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Error Analysis - PHY133 Classical Physics I Laboratory...

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PHY133 - Classical Physics I Laboratory Error Analysis Statistical or Random Errors Every measurement an experimenter makes is uncertain to some degree. The uncertainties are of two kinds: (1) random errors, or (2) systematic errors. For example, in measuring the time required for a weight to fall to the floor, a random error will occur when an experimenter attempts to push a button that starts a timer simultaneously with the release of the weight. If this random error dominates the fall time measurement, then if we repeat the measurement many times ( N times) and plot equal intervals (bins) of the fall time t i on the horizontal axis against the number of times a given fall time t i occurs on the vertical axis, our results (see histogram below) should approach an ideal bell-shaped curve (called a Gaussian distribution) as the number of measurements N becomes very large. The best estimate of the true fall time t is the mean value (or average value) of the distribution t : t = ( Σ N i=1 t i )/ N . (1) If the experimenter squares each deviation from the mean, averages the squares, and takes the square root of that average, the result is a quantity called the "root-mean-square" or the "standard deviation" σ of the distribution. It measures the random error or the statistical uncertainty of the individual measurement t i : σ = [ Σ N i=1 ( t i - t ) 2 / ( N -1) ]. (2) About two-thirds of all the measurements have a deviation less than one σ from the mean and 95% of all measurements are within two σ of the mean. In accord with our intuition that the
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uncertainty of the mean should be smaller than the uncertainty of any single measurement, measurement theory shows that in the case of random errors the standard deviation of the mean, σ m , is given by: σ m = σ / N , where N again is the number of measurements used to determine the mean. Then the result of the N measurements of the fall time would be quoted as t = t ± σ m . Whenever you make a measurement that is repeated N times, you are supposed to calculate the mean value and its standard deviation as just described. For a large number of measurements this procedure is somewhat tedious. If you have a calculator with statistical functions it may do the job for you. There is also a simplified prescription for estimating the random error which you can use. Assume you have measured the fall time about ten times. In this case it is reasonable to assume that the largest measurement t max is approximately +2 σ from the mean, and the smallest t min is -2 σ from the mean. Hence: σ ¼ ( t max - t min ) is an reasonable estimate of the uncertainty in a single measurement. The above method of
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