computer - 32 Experiment IV: Computer Analysis of...

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32 Experiment IV: Computer Analysis of Experiments II and III Goals Learn the basic concepts in error analysis Use a computer spreadsheet program, Excel, to analyze data from Experiments II and III Introduction and Background Experimental Errors : Experimental error or uncertainty is inherent in any experimental result at some level, however small. It is set by a combination of the design of the experiment, the quality of the apparatus, and the care and skill of the experimenter. Generally, it is possible to separate the sources of experimental errors into two categories: random and systematic . Random Errors : The existence of random errors in a measurement can be inferred if repetition of the measurement does not give the same result each time. By definition, random errors are those that tend to average out upon repetition of the measurement. Hence for random errors, the more repetitions of a measurement, the less uncertainty there is in the resulting average. From the mathematics of statistics, it can be shown that the uncertainty due to random error in the average of N measurements decreases as N when N is large. For example, 400 measurements should give an average with half the uncertainty due to random errors as compared to 100 measurements. The first step in treated the random error in a large number N of repeated measurements is to calculate the average: ) ... ( 1 3 2 1 N y y y y N y + + + + = (4-1) which is the desired result and sometimes called the mean in statistics. To establish the uncertainty in the average, y , the usual procedure is to first calculate what is called the standard deviation, σ , of the measurements. If no one measurement is more accurate than any other, then σ is defined by 2 2 2 2 1 ) ( ... ) ( ) [( 1 1 N y y y y y y N - + + - + - - = s (4-2) The physical significance is that any one additional y measurement has about a 2 in 3 chance of falling between s ± of the average y . Statistical analysis further shows that the average y has a 2 in 3 chance of falling within s ± of the true value. Thus it is common to append s ± as a measure of the uncertainty due to the randomness in the measurements. The standard deviation for the average (mean) is N m s s = (4-3) and the final result is often expressed as
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33 N y s ± (4-4) Systematic Errors : A systematic error is one which tends to repeat and thus create a shift in the average from the true value. A systematic error cannot be averaged out by repeated measurements. Systematic errors may result from the experimenter, the apparatus, or the poor design of the experiment. Because they are not revealed by repeated measurements, care must be taken to investigate and account for all possible sources of errors. This can be very difficult to do. Such
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This note was uploaded on 11/10/2011 for the course PHY 2053 taught by Professor Lind during the Fall '09 term at FSU.

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computer - 32 Experiment IV: Computer Analysis of...

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