Kevin Lehmann
Gaussian Distributions
Kevin Lehmann, Princeton University
The author generates a set of data from a Gaussian distribution to illustrate the properties of the distribution such as the mean of a set of data, the confidence level for the mean, the chi-squared function, and the Student t Distribution The document concludes with a discussion of the mean absolute deviation and a survey of the Moments of a distribution and how the mean absolute deviation and the moments are used to characterize the shape of the distribution. Variance, skew, and kurtosis are the three moments discussed in this document.
Mean Versus Median
Kevin Lehmann, Princeton University
This worksheet provides a comparison of the mean and median values for both theoretical distributions and for data sets sampled from Gaussian and Lorentzian distribution functions. The document shows that the mean value provides a moderately better estimate of the central value than the median for the case of a Gaussian. However, in the case of a Lorentzian, due to its slow fall off for large displacements from the central value, the mean is almost useless as a statistic, while the median functions quite well. The document also introduces the idea of finding the optimal estimate by using the method of maximum likelihood. This document requires Mathcad 6.0+ including upgrade through patch 'e' .
Rejection of Data
Kevin Lehmann, Princeton University
The document provides a detailed presentation to the theory of rejection of data using a Gaussian distribution. The document discusses the conditions under which the Q-test is used. The exercises in the document give students opportunities to practice the concepts. The document provides a numerical example of how statistical methods can reduce the errors in information extracted from measurements with real, as oppose to Gaussian, noise characteristics.
The Morse Oscillator
Kevin Lehmann, Princeton University
In this worksheet, we find a presentation of the vibrational motion of a diatomic molecule held together with a potential function of a special form known as the Morse Potential. Both the classical and quantum motion of the oscillator will be studied, and explicit expressions for eigenenergies and wavefunctions are given The effect of rotation is also discusses. The document contains embedded 20 exercises and 4 advanced problems for users to test their mastery of the topic.