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Quantifying the Uncertainty
The number we write as the uncertainty tells the reader about the instrument used to make the measurement. (As stated above, we assume that the instrument has been used correctly.) Consider the following examples.
Example 1: Using a ruler
The length of the object being measured is obviously somewhere near 4·3cm (but it is certainly not exactly 4·3cm).
The result could therefore be stated as
4·3cm ± half the smallest division on the ruler
In choosing an uncertainty equal to half the smallest division on the ruler, we are accepting a range of possible results equal to the size of the smallest division on the ruler.
However, do you notice something which has not been taken into account? A measurement of length is, in fact, a measure of two positions and then a subtraction. Was the end of the object exactly opposite the zero of the ruler? This becomes more obvious if we consider the measurement again, as shown below.
Notice that the left-hand end of the object is not exactly opposite the 2cm mark of the ruler. It is nearer to 2cm than to 2·1cm, but this measurement is subject to the same level of uncertainty.
Therefore the length of the object is
(6·3 ± 0·05)cm - (2·0 ± 0·05)cm
so, the length can be between
(6·3 + 0·05) - (2·0 - 0·05) and (6·3 - 0·05) - (2·0 + 0·05)
that is, between
4·4cm and 4·2cm
We now see that the range of possible results is 0·2cm, so we write
length = 4·3cm ± 0·1cm
In general, we state a result as
the smallest division on the measuring instrument
Example 2: Using a Stop-Watch
Consider using a stop-watch which measures to 1/100 of a second to find the time for a pendulum to oscillate once. Suppose that this time is about 1s. Then, the smallest division on the watch is only about 1% of the time being measured. We could write the result as
T = 1s ± 0·01s
which is equivalent to saying that the time T is between
0·99s and 1·01s
This sounds quite good until you remember that the reaction-time of the person using the watch might be about 0·1s. Let us be pessimistic and say that the person's reaction-time is 0·15s. Now considering the measurement again, with a possible 0·15s at the starting and stopping time of the watch, we should now state the result as
T = 1s ± (0·01+ 0·3)s
In other words, T is between about 0·7s and 1·3s.
We could probably have guessed the answer to this degree of precision even without a stop-watch!
Conclusions from the preceding discussion
If we accept that an uncertainty (sometimes called an indeterminacy) of about 1% of the measurement being made is reasonable, then