Average Values and Their Uncertainties
Quick Summary
When presenting the averages of a set of trials:
- The uncertainty is half the largest range of the set of trials rounded to 1 significant figure.
- The average value is rounded to the decimal place where the uncertainty starts.
Procedural or experimental uncertainty refers to the variation in results when repeating the same measurement several times. It is due to the inability to completely control all factors affecting an experiment. This is why we take three trials and find the average whenever possible. The simplest way to get a rough estimate of the procedural uncertainty is to take half of the range of one set of trials. We usually use the largest, typical range of all the sets of three trials.
Determining the uncertainty in the averages of a set of data: Looking at the example below, we see that the set of trials for 210 Hz has the largest range of all the sets of trials. The range in these three trials is (38.4 - 38.1 cm) = 0.3 cm. The uncertainty is 0.3*(1/2) = 0.15 cm ~ 0.2 cm since we round all estimated uncertainty to 1 significant figure.
Rounding the Averages: Again, using the data below as an example, we see that the first set of trials (17.0, 16.9, 16.9) has an average of 16.933333333. Since we found above that the uncertainty in the data is 0.2, we know that we should round our average to 16.9, the place where the uncertainty starts. Rounding to 16.93 means that you are claiming that you know the value of the hundreths place, but if the uncertainty starts in the tenths place, you obviously cannot be sure of anything in the hundredths place.
Frequency and Distance from the End of the Tube to the Water Level (First Harmonic)
Frequency
(±1 Hz)Distance from End of Tube to Water Level (±0.2 cm)
Trial 1
Trial 2
Trial 3
450
17.0
16.9
16.9
390
19.6
19.9
19.7
330
23.5
23.7
23.5
270
29.3
29.4
29.4
210
38.4
38.2
38.1
150
54.5
54.4
54.4
Table 1: The frequency emitted by the frequency generator and the point of resonance from the water level to the end of the tube. The uncertainties of the frequency and the distance are the instrumental uncertainty of the frequency generator, and the ruler respectively.
The Frequency and the Mean Distance from the End of the Tube to the Water Level (First Harmonic)
Average Distance From End of Tube to Water Level (±0.2 cm)
450
16.9
390
19.7
330
23.6
270
29.4
210
38.2
150
54.4
Table 2: The frequency and the average distance from the end of the tube to the water surface. The uncertainty of the average distance was taken from half the largest range of all trials.