CHAPTER 4 ESTIMATES OF MEAN AND ERRORS

4.1 METHOD OF LEAST SQUARES I n Chapter 2 we defined the mean  of the parent distribution and noted that the most probable estimate of the mean  of a random set of observations is the average x of the observations. The justification for that statement is based on the assumption that the measurements are distributed according to the Gaussian distribution. In general, we expect the distribution of measurements to be either Gaussian or Poisson, but because these distributions are indistinguishable for most physical situations we can assume the Gaussian distribution is obeyed. Method of Maximum Likelihood Assume that, in an experiment, we have observed a set of N data points that are randomly selected from the infinite set of the parent population, distributed according to the parent distribution. If the parent distribution is Gaussian with mean  and standard deviation , the probability dP i for making any single observation x i within an interval dx is given by dP i = P i dx (4.1) with probability function Pi = P G (x i, ,  ) [see Equation(2.23)]. For simplicity, we shall denote the probability P i for making an observation x i by

Because, in general, we do not know the mean  of the distribution for a physical experiment, we must estimate it from some experimentally derived parameter. Let us call the estimate  ', What formula for deriving  ' from the data will yield the maximum likelihood that the parent distribution had a mean equal to  ? If we hypothesize a trial distribution with a mean  ' and standard deviation  ' = , the probability of observing the value xi is given by the probability function Considering the entire set of N observations, the probability for observing that particular set is given by the product of the individual probability functions, P i (  '), where the symbol denotes the product of the N probabilities P i (  '). The product of the constants multiplying the exponential in Equation (4.3) is the same as the product to the N th power, and the product of the exponentials is the same as the exponential of the sum of the arguments. Therefore, Equation (4.4) reduces

According to the method of maximum likelihood, if we compare the probabilities P("",') of obtaining our set of observations from various parent populations with different means "",' but with the same standard deviation (J"' = (J", the probability is greatest that the data were derived from a population with "",' = ""'; that is, the most likely population from which such a set of data might have come is assumed to be the correct one. Calculation of the Mean The method of maximum likelihood states that the most probable value for "",' is the one that gives the maximum value for the probability P("",') of Equation (4.5). Because this probability is the product of a constant times an exponential to a negative argument, maximizing the probability P("",') is equivalent to minimizing the argument X of the exponential, (4.6) To find the minimum value of a function X we set the derivative of the function to 0, (4.7) Estimates of Mean and Errors 53 and obtain dX = _1. ~ ~(Xi - "",')2 = ~ (Xi - """) = ° d"",' 2 d"",' (J" (J"2 (4.8) which, because (J" is a constant, gives (4.9) Thus, the maximum likelihood method for estimating the mean by maximizing the probability P("",') of Equation (4.5) shows that the most probable value of the mean is just the average x as defined in Equation (1.1). Estimated Error in the Mean What uncertainty (J" is associated with our determination of the mean "",' in Equation (4.9)? We have assumed that all data points Xi were drawn from the same parent distribution and were thus obtained with an uncertainty characterized by the same standard deviation (J". Each of these data points contributes to the determination of the mean "",' and therefore each data point contributes some uncertainty to the determination of the final results. A histogram of our data points would follow the Gaussian shape, peaking at the value "",' and exhibiting a width corresponding to the standard deviation (J". Clearly we are able to determine the mean to much better than ± (J", and our determination will improve as we increase the number of measured points N and are thus able to improve the agreement between our experimental histogram and the smooth Gaussian curve. In Chapter 3 we developed the error propagation equation [see Equation (3.13)] for finding the contribution of the uncertainties in several terms contributing to a single result. Applying this relation to Equation (4.9) to find the variance (J"~ of the mean "",', we obtain (4.10) where the variance (J"r in each measured data point Xi is weighted by the square of the effect a"",' / aXi' that that data point has on the result. This approximation neglects correlations between the measurements Xi as well as second- and higherorder terms in the expansion of the variance (J"~, but it should be a reasonable approximation as long as none of the data points contributes a major portion of the final result. If the uncertainties of the data points are all equal (J"i = (J", the partial derivatives in Equation (4.10) are simply ~~; = a~i (~~Xi) = ~ (4.11