Measurement and Uncertainty

Measurement and Uncertainty
When recording data, each entry should be given a corresponding estimated error, or uncertainty. The uncertainty gives the reader an idea of the precision and accuracy of your measurements. Use the following method for finding the uncertainty associated with any measuring device used in lab.
First, find the least count, or the smallest printed increment, of the measuring device. On the meter sticks, the least count is 1 mm. On the double pan balances, the least count is 0.1 g. On the small graduated cylinders, the least count is 25 ml. If you are using the full precision of the instrument, you are probably safe in saying that your measurement is within one least count of the measured value, in either direction.

Figure 1
For example, say you are measuring the object in Figure 1. If you use the meter stick to measure an object's length as being around 86 mm, that means you are pretty sure that the actual value is between 85 mm and 87 mm. Therefore, you should represent your data like this: l = 86 mm ± 1 mm
In this case, your uncertainty is ± 1 mm. However, you may feel that you are able to attain more precision than is indicated by the least count. In that case, you should do some estimating. By estimating, you divide the least count of your measuring device into imaginary increments. In this lab, it is recommended that you divide the least count into five imaginary increments. This is called the one-fifth rule. There will be some occasions when the one-fifth rule seems too generous. If you feel that your confidence in the last significant figure of the measurement is greater than this, then of course it would be more appropriate to use, say one-tenth of the least count. Similarly, if your confidence in the last significant figure is lower, then you might use half the least count. At any rate, you should use good judgment in estimating the error. Always think in terms of having to justify your estimates to your instructor! The following data points were estimated to the nearest half of the least count and the nearest fifth of the least count, respectively. l = 85.5 mm ± .5 mm l = 85.6 mm ± .2 mm
All three of these length measurements are correct, but they represent varying degrees of precision. In each case, the uncertainty was decreased, indicating greater accuracy of measurement. You should always try to be as precise as possible, and by estimating, you were able to attain a third significant digit (see appendix B for more about significant digits).
There are two rules to follow to help make sure you have determined your uncertainty correctly. Note that these rules were followed in each case above. * Rule 1: The uncertainty should be to the same precision as the measured value. * Rule 2: The measured value should be evenly divisible by the uncertainty.
In the case of a digital device, the uncertainty is equal to the smallest digital increment, and no estimating may be performed. For example a stopwatch reading may be represented as t = 12.63 s ± .01 s.
Vernier Caliper
Another example of an instrument that requires no estimating is the Vernier caliper, which is used when greater precision is needed that what the meter stick can provide. The Vernier caliper is a device that employs an auxiliary (Vernier) scale which permits more accurate measurements of small fractions of a main scale division. The Vernier scale consists of equally spaced divisions which have the same total length as (n -1) main scale divisions. As an example, consider the caliper shown in Figure 2:

Figure 2
The main (stationary) scale is divided into millimeters. The Vernier (moveable) scale is 9 mm long and consists of 10 divisions. Each division on the Vernier scale, then, is 0.9 mm long or 0.1 mm shorter than a main scale division. Note that the first division on the Vernier scale is 0.1 mm from the first division on the main scale; the second Vernier division is 0.2 mm from the second main division scale division, etc. If the Vernier is adjusted so that the first divisions of the two scales are aligned, the distance between the jaws of the caliper is 0.1 mm. If the jaws are opened further, until the second divisions on the scales are aligned, the distance between the jaws will be 0.2 mm.
In general, the number of the Vernier line which is aligned with a line on the main scale is equal to the number of tenths of a millimeter between the caliper jaws. If the separation between the jaws is more than one millimeter, the reading of the Vernier is added to the reading of the main scale up to the zero point of the Vernier scale. The uncertainty of the Vernier is the same as the least count, which is 0.1mm. Figure 2, therefore, shows a reading of 7.5 mm ± .1 mm, or 0.0075 m ± .0001 m.
Micrometer Caliper
If one requires greater precision than can be obtained with a Vernier caliper, a micrometer caliper is often used. This instrument consists of an integral number of equally spaced divisions. Consider as an example the micrometer caliper shown in Figure 3.

Figure 3
The screw of this instrument has 2 threads per millimeter so that it advances .5 mm for each revolution. Since there are 50 divisions around the rim, the least count and the uncertainty of this instrument is 1/50 x 0.5 mm = 0.01 mm. In reading the rim of the instrument, one must determine whether the barrel is on its first or second revolution after a main scale division (one can determine this by simple inspection).
Metric Dial
Another useful measuring device is the metric dial (see Figure 3). The least count of the big needle is 0.01 mm. There are 100 increments in a full circle of the needle. The small needle counts the number of times the large needle has made a complete circle, so its least count is 1 mm. There is a set screw on the side of the dial that will allow one to zero the markers to the needle, then retighten.

Figure 4