Measurement Uncertainty Quiz

Caution:  This quiz is not a standardized diagnostic instrument and has not been tested for reliability and validity.  A similar but more complete diagnostic test is being developed, but the subject of measurement uncertainty does not lend itself well to a right/wrong test, so even experts may disagree about which answers are "right" on such a test.  The questions found below are a sample of the open-ended questions that were given to approximately 100 introductory physics students and 30 experts (graduate physics students and teachers).  The most common responses have been edited and presented here as multiple-choice options.  This quiz is most valuable as an opportunity for discussion with students about why one answer might be better than the others.

1) Rank the following measurements in order from the most precise to the least precise based on the relative uncertainty implied by each value:   9.7 m,   13 m,   1.45 m,   2.1 m,   0.005 m
(Use > or = , so that A > B means A is more precise than B, and A = B indicates equal precision)

a)  0.005 > 1.45 > 9.7 = 2.1 > 13
b)  0.005 > 1.45 > 2.1 > 9.7 > 13
c)  1.45 > 9.7 = 13 = 2.1 > 0.005
d)  1.45 > 9.7 > 2.1 > 13 > 0.005
2) A group of students are told to use a meter stick to find the length of a hallway.  They make 6 independent measurements:  4.402 m,   4.217 m,   4.345 m,   4.925 m,   4.372 m,   4.289 m.  How should they report their best estimate of the length of the hallway?
a)  L = 4.33 ± 0.03 m
b)  L = 4.43 ± 0.25 m
c)  L = 4.325 ± 0.073 m
d)  L = 4.425 ± 0.104 m
3) A student uses a protractor to measure an angle to be A = 82o ± 1o.  What should she report for sin(A)?
a)  sin(A) = 1.0 ± 0.2
b)  sin(A) = 0.99 ± 0.02
c)  sin(A) = 0.990 ± 0.002
d)  sin(A) = 0.9903 ± 0.0024
The following text applies to questions 4-6:  A simple pendulum is known to have a period of oscillation, T = 1.55 s.  Student A uses a digital stopwatch to measure the total time for 5 oscillations and calculates an average period T = 1.25 s.  Student B uses an analog wristwatch and the same procedure to calculate an average period for the 5 oscillations and finds T = 1.6 s.

4) Which period is more accurate, and why?

a)  Student A's period of 1.25 s because a digital stopwatch is more reliable.
b)  Student A's period of  1.25 s because the stopwatch can measure to 0.01 s.
c)  Student B's period of 1.6 s because it is closer to the known period than A's value.
d)  There is not enough information to answer this question.
5) Which measurement is more precise, and why?
a)  Student A's period of 1.25 s because a digital stopwatch is more reliable.
b)  Student A's period of 1.25 s because the stopwatch can measure to 0.01 s.
c)  Student B's period of 1.6 s because it is closer to the known period than A's value.
d)  There is not enough information to answer this question.
6) What is the most probable source of error that could explain the difference in the results?
a)  Human reaction time in starting and stopping the timing devices.
b)  The stopwatch may run too fast; not calibrated properly.
c)  The amplitude of oscillation may have been too large for one pendulum.
d)  Student A mistakenly measured 4 oscillations instead of the intended 5.
The following text applies to questions 7-9: A student performs a simple experiment to find the average acceleration of a falling object.  He drops a baseball from a building and uses a string and meter stick to measure the height the ball was dropped.  He uses a stopwatch to find an average time of fall for 3 trials from the same height and reports the following data:
   h = 5.25 ± 0.15 m,    t = 1.14 ± 0.06 s.

7) Use the equation a = 2h/t2 to determine the average acceleration and its uncertainty.

a) 8.08 ± 0.1 m/s2
b) 8.08 ± 0.88 m/s2
c) 8.08 ± 0.06 m/s2
d) 8.1 ± 0.9 m/s2
8) Comment on the accuracy of the acceleration result. Do you think the student made any mistakes?
a)  The uncertainty is high; probably a mistake in height measurement or reaction time with stopwatch.
b)  Although a < g, the result seems reasonably accurate since air resistance would reduce the ball's acceleration.
c)  The result does not agree with 9.8 m/s2, so the student must have made a mistake.
d)  The result can only be as accurate as the measurements; cannot tell if a mistake was made.
9) What one suggestion would you tell this student to improve the accuracy of the experimental result?
a)  Measure height better, maybe use a long tape measure that does not stretch.
b)  Improve the precision of the time with more trials or automatic device.
c)  Get an assistant to time when the ball hits the ground.
d)  Reduce or eliminate air resistance.
10)  A student measures the ratio of the circumference of a circle divided by its diameter to be 3.3 ± 0.1, where the reported uncertainty is the standard error found from repeated measurements.  Does this result agree with the known value of  PI = 3.14159...?
a)  No, the uncertainty range of the measured result does not include the value of PI.
b)  The probability of agreement is about 5%.
c)  The relative error is 5%, which is small enough to be considered acceptable.
d)  We need to know how the measurements were made to answer this question.