26, 24, 26, 28, 23, 23, 25, 24, 26, 25

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1 The ormal Distribution Introduction Chapter 5 in the text constitutes the theoretical heart of the subject of error analysis. We start by envisioning a series of experimental measurements of a quantity. Those measurements are then collected into a histogram for easier viewing. Initially we work with a variable which has only discrete (integer) values, but we quickly extend to variables which can take on continuous values. This leads us to the topic of binning in the histogram. Having seen the plots of histograms, we next attempt to predict their theoretical shape or distribution function. For error analysis, the theoretically established distribution function is the normal distribution or the Gauss distribution, better know colloquially as the bell curve because of its shape. Once we have the mathematical distribution function formula, we can then prove all the statements which we introduced in the previous chapter: the mean is the best estimate of the true value of a quantity, the standard deviation of the mean σ x = σ x / is the best estimate of the uncertainty etc. Histograms and Distributions As usual, we assume that there is some physical quantity x which we want to know, and that we make a series of independent measurements of that quantity using the same experimental technique. The text trivial example is a length quantity x, and say we measure the length 10 different times to obtain the following values (in cm): 26, 24, 26, 28, 23, 23, 25, 24, 26, 25 Just looking at these numbers, or even worse 100 such numbers, does not immediately tell us too much about the best value of x, or how good was our experimental technique. A better way to represent the measurements is to arrange them in order (from 23 to 28 in this case), and count how many times a given length value was obtained. At first we might put these in a table as 1 Histogram table of discrete measurements different values, x k number of times value found

2 Lecture 4: Chapter 5, The ormal Distribution 2 Histograms and Distributions Weight and Frequency Factors The number of times each different value is measured is the weight of that measured value, which we can symbolize as n k. Hence we can re-write our best estimate mean value x as i x i x = = k x k n k This expression on the right is called the weighted mean formula. Very clearly the sum of all the weights n k must add up to the total number of measurements. We can define a normalized weight for frequency factor F k = n k and then our mean value formula is i x i x = = k x k n k x k F k = k These are called the normalized weights because the sum of F k is unity. Continuous Distributions In our example above we considered x to be a length but we took only integer values. aturally, lengths are not integers in general, so what to we do for histograms in such cases which are actually the majority of measurements? For example, consider the set of 10 measurements to be 26.4, 23.9, 25.1, 24.6, 22.7, 23.8, 25.1, 23.9, 25.3, 25.4 Obviously there is no point in make a table of 10 entries, one for each measurement. Instead, we divide up the measurements into intervals of bins, and count how many measurements fall into a bin. As another example, a professor giving the results of an exam will normally group the grades in decades (90 100, 80 89, 70 79,...). In this length case we can bin in 1 cm intervals: Histogram table of continuous measurements different values, x k number of times value found

3 Lecture 4: Chapter 5, The ormal Distribution 3 Histograms and Limiting Distributions Histogram Bin Size In the case of continuous variables, or even in some cases of discrete variables, one has to exercise some care about the binning size. If one makes the bins too small, then one will have either 0 or 1 entries in every bin which is pointless. Similarly, if the bins are too large, then all the entries might fall into one or two bins which would be equally meaningless. In general, as our number of measurements becomes very large, then we can let our bin sizes get smaller. But with small value sof, then we are forced to have larger bin sizes. Limiting Distributions According to error theory, every measurement process has a theoretical limiting distribution function, meaning that one made an infinite set of repeated measurements, then one would have a certain shaped histogram f(x). A histogram of any finite set of measurements will approximate the shape of the limiting distribution. The width of that distribution tells us about the precision of the measurement technique. The distribution function f(x) is a probability function. It tells us that if we make a set of measurements for the quantity x, then the fraction of those measurements which fall say between two limits a and b will be given by a definite integral Probability of finding a x b = f(x)dx a Since the value of x must be somewhere, if we extend the limits to ±, then the integrated probability must be units. So the distribution function is normalized Probability of finding x + = b f(x)dx = 1 Of course you might wonder about extending the limits to ±. For example, how could a length measurement be negative? In principle this does not matter since we have not set what f(x) is for any given measurement. This we will do shortly, but for now we explore the general use of the limiting distribution function.

4 Lecture 4: Chapter 5, The ormal Distribution 4 General Use of the Limiting Distribution Function The Mean Value and Standard Deviation We have said that the mean value of a set of measurements is the best estimate of the true value. In terms of the distribution function, we calculate the mean value as x = xf(x)dx Similarly the standard deviation σ x is given by σ 2 x = (x x)2 f(x)dx You should be clear on the practical meaning of the variable σ x. A small value of σ x means that the distribution function is narrow and that the measured values cluster closely about the best value. Thus the experimental method has high precision. On the other hand, a larger value of σ x means that the experimental method has low precision since the measured values have a high probability of being away from the mean or best value. The ormal Distribution Function We now given you the normal distribution function explicitly. We assume that there is a quantity x to be measured whose true value is X. The measurement process has a certain precision such that the width of the distribution function is σ. The errors in the measurement process are assumed to be independent and random. In this case the distribution function becomes the normal or Gaussian distribution funtion represented as f(x, X, σ) = 1 σ /2σ2 exp (x X)2 The coefficient in front of the exponential factor gives the normalization condition that the definite integral of f(x, X, σ) over the interval x + is unity.

5 Lecture 4: Chapter 5, The ormal Distribution 5 The ormal Distribution Function Using the ormal Distribution Function We readily evaluate x and σ x with the normal distribution function. This is done in the text, and the results should be no surprise to you: x = 1 σ x /2σ 2 dx = X exp (x X)2 σx 2 = 1 σ (x x)2 exp (x X)2 /2σ 2 dx = σ 2 What we are showing here is that after a very large set of measurements we would expect the mean value of the measurements to be the true value of the quantity. Similarly, we would expect that the standard deviation of our own set of measurements would the the standard deviation of the distribution function of the measurement process. Confidence Intervals ow we are ready to prove our previous assertions about quantitative meaning of the σ variable. We have said that about 2/3 of the time a given measurement result will be within ±σ of the true result. This we can prove with the normal distribution function: Probability of X σ x X + σ = 1 σ X+σ /2σ 2 dx = 0.68 X σ exp (x X)2 Physicists very often quote the two sigma (2σ) band for their result. In that case Probability of X σ x X + σ = 1 σ X+2σ /2σ 2 dx = 0.95 X 2σ exp (x X)2 So there is only a 5% chance of the true value X being outside of the two-sigma band. A typical set of probabilities is contained in the following table Hence if ormal Distribution Probabilities Sigma fraction Probability (%) you quote a result at the three sigma level, you are saying that you are 99.7% sure that the true value is within that range.