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Test Your Might! A Quiz on Error Analysis Concepts
Before you start, remember the rule of significant figures: the result of a calculation should have no more significant digits than the least precise number used. If you're calculating uncertainties, these usually have only one significant figure.
Absolute vs. Relative Uncertainties
1) A resistor is a circuit unit that impedes the passage of current through it. If a resistor has a resistance of 1.5 x 106 W± 5%, what is the uncertainty in ohms?
2) A Van de Graff generator holds a charge of 12 ± 1 coulombs (C). Express the uncertainty as a relative (percent) error.
Random vs. Systematic Errors
3) While trying to measure the magnetic field produced by a current-carrying coil, your GSI reminds you that the Earth's magnetic field may be a factor. Will this cause a random error or a systematic error in your measurements?
4) Your GSI also reminds you that the electrical system in the building is poorly insulated, so if you're working near a wall there will be 60 Hz oscillating electric fields playing around your apparatus. Is this more likely to cause a systematic error, a random error, or neither?
Estimating Uncertainties
5) A voltmeter gives a digital readout down to hundredths of a volt (for example, 4.51 V). What is the minimum uncertainty on the reading?
6) If the display fluctuates between 4.42 V and 4.59 V as you watch it, what should you write down as the best estimate of the reading and the uncertainty?
7) If a measurement has two sources of uncertainty, one larger than the other, which one do you use? (For example, using a ruler graded in 1mm intervals to measure the position of a spark table dot that is 2mm wide.)
Error Propagation
8) A capacitor is a device that stores charge. The relationship between the charge on the capacitor and the voltage measured across the capacitor is Q (charge) = C (capacitance) × V (voltage),where the constant C depends on the size and shape of the capacitor. If we measure the voltage across a capacitor to be 8.6 ± 0.1 V, and the capacitor is rated to be 0.2 × 10-6 farads ± 1%, what is the charge on the capacitor? (Using units of volts and farads gives you a charge in coulombs.) What is the uncertainty, in coulombs?
9) Two resistors connected end-to-end have a total resistance equal to the sum of their individual resistances. If we connect two resistors in this manner, each rated at 100 W± 5%, what is the total resistance and its uncertainty?
10) The voltage across a capacitor discharging through a resistance is given by V(t) = Vie-t/RC. You know that the initial voltage is 9.00 ± 0.01 V (because you charged it with a fresh battery), the resistance in the circuit is 1.2 × 108 W ± 10%, and the capacitor has a strength of 2.0 × 10-6 farads. If you complete the circuit, wait 10.0 ± 0.5 seconds, then disconnect it, what will the final voltage be, with uncertainty?
Dealing with Random Errors
We build a circuit and measure its resistance, with the results R = 5.9, 5.0, 6.2, 5.4, and 6.2 W. The random errors in the experiment are clearly larger than the ± 0.1 inherent in your measuring device, perhaps because your circuit is being influenced by external electric fields.
11) What is your best guess at the true resistance of the circuit?
12) What is the uncertainty on each individual measurement?
13) What is the uncertainty associated with your best estimate of the true resistance?
14) How many more measurements should we make to reduce the uncertainty of question 13 by about half?
15) Is including additional measurements likely to change the standard deviation of the data significantly?
16) If we repeat the experiment later on, measuring the circuit five times, what is the probability that our new result will fall within the range of uncertainty given in question 13?
Created by Ben Mathiesen
Last Updated by Andy Pawl
Sept. 2003
apawl@umich.edu