 In the lab, we write an uncertainty almost every time we make a measurement.  Our notation for measurements and their uncertainties takes the following form:  (measured value  uncertainty) proper units  where the  is read `plus or minus.'

 There are two reasons why a measurement might vary, one is because the instrument does not give the same reading every time and the other is because the quantity itself is varying.  Device  If we measure the mass of a steel ball on a top pan balance that measures to the nearest 0.1g then we may find that the reading varies between 52.3g and 52.4g.  This could be because the actual value is somewhere between the two and the electronics in the balance is causing the value to flicker between one and the other. This is an instrument uncertainty.  Method  If we measure the time taken for a ball to drop 0.4m we might get measurements ranging from s to s. This is not due to some random fluctuation in the clock but it's because the ball does not drop the same way every time. This is an uncertainty in the method.

 Reading uncertainty is related to the instrument itself  When determining an uncertainty from a measuring device, you need to first to determine the smallest quantity that can be resolved on the device.  For digital devices such as a digital scale, or a digital stopwatch, or a digital timer, the reading uncertainty is the least reading you can get on this device. Some examples....

 The minimum reading that the adjacent digital scale can read is:  0.1g  Therefore, the measurement is expressed as:  14.8 g  0.1 g ( the measurement is between 14.7 and 14.9)  What about this one?

 For the adjacent timer, what is the reading uncertainty?  How do you express the measurement shown?

 How can you express the reading on the adjacent digital ammeter?

 A stop-watch that measures to 1/100 of a second is used to find the time for a pendulum to oscillate 15 times. Suppose that this time is about 13 s.  The smallest division on the watch is 0.01s, so the time is recorded as ∆t = s ± 0·01s  WHAT IS WRONG HERE?  the reaction-time of the person using the stopwatch was neglected!  If the reaction time is about 0·15s, both at the starting and stopping time of the watch. So, a total of 0.3 s  0.3 s is >> 0.01 s  So, the time is more likely to be recorded as ∆t = 13.0 s ± 0·3s  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)

 For Analog devices, the reading uncertainty is half the least division interval on the instrument.  For the adjacent ammeter, the least division is 1 A.  The reading uncertainty for this ammeter is 0.5 A  The measurement is expressed as:  2.8 A  0.5 A  If the measurement was 2 A, how would you express it?  2.0 A  0.5 A

 Determine the reading uncertainty for this ammeter.  Express the shown measurement along with the uncertainty.  3.6 A  0.1 A

 Write the adjacent measurement along with the uncertainty.  Pay attention to the meniscus

 The ruler is an analog device with the minimum reading interval being 1 mm  The reading uncertainty should be 0.5 mm  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:  we are talking about two uncertainties in the measurement: one at the left side of the object and one at the right side  So, it is more likely that the uncertainty is 2  0.5 mm = 1mm or 0.1 cm (0.5 mm for each side)  This becomes more obvious if we consider the measurement again, as shown in the adjacent figure.

 Repeating measurement to find uncertainty  One reason for repeating the measurement of a quantity is so we can take an average to reduce the random error.  To reduce the random error, we need to do lot of repeats ( 3 minimum).  A simple (and acceptable) way of finding the uncertainty is to take the half difference between the maximum and minimum values (½ the range) and use this as the uncertainty in the average.

t 1 (s)  0.01 st 2 (s)  0.01 st 3 (s)  0.01 st 4 (s)  0.01 st 5 (s)  0.01 s  The following values were obtained when a student measured the time taken for a ball to fall a short distance:  Calculate the average time and the related uncertainty.  Average time = s  Uncertainty ∆t= range/2 = ( )/2 = 0.02  Measurement and uncertainty should have the same precision  Average time = 2.56 s  0.02 s