We have used the terms uncertainty and error; they are not quite the same thing. In general, an ‘error’ is just a problem that causes the reading to be different from the true value (although a zero error can have an actual value). The uncertainty, however, is an actual range of values around a measurement, within which you expect the true value to lie. The uncertainty is an actual number with a unit.
For example, if you happen to know that the true value of a length is $21.0 cm$ and an ‘error’ or problem causes the actual reading to be $21.5 cm$, then, since the true value is $0.5 cm$ away from the measurement, the uncertainty is $ \pm 0.5\,cm\,$. 
But how do you estimate the uncertainty in your reading without knowing the true value? Obviously, if a reading is $21.5 cm$ and you know the true value is $21.0 cm$, then the uncertainty in the reading is $0.5 cm$.
However, you may still have to estimate the uncertainty in your reading without knowing the true value.
So how is this done?

First, it should be understood that the uncertainty is only an estimate of the difference between the actual reading and the true value. We should not feel too worried if the difference between a single measurement and the true value is as much as twice the uncertainty. Because it is an estimate, the uncertainty is likely to be given to only one significant figure. For example, we write the uncertainty as $0.5 cm$ and not $0.50 cm$.
The uncertainty can be estimated in two ways.
Using the division on the scale – Look at the smallest division on the scale used for the reading. You then have to decide whether you can read the scale to better than this smallest division. For example, what is the uncertainty in the level of point B in Figure P1.2? The smallest division on the scale is $1 mm$ but is it possible to measure to better than $1 mm$? This will depend on the instrument being used and whether the scale itself is accurate. In Figure P1.2, the width of the line itself is quite small but there may be some parallax error that would lead you to think that $0.5 mm$ or $1 mm$ is a reasonable uncertainty. In general, the position of a mark on a ruler can generally be measured to an uncertainty of $ \pm 0.5\,mm$. In Figure P1.8, the smallest division on the scale is $20 g$. Can you read more accurately than this? In this case, it is doubtful that every marking on the scale is accurate and so $20 g$ would be reasonable as the uncertainty.

You need to think carefully about the smallest division you can read on any scale. As another example, look at a protractor. The smallest division is probably $1{\,^ \circ }$ but it is unlikely you can use a protractor to measure an angle to better than $ \pm 0.5{\,^ \circ }$ with your eye.
Repeating the readings – Repeat the reading several times. The uncertainty can then be taken as half of the range of the values obtained; in other words, the smallest reading is subtracted from the largest and the result is halved. This method deals with random errors made in the readings but does not account for systematic errors. This method should always be tried, wherever possible, because it may reveal random errors and gives an easy way to estimate the uncertainty. However, if the repeated readings are all the same, do not think that the uncertainty is zero. The uncertainty can never be less than the value you obtained by looking at the smallest scale division.
Which method should you actually use to estimate the uncertainty? If possible, readings should be repeated and the second method used. But if all the readings are the same, you have to try both methods!
The uncertainty in using a stopwatch is something of a special case as you may not be able to repeat the measurement. Usually, the smallest division on a stopwatch is $0.01 s$, so can you measure a time interval with this uncertainty? You may know that your own reaction time is larger than this and is likely to be at least $0.1 s$. The stopwatch is recording the time when you press the switch but this is not pressed at exactly the correct moment. If you do not repeat the reading then the uncertainty is likely to be at least $0.1 s$, as shown in Figure P1.7. If several people take the reading at the same time, you are likely to see that $0.01 s$ is far too small to be the uncertainty.
Even using a digital meter is not without difficulties. For example, if a digital ammeter reads $0.35 A$, then, without any more information, the uncertainty is $ \pm 0.01A$, the smallest digit on the meter. But if you look at the handbook for the ammeter, you may well find that the uncertainty is $ \pm 0.02$ or $ \pm 0.03A$ (although you cannot be expected to know this).

Questions

 

4) Figure P1.8 shows a lever-arm balance, initially with no mass in the pan and then with a standard $200g$ mass in the pan.
Explain what types of error might arise in using this equipment.

5) Estimate the uncertainty when a student measures the length of a room using a steel tape measure calibrated in millimetres.

6) Estimate the uncertainty when a girl measures the temperature of a bath of water using the thermometer in Figure P1.9.

7) A student is asked to measure the wavelength of waves on a ripple tank using a metre rule that is graduated in millimetres. Estimate the uncertainty in his measurement.

8) Estimate the uncertainty when a student attempts to measure the time for a single swing of a pendulum.

9) What is the average value and uncertainty in the following sets of readings? All are quoted to be consistent with the smallest scale division used.
a: $20.6, 20.8$
b: $20, 30, 36$
c: $0.6, 1.0, 0.8, 1.2$
d: $20.5, 20.5$.