We have seen how results that are calculated from several measured quantities depend on the uncertainties of each of the measured quantities. There may be several factors that affect the uncertainty of a calculated result. When treating a calculated result, it is important to consider all of the factors that contribute to the uncertainty. Similarly, even a single measurand might have several factors that contribute to its uncertainty, because measurements are often fairly complicated comparisons of the measurand to a standard. All of the factors that contribute to the uncertainty in a single measured value make the uncertainty budget.

Example: Suppose you are using a stopwatch to measure the period of a 1 m long pendulum. The following factors might affect the uncertainty of the period. 1. Reaction time - You watch the pendulum swinging and push a button on the stopwatch when it appears to be in the right position. The measurement will depend on how well synchronized your button pushing is with the motion of the pendulum. This will depend on your skills as a stopwatch operator. The uncertainty due to reaction time might be about +0.1 s. 2. Calibration - Even with a perfect stopwatch operator, the measurement from the stopwatch will still be uncertain because the stopwatch itself has some uncertainty. The stopwatch has been calibrated at some point. The uncertainty in the calibration might be about 10.2%, which in this case corresponds to +0.04 s. 3. Apparatus - You have seen in Experiment #37 that the period of a pendulum is dependent on the pendulum's length. In building the pendulum, the length had to be measured. Of course the length measurement isn't exact either. If the uncertainty in the length is +0.1 cm, this will cause the uncertainty in the time to be 10.01 s. 4. Initial conditions - The period of a pendulum also depends slightly on the amplitude (but not if the amplitude is very small). The amplitude won't be exactly the same in any two trials. This might cause an uncertainty of 10.05 s in the period. 5. External factors - The surroundings of the pendulum might have some effect on it, such as if there is a breeze in the room or if the air pressure is unusually high. This might cause an uncertainty of 10.001 s in the period These effects are summarized in the table on the next page. Each of the factors in the uncertainty budget contributes to the total uncertainty of the measurement. The total uncertainty is total =101+07+03 The sum here adds up all of the uncertainties of the individual factors in the uncertainty budget. For the pendulum example, the total uncertainty is total =V|0.1s + 0.04 s )+ (0.05s] +(0.001s)=0.119sFactor Uncertainty Reaction time 0.1 s Stopwatch calibration 0.04 s Pendulum length 0.01 s Pendulum amplitude 0.05 s Wind/air effects 0.001 s Total 0.119 s Table 1: Uncertainty budget for pendulum period experiment