p4bl_09statististics - Physics 4BL Laboratory 1...

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Physics 4BL Laboratory 1 Uncertainties and statistics I Background Information I.1 Uncertainty in Measurements Any experimentally measured quantity x cannot be known exactly due to error in our measurement. Therefore, when a measured quantity is reported, for example in a scientific journal or an academic lab report, one presents the measured quantity as x = x best ± δx, where x best is our best guess at the quantity x, and δx quantifies the uncertainty in our measure- ment. Uncertainties in measurement are often classified into two distinct categories, systematic and random. The measurements performed in this laboratory introduce us to analysis methods for treating the latter type, which is statistical in nature. As a result, our confidence level on the outcome is improved by making more measurements, and the exercises here demonstrate how to treat this aspect quantitatively. It is worth emphasizing that appreciating the impact of random uncertainties is fundamental to the process of measuring any physical quantity. Whether it seems relevant for a given situation is quantitative: is the confidence level in the measured outcome small or large when compared to what is needed? Lets assume N measurements of x are made, yielding values x 1 ,x 2 ,...,x N . A large number of such measurements, called an ensemble, has certain useful properties such as the mean value and the spread of the values around it, called the standard deviation. In nearly all situations our best estimate of x is the mean value of these measurements: x = 1 N N X i =1 x i (1) As more measurements are made, we expect that our measured mean value x will approach the “true” value X, which is defined as the mean value if an infinite number of measurements were made. That is, X = lim N →∞ x = lim N →∞ 1 N N X i =1 x i . What we seek is a quantitative measure of how far the measured mean x may deviate from the “true” value X. Standard practice is to use a quantity called the standard deviation of the mean, denoted by σ x as such a measure. Then, when one publishes (as in your lab reports for this class) a measured quantity, it is written as x = x ± σ x signifying that we are confident X, the quantity we ultimately want to know, lies within some range of values around our measured x , due to random error. In the first part of the present laboratory, you will examine how to calculate our measure of random error, called the standard deviation of the mean, and how it is affected by changing the ensemble size, N. 1
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When performing an experiment, we seek to determine quantities that are measured directly, and also quantities that are calculated from something that is directly measured. The fact that error exists in the latter type is referred to as propagation of error , and you will study this in the second part of the present laboratory. Consider a quantity F that is a function of 2 directly measured quantities, x and y: F = f ( x,y )
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This note was uploaded on 01/14/2010 for the course PHYS 4BL taught by Professor Slater during the Winter '07 term at UCLA.

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p4bl_09statististics - Physics 4BL Laboratory 1...

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