On many occasions the quantities we measure (length, angles, currents, etc.) do not fit exactly onto the graduations of our scales. The ruler shown here is being used to measure the width, w of the shaded region. Notice that the width is a bit more than three units but not exactly 3.2 units (which is the next scale graduation in this case).

In such cases we have to estimate between the graduations. There are two ways to approach this situation. The first is to estimate the value with a single reading, and the second is to independently measure the width several times and make an average.

Estimating and Accuracy of Scale Divisions

When estimatimg, it is important that our estimation be as accurate as possible. However, no matter how well we can make the estimate, there is always some uncertainty introduced in our reading. There are two independant sources of this uncertainty. The first arises from the estimating process itself, while the second is due to the precision of the measuring device.

Estimating:

In the present case the simplest estimate is to assign a value half way along the interval. We could then report the reading as 3.1 +/- 0.1 units. This is based on the observation that the measurement is larger than 3.0 units, smaller than 3.2 units, together with the fact that adding or subtracting o.1 units would bring us to at least one of the nearby scale divisions.

Still we might do better than this. In the expanded view we can see that the edge of the figure only reaches about a third of the way beyond 3.0 units. So one third of the interval is 0.0666... units (which we round to 0.07 here), giving an estimate for w as w = 3.07 +/- .07 units. In this case adding or subtracting 0.07 units would bring us to a known scale division. You might argue that adding 0.03 units will bring us up to 3.1 units which we are confident is too large, so that an estimate of our uncertainty could be as small as 0.03 units. While this is true this small uncertainty would have to be viewed as the most optimistic report of our estimating process. A reasonable compromise could be found by reporting w to be w = 3.07 +/- 0.05 units.

Precision of the device:

Any measuring device is only as accurate as the precision of the maunfacturing process allows. In this picture two rulers from the same maunfacturer are shown side by side. While the difference may seem exagerated some differences will always be part of any manufacturing process. Most manufacturers of quality measuring instruments adhere to a convention about precision (accuracy) of their devices. The level or degree of precision in the manufacturing and quality control process is indicated by the size of the scale divisions shown on the device. The uncertainty is assumed to be one half (1/2) of the smallest scale division shown. For our ruler this means that when you compare any two rulers made in this manufacturing process, the largest difference in overall length should never be greater than 0.1 unit. If this is the case then no measurement with this ruler should be assumed to be more accurate than 0.1 unit. Now you understand why the ruler does not show more scale divisions. If this device was more accurate than 0.1 unit, the manufacturer would have included scale divisions in between the graduations shown.

This fact means that the result stated above is pretty optimistic about the precision of the measurement made. Should we return to the original statement for w? Perhaps not. The largest uncertainty arrises when the measurement uses the entire length of the ruler. Also random chance would suggest that we probably do not have the worst ruler in the manufacturing lot. So our estimated measurement is probably better than w = 3.1 +/- 0.1 units, but not as good as w = 3.07 +/- 0.05 units.

When the device has a digital readout, such as a digital voltmeter, responsible manufacturers indicate the precision by the number of digits displayed. The general rule is plus or minus one of the least significant figure displayed. The most reputable manufacturers do better than this keeping the uncertainty to one half of the laest significant figure. But unless you have access to the users manual or prior experience with the product this is not a safe assumption.

General rule of thumb: It is not uncommon when making a single measurement to avoid all this painstaking process and simple assign one half the smallest scale division as the uncertainty. For instance, the smallest scale division of common pan balances is 0.1 g. In this case we would assume an uncertainty of +/- 0.05 g.

Making Multiple Measurements

There is another approach to this problem. While it takes longer is highly recommended when we care about the accuracy of the result. For this method, simply measure the width many times and average the results. When several measurements of a quantity are taken the uncertainty can be estimated by computing an average deviation. The mathematics for this is developed in the next two sections; finding the mean and the average deviation.

When making repeated measurements you should try to make each trial independent of the previous one as much as possible. Have different people make the measurement, use several different devices (new rulers for example). Make each measurement as accurately as possible and start each measurement from scratch (pick up the ruler between measurements for example).