The first thing to note is that the differences in the number of atoms in different signatures, when using a top balance that has a resolution of two decimal places, is negligible. We can assume that all signatures have approximately the same number of atoms. There is a way of making the average more reliable but I'll come to that later.

The task is to find the number of atoms in a signature. We are to assume that the pencil we're signing with writes in pure graphite. Now, our scale only reads to two decimal places. That's obviously not accurate enough to measure the mass of a pencil mark of a single signature. The obvious solution is to do more signatures; say a hundred of them. Then we divide the measurement of the mass by a hundred to get an estimate of the mass of a single signature.

I did mention that there are negligible differences in the signatures of different people. Some might have signatures of slightly smaller or larger masses. The solution to this is to get a hundred different people to sign their name on the paper. This way, you can get the average mass of the signatures of a hundred people. Or you could get fifty people to sign their name twice, which would be easier to do, but a less accurate and varied average. But, the differences would be negligible, and not detectable by the top balance that we'd be using, so it doesn't really matter. I can just sign my own name a hundred times.

First of all you should calibrate the top balance, and then take an accurate measurement of the mass. All masses should be recorded in the same place with the same conditions. Then, after the signatures have been done on the paper, the mass should recorded again. The mass of the graphite on the paper will be the second mass measurement minus the first one. This value would then be divided by a hundred to get the mass of a single signature.

Let's say that the mass of a hundred signatures came out to be 0.02g. The mass of a single signature is then 0.02g / 100 = 2 x 10-4g. There are 6.02214086 x 10^23 atoms in 12g of carbon. To get the number of atoms in 2 x 10-4g of carbon, you'd divide that mass by 12g, and then multiply it by 6.02214086 x 10^23. This gives us an answer of 1.003690143 x 10^19. That's the number of atoms in my signature. That's it.

However, there's another problem we haven't addressed. Percentage inaccuracies. The top balance that we are using only measures to two decimal places. This gives us an error margin of +/- 0.005g. We took two measurements so we do 2 x (0.005g / 0.02g) x 100 = 50% error. That's quite a large error margin, but it's to be expected for a measurement so precise.

This percentage inaccuracy gives us a +/- range of the actual answer. We take our answer and multiply it by 0.5 and 1.5 to get the lower and upper ranges, respectively. This gives us:

5.018450715 x 10^18 to 1.505535215 x 10^19

Again, this range used example data to achieve a numerical value. The actual answer would have to be determined by actually carrying out the above method.