply checking how accurate the tool that is being used to do the measuring is.
One can also calculate "uncertainty" of a group of measurements by finding the standard deviation which means that any more measurements made will have a 70% chance of being within one standard deviation of the "true value".
Finally, there is also standard error which is also a measure of "uncertainty", but I'm not sure what it says.

Which "uncertainty" do I have to find in the following problem and would I go about finding it?

A particular 1.00-kg bag of rice consists of 2.00% "supergrains", each of which contains 5ppm of arsenic and the rest of the grains contain no arsenic. What is the average concentration?

A grain of rice weighs 20 mg. What is the uncertainty in the concentration due to sampling if a 10.00 g sample is taken for analysis from this bag?

If the entire contents of the bag were ground to spherical particles 100 micrometer in diameter, what would the uncertainty be if a 10.0 g sample were taken for analysis?

The first part is pretty straight forward. I calculated that the average concentration of arsenic in this bag of rice is 100ppb.

It's the next two questions I don't understand partly because as I said in the beginning, I'm not sure what "definition" of uncertainty they are asking for/about. How would I go about finding it?

Please correct any assumptions or definitions I've made if they are incorrect in any way. I'm really trying to learn and I know how important precision of language is.

Answer 1

The problem is that the term "uncertainty" is colloquially used to refer to lots of different things where it probably shouldn't be. For a guide to definitions of various terms in metrology see the VIM. The GUM provides a somewhat practical guide to actually computing the uncertainty.

In particular, if you compare a single measurement to some nominal "true" value, as in your first point. This is not the uncertainty. This would be the measure accuracy or error.

The uncertainty is a description of the spread of a set of measurements. It also requires that a distribution and confidence level of the measurements is given. Commonly a normal distribution at 95% (~2$\sigma$) is used.

The standard uncertainty, also sometimes called the standard error is just the standard deivation of the set of measurements, i.e. the uncertainty at one $\sigma$.

I your situation you are dealing with a binary situation (Supergrain or not), which is not what I'm very familiar with. In such a situation your distribution is bi-nominal, although for larger sample sizes you can use the central limit theorem to approximate to a normal distribution.

In this case wikipedia says the standard uncertainty is given by $\sqrt{\frac{1}{n} p(1-p)}$ where n is the number of grains and p the probability (2%). Scale this up to get whatever your desired confidence level is. For the third question I feel you are missing some information to work out how 100 $\mu$m balls effect the number of particles in your sample. Do you have some information on the density?