Worksheet 1: Uncertainty analysis¶

In this first lab we reinforce the concept of uncertainties based on simple measurements of length and volume of objects and their masses with different tools. Everything we measure has uncertainties, and to be able to say whether a measurement can be reproduced or agrees with theoretical predictions, we need to carefully take them into account.

It is important to distinguish between statistical/random and systematic uncertainties. They are fundamentally associated with the terms of precision and accuracy. Statistical uncertainties can be reduced by repeated measurements up to a degree, while systematic uncertainties require improvement of measurement procedure or measurement devices. See this introduction for more details.

The standard deviation of a single measurement describes its average uncertainty. The average of multiple measurements is taken as the best estimate, and its reduced uncertainty is given by "standard deviation of the mean". When we measure individual quantities with uncertainties, and those quantities enter through a mathematical formula in a derived quantity, we use uncertainty propagation to compute the uncertainty of the derived quantity. For example when we measure the volume with a ruler, each dimension measurement will have an uncertainty, which needs to be propagated in the volume $V=w\cdot h \cdot l$ with width, height and length measurements for a rectangular object.

Measurement 1 and 2¶

For measurement 1 and 2 we use one of the handheld digital multimeters (DMM) in figure 1, the benchtop digital multimeter in figure 2, and the analog voltmeter in figure 3 to measure the voltage of a dry-cell batter (figure 4). We also perform a calibration using the benchtop power supply in figure 5.

Warning

To protect the analog voltmeter, always choose a range that is higher than the highest voltage you might expect in the circuit.

*Figure 1:* Two different models of digital multimeters (DMM). The MN35 manual and BM8300L manual provide further information and measurement accuracies.

*Figure 2:* RIGOL DM3068 benchtop digital multimeter. The manual provides further information and measurement accuracies, see page 47 for DC voltage measurement and uncertainties on page 145. For simplicity you can assume a systematic uncertainty of $0.5\%$ of the measurement reading.

*Figure 3:* Two models of analog voltmeters by Pasco. The manual shows the accuracy, $2\%$ of the full scale, either $3\,\text{V}$ , $15\,\text{V}$ or $30\,\text{V}$ . Iclude this uncertainty in addition to a read-off uncertainty.

*Figure 4:* Alkaline-manganese dioxide dry-cell battery with a typical voltage of $1.5\,\text{V}$ , see the data sheet.

*Figure 5:* Benchtop power supply. The quickstart guide provides introductory information, and the datasheet accuracy information. For simplicity, you can assume an uncertainty of $0.03\,\%$ plus $10\,\text{mV}$ for voltages and $0.03\,\%$ plus $10\,\text{mA}$ for currents.

Measurement 3: Human resistance¶

We distinguish between systematic and random/statistical uncertainties. The uncertainties typically experienced in this electricity and magnetism lab are systematic, given as uncertainties that manufacturers specify for their measurement devices, limiting the accuracy. In general we cannot increase the precision of the measurement by taking multiple readings off the measurement device (e.g. the voltage the DMM displays).

While there are always random uncertainties, they are small compared to the systematics in this lab. In mechanics it is easier to demonstrate meaningful random uncertainties. For example, if you measure the length of an object's outline using a piece of cord (worksheet 1 mechanics), your result will slightly vary from how a different person would align the cord with the outline. This is a random uncertainty and can be reduced through repeated measurements by different people. Additionally there is a systematic uncertainty given by how accurately you can measure the length of the rope (using a meter stick). In this case the random uncertainties could be as large or larger than the systematic (which is half a mm for a ruler). But the random uncertainties can be reduced by taking more measurements until they can be neglected compared to the systematic uncertainty.

In this electricity and magnetism lab it is much more difficult to experience random uncertainties of a significant size. Measurement 3 tries to model the random aspect in measuring the resistance between two hands of a human. Different people will show a different resistance, based on their skin condition, and conductivity through the body.

The difficulty is that there are a lot of systematic effects that are still large compared to a residual random effect. For example, the pressure you apply to the probe will have a large impact on how big of a resistance is being measured. Or whether your hands are more or less dry. These are effects that cause a systematic deviation/uncertainty, but they are very difficult to hold constant, to control. The uncertainty associated with applying different pressure to the probe cannot be reduced by taking more and more measurements. This means that when you average the measurements from multiple people with the intention to improve the random uncertainty, you would have to hold these systematic effects constant, i.e. everyone would have to apply the same pressure. This would be very difficult to do in this lab, so that your computed standard deviation of the mean for the average of multiple people's result will be wrong by such uncontrolled systematic effects.

Measurement 4¶

Using the handheld DMM (figure 1) and the benchtop DMM (figure 2), the resistances of R3 and R4 on the PASCO board in figure 6 are measured.