The number we write as the uncertainty tells the
reader about the instrument used to make the measurement. 

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
seems to be 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
yet been taken into account? 

In situations like this one has a tendency to concentrate on the right
hand end of the ruler. 

A measurement of length is, in fact, a measurement
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
here. 





We now notice that the lefthand 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)cm 

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 



so, in general, we state the result of a measurement as 

reading ± the smallest division on the measuring instrument 



Example 2: Using a StopWatch 

Consider using a stopwatch 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. 

This means that the smallest division on the watch is only about 1% of the
time being measured. 

We could therefore 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 reactiontime of
the person using the watch might be about 0.1s. 

Let us be pessimistic and say that the person's reactiontime 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 stopwatch! 



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 stopwatch is useful for measuring
times of about 30s or more (for precise measurements
of shorter times, an electronically operated watch
must be used) 

