ErrorUncertaintyAppendix07

# ErrorUncertaintyAppendix07 - ERRORS AND UNCERTAINTY...

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1 ERRORS AND UNCERTAINTY LIMITS (revised January 8, 2007) Purpose and Scope: This Appendix provides a framework to analyze and propagate experimental errors and determine uncertainty limits in laboratory measurements. These guidelines apply, in general, to the lab components of Physics 211 and 213 and to Physics 241 and 242 lab courses (which include more rigorous calculus-based error propagation techniques). Errors as Uncertainties: “In science, the word error does not carry the usual connotations of the terms mistake or blunder : Error in a scientific measurement means the inevitable uncertainty limit that attends all measurements. As such, errors are not mistakes; you cannot eliminate them by being very careful. The best you can hope to do is to ensure that the errors are as small as reasonably possible and to have a reliable estimate of how large they are.” (Taylor, An Introduction to Error Analysis, 2 nd Ed. , p. 3) The Skeptic, as the member of the research team tasked with focusing on error, should carefully observe and document all measurements and the associated equipment, but the entire team should be involved in the process of error analysis and should record their deliberations in the Analysis and Discussion paragraph of the laboratory report. The analysis of error should consider both accuracy and precision. Accuracy is a measure of how close your average result is to some accepted or “true” value. Precision , on the other hand, is a measure of how close your experimental results are to each other. A good way to illustrate this is to consider a game of darts. The bulls-eye represents the accepted value. Figure 1 shows the differences between precision and accuracy. In (A) you hit the bulls-eye with all five darts. This corresponds to high precision and high accuracy. On your next turn (B) all the darts hit around the same position so you have high precision, but unfortunately you are not close to the center of the target so you have low accuracy. The graphic (C) shows low precision, since your darts are all over the board, but high accuracy since the average of the scattered shots fell near the center. Finally, you are shooting poorly if you have a game like (D): Here you are not close to the bulls-eye, even on average, and your darts are spread out over the board. This is low precision and low accuracy. Depending on what you are measuring, you might not know the true value and therefore cannot know if your average result is accurate. But you can assess precision by observing how the individual results scatter. When you make multiple measurements ( N times) you must first find the average of these values, which is indicated by the angled brackets: N t t t t t N t N N i i + + + + = = = midhorizellipsis 3 2 1 1 1 (1)

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2 x x x x x x x x x x x x x x x x x x x x (A) (C) (D) (B) High precision High accuracy Low precision High accuracy Low precision Low accuracy High precision Low accuracy Fig. 1: The difference between accuracy and precision using a dart board.
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