Chemistry - Is it better to use a smaller, more accurate measuring cylinder several times or a larger, less accurate one for the same volume?
Solution 1:
You have three types of errors:
The errors in the accuracy of your measurement. Assuming you are reading the water levels consistently (at eye level at the bottom of the meniscus), these should be random.
To add random, uncorrelated errors, the standard method is to sum their squares and take the square root. So, for the 0.1 mL average error for the 10 mL cylinder:
$$\text{average error for seven combined measurements} = \sqrt{7 \times (0.1 \text{ mL})^2} = 0.26 \text{ mL}$$
As you correctly intuited, it wouldn't make sense to simply multiply the 0.1 mL by 7, since this would be saying you have either a +0.1 mL error every time, or a –0.1 mL error every time, but that's not how random errors work—they are just as likely to be positive as negative. In addition, they don't all have a magnitude of 0.1 mL; rather, their magnitudes follow a distribution. Thus essentially what you have is a random walk, with a random distribution of step sizes, where the average step size is 0.1. The square-root-of-the-sum-of-the-squares method gives you the average distance you'd end up from your starting point with 7 of these steps
Error in the calibration of the cylinder. This would be a systematic error, and it would add to the error of each measurement. Higher qualities of cylinders (e.g., Class A glassware) have lower errors.
Error in the consistency with which the cylinder delivers the measured amount. These cylinders are probably calibrated as "TD", which means "to deliver". That is, it is assumed not all the water will drain out, so the markings on the cylinder are calibrated to account for that. However, there will be some random error in how closely the amount that drains out corresponds to the assumed amount that drains out. The only way to determine this error would be to contact the manufacturer (or to test it yourself using a scale). In addition, if your cylinder is not clean, more water may adhere to the walls, which could cause a systematic error in the amount of water delivered.
Note also the comment by Andrew Morton that there's an additional important practical consideration, which is that if you have to use seven measurements, you could miscount and thus end up being off by $\pm$10 mL! And this error is much easier to make when measuring with a graduated cylinder than, say, a tablespoon, because each measurement with the cylinder takes focus to get the total volume to 10.0 mL, and that focus can cause you to lose count. There are of course ways to address this, e.g., making a mark on a piece of paper for each 10.0 mL added.
Solution 2:
Keep in mind that a graduated cylinder is never meant for accurate volume transfer. You would probably use a Class-A 100 mL buret to transfer 70 mL to a vessel if accuracy were that critical say, 70.0 mL was desired. All these general chemistry rules of thumbs are not useful for real analytical work. Experimental numbers matter the most.
Let us assume you don't have a 100 mL burette. This is how a practical scientist would do it. You would like know whether 7 transfers from a 10 mL cylinder are better or 1 transfer from a 100 mL cylinder is better? You calibrate your cylinders. You know the density of the liquid. You would transfer 10 mL, 7 times and weigh it on an analytical balance. Note the mass and from the mass-density-volume relationship, determine the volume.
Then repeat the same with a 100 mL cylinder, and transfer 70 mL in one go. Measure the mass and hence volume. Whatever delivers the right volume, is the way to go.
Practically, 7 transfers will generate more errors because you do not know the drainage time of the cylinder, i.e., how long should you wait till all the film of liquid is out of the cylinder as the last drop of liquid. Also see the rules for the propagation of error.
By error propagation rules, uncertainties cannot be added like you have shown. It is an incorrect approach. The cylinder with 0.1 mL error will not add up to 0.7 mL by any means. if you make 7 additions.
Solution 3:
Meanwhile, the mark scheme says that one must use the 100 ml cylinder, for the 10 ml cylinder would give a total 7*0.1/10=7% percentage uncertainty, compared to the 1.4% from the 100 ml cylinder. (1.4% < 7%)
Let's simplify the problem and say the 10 mL cylinder has a +1% systematic error (and there is no random error), i.e. delivers 10.1 mL when filled to the 10 mL line. So filling it up to 10.1 mL seven times will deliver 70.7 mL instead of 70.0 mL. That is a 1% systematic error.
So the relative systematic error does not change comparing delivering a single volume to delivering the sum of multiple measurements. The absolute error, of course, increases (i.e. the difference between the intended and the actual volume).
In the calculation of the marking scheme, the denominator is 10 (mL), but it should be 70 (mL). If you look at the other answers, which are excellent, that is not the only thing the marking scheme got wrong. The bigger problem is that the question does not make sense in face of the reality of measuring volumes in the lab.