The difference in using a predetermined error or standard deviation for calculations

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This is a simple question regarding the study of Linear Motion by measuring the displacement and time.

When calculating displacement by observing a body committing linear motion (its velocity is constant) using proper lab equipment, I do not understand which of the following two errors for displacement I should use while I get my data from the experiment.

Firstly, the error can be orders of magnitude equal to the orders of magnitude of most accurate measurement unit. For example if we were to use a measuring tape that measures up to millimeters, the error will be (this is a predetermined error): $$\sigma_{\text{tape}}=0.001\;m$$

However, the error in displacement can also be calculated from the following (for 10 repeats with different displacements): $$\sigma_x=\sqrt{\sum_{i=1}^{10}\frac{(x_i-\bar{x})^2}{(10-1)}}$$

So, which is the difference between these two? Are they used in different occasions?

It is a little hard to see the purpose of the second error in this example. You probably got the same answer 10 times in a row.

It is easier with a more complicated measurement. Surveryors used to measure distance with a very precisely calibrated chain. You count some links and stretch the chain from two movable points to make a certain distance. But a chain sags, so you have to take that into account. You have to pull the points apart with just the right standard force. Then the sag will be just what you expect, and the separation between the ends will be right.

You can do your best to use the same force every time, but you likely will not get it just right. You probably will pull a little to hard as often as you don't pull quite hard enough. In this case, averaging helps cancel the too hards and not hard enoughs so you get a better answer in the middle.

The standard deviation is a measure of how repeatable the measurement was. One surveyor may pull way too hard or way to easy, even though he gets it about right on average. Another may be much more uniform.

These random variations in measured values hide the true value. The guy who is more uniform is going to have a better idea of the true value, because he has less randomness. Even though both get about the same average measurement, you expect the uniform guy's value to be closer to the true value. Each guy's standard deviation is a measure of how close you to the truth you expect his answer to be.

A fixed error comes from counting links. If you are trying to figure out how far apart two points on the ground are, you have to choose between a fraction of a link too short and a fraction of a link too long.

Suppose you have a chain with links 1 inch long, and you are trying to measure a distance on the ground to the nearest foot. You might pull not hard enough one time. A few extra links fit between the marks on the ground, and you get a distance a few inches too long. The next time you might pull too hard and get a distance a few inches too short. Your measured values are all an integer number of inches. Sometimes you notice that your measurement was a fraction of a link too long and sometimes a fraction too short. But these errors tend to cancel when you average. You don't need to do anything to account for the size of the links.

Suppose you are trying to measure a distance to the nearest millimeter. If your forces are all very uniform, you cannot do it. You get the same answer every time, and that answer is perhaps a fraction of a link too long every time.

If your forces are less uniform, you get answers that are sometimes a fraction of a link too big and sometimes a fraction too small. Averaging tends to cancel these errors. Many measurements produce a better cancellation. You can see this in the standard deviation. Many measurements produce a smaller standard deviation. If you do enough measurements, the standard deviation becomes smaller than 1 mm.

However, it isn't likely that you can really make this last case work. You have to control your forces very carefully to be sure you aren't pull just a little too hard each time. If you can do this, you get the the same answer every time. If you can't you get variation in forces, but you can't be sure that the forces are right.

Usually you can't measure more precisely than the precision of the instrument you use to do the measurement.

• So, there's no clear answer to that? If we let someone do the experiment, compare their standard deviation with the error 0.001m, their standard deviation could either be bigger or smaller than that. If we do this many times, we will get a much better and accurate answer. So, the standard deviation is unrelated to the 0.001 m error completely. The standard deviation measures the accuracy of the measurement, and the 0.001 m error is just an error because of the equipment. Right? – Gradient Entropy Feb 28 at 17:16
• Pretty much. There is a quibble. Precision is a better word than accuracy here. A result is precise if the standard deviation is small. It is accurate if it matches the true value well. – mmesser314 Mar 1 at 6:00
• Marking your answer as acceptable, but one more question. In what instances is the standard deviation used as an error in calculations where the error of the equipment itself isn't used? – Gradient Entropy Mar 2 at 16:17