Category: Statistical theory. These pages describe some of the more mathematical and/or technical aspects of Statistics. Articles are arranged by date with the most recent entries at the top. You can find the theme and closely related categories and other resources at the bottom of this page.
Stats: Can the standard deviation be more than half of the range? (June 22, 2007). Dear Professor Mean, I was trying to work with some simple data sets to see how large I could make the standard deviation relative to the range. I know the standard deviation can never be larger than the range, but I can't seem to get it to be larger than half the range.
Stats: Compound interest and powers (February 11, 2007). In some of my mathematical calculations, I end up computing an expression that involves a number very close to one raised to a very large power. This term can often be approximated by an exponential function, but I can never quite remember the relationship. An example involving compound interest may help me remember better in the future.
Stats: Mathematical and statistical challenges (December 13, 2006). A regular poster on the EDSTAT-L list (DR) mentioned an interesting page on the IBM website, www.research.ibm.com/ponder, that offers a monthly puzzle on mathematical topics.
Stats: Testing for bimodality (May 3, 2005). I have talked about this topic before and it is a rather tricky thing. A recent discussion of tests of bimodality on edstat-l, though, yielded a few promising leads relating to the Dip test, which is described in: The Dip Test of Unimodality, Hartigan JA, Hartigan PM. The Annals of Statistics, v13(1):70-84, 1985.
Stats: A surprising application of the harmonic mean (February 1, 2005). The radio show, Car Talk, has a puzzle that they read every week on the show. Usually, it is some unusual or unexpected problems with automobiles, but Ray and Tom Magliozzi also will toss in a mathematical puzzle from time to time. A recent car talk puzzler, www.cartalk.com/content/puzzler/transcripts/200505/index.html, discusses a family that has two cars, one which gets 10 miles per gallon and the other gets 100 mpg.
Stats: Simpson's Paradox (December 22, 2004). Someone wrote to the Evidence Based Health email discussion group about a theoretical situation where someone had an estimated risk of disease based on a study that showed the degree to which factors a, b, and c might influence disease status. Suppose in a different study, a factor d was shown to double the risk of disease. What could you then say about the probability of disease among a patient who has a, b, c, and d? You would think that someone with a, b, c, and d should have a greater risk of disease than someone with just a, b, and c. The answer unfortunately, is that nothing is predictable here, and it is possible for someone with a, b, c, and d to have a lower risk even though a study looking at d showed a doubling of risk. You have to watch out for Simpson's Paradox.
Stats: Searching for bimodality (August 4, 2004). One of the people I work with is always looking for hidden subgroups in his data. For him, this is a starting point for exploring for genetic variations. That's an admirable activity, but it is remarkably difficult to thing to do in practice. The first step is to see if the distribution of values for some measurement has a bimodal distribution. A second mode is an indication of a subgroup of patients that may have a genetic variation from the rest of the patients.
Stats: Missing values (June 22, 2004). Someone here at the hospital asked me how to do a reliability analysis on a 20 item measure where a large number of participants left a single item blank. There are several approaches that work, but you need to exercise a bit of caution.
Stats: Degrees of freedom, Part 2 (April 15, 2004). I received an email inquiry about degrees of freedom. I explain the concept briefly, but this person wanted a more detailed answer to the question, why do we use n-1 in the calculation of the standard deviation and not n?
Stats: Maximum likelihood estimation (May 6, 2003). Dear Professor Mean: What is maximum likelihood estimation and how does it work?
Stats: Stein's paradox (January 27, 2000). Dear Professor Mean: What is "Stein's Paradox?"
Stats: Degrees of Freedom (September 3, 1999) Dear Professor Mean, In your Simple Descriptive Statistics class, you described the standard deviation as the square root of the average squared deviation. If it is an average, how come we divide by the degrees of freedom (n-1) rather than n. Is this just a conspiracy among statisticians to make this stuff harder to understand.
Theme and closely related categories:
- Approximation Theorems of Mathematical Statistics Description: Serfling's book provides all the mathematical theory needed to establish that something is asymptoticly normal. This book is for students who want more mathematical details.
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This webpage was written by Steve Simon on 2007-06-16, edited by Steve Simon, and was last modified on 2008-07-08. Send feedback to ssimon at cmh dot edu or click on the email link at the top of the page.