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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 43cm (but it is certainly not exactly 43cm).

The result could therefore be stated as

43cm 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 21cm, but this measurement is subject to the same level of uncertainty.

Therefore the length of the object is

(63 005)cm - (20 005)cm

so, the length can be between

(63 + 005) - (20 - 005) and (63 - 005) - (20 + 005)

that is, between

44cm and 42cm

We now see that the range of possible results is 02cm, so we write

length = 43cm 01cm

In general, we state a result as

reading 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 001s

which is equivalent to saying that the time T is between

099s and 101s

This sounds quite good until you remember that the reaction-time of the person using the watch might be about 01s. Let us be pessimistic and say that the person's reaction-time is 015s. Now considering the measurement again, with a possible 015s at the starting and stopping time of the watch, we should now state the result as

T = 1s (001+ 03)s

In other words, T is between about 07s and 13s.

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

a) a ruler, marked in mm, is useful for making measurements of distances of about 10cm or greater.
b) a manually operated stop-watch is useful for measuring times of about 30s or more (for precise measurements of shorter times, an electronically operated watch must be used)


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