Mean Versus Median©
Kevin Lehmann
Department of Chemistry
Princeton University
Princeton, NJ 08544
United States
mail to: Lehmann@princeton.edu



Abstract
This document is the third in a series written by K. Lehmann. In this worksheet, we find 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. Application of this method to a Gaussian distribution leads to the expression for the mean value, i.e.the mean value is the best estimate for the central value for a Gaussian distribution. For the Lorentzian distribution, however, the maximum likelihood method leads to a set of coupled nonlinear equations for the parameters. This is the typical situation. One can solve these equations numerically using the built in functions of Mathcad. We find, however, that in this case this optimal estimate gives a standard deviation around the correct value only slightly smaller than that provided by the median value. Thus the median value is almost the optimal estimate of the center of a Lorentzian distribution.Further, the document shows that, in this case, the central value predicted by the sample median and the maximum likelihood methods are highly correlated; they are typically much closer to each other than to the 'true' value of the Lorentzian distribution from which the data was sampled. This document requires Mathcad 6.0+ including upgrade through patch 'e' . The document is suitable for advanced undergraduates and graduate students.
Keywords
Audiences: Upper-Division Undergraduate, Graduate Education / Research
Pedagogies: Computer-Based Learning
Domains: Analytical Chemistry, Laboratory Instruction, Physical Chemistry
Topics: Chemometrics, Mathematics / Symbolic Mathematics
Documents
File NameDescriptionSoftware TypeSoftware Version
MeanVsMedian.mcd Mathcad Computational Document Mathcad 6
MeanVsMedian.pdf Read-Only Document
Comments to: Kevin Lehmann at Lehman@princeton.edu
©Copyright Kevin Lehmann, 1998. All rights reserved. You are welcome to use this document in your own classes but commercial use is not allowed without the permission of the author.