I blow hot and cold on the levels of measurement stuff discussed recently on EdStat-L. On the one hand, you DO need to pay attention to what kind of data you have in deciding what technique to use. I think of the wonderful example in Brase and Brase where they ask the students to do a chi-square test on a table in which the cells give weights of nuts rather than counts (so you can get any level of significance or nonsignificance you want by simply changing the units of weight). On the other hand, there are other circles where "levels of measurement" seems to have become an esoteric religion, quite contrary to science in its outlook on the world.
With my own students, I like to distinguish between what I call categorical and measurement data, which I present as two ends of a continuum. Each technique presented comes with some clue as to the type of data it is intended for. In the one-variable case, we have
for measurement data
means, medians, modes (maybe), variances, ranges, standard deviations, quantiles, stem and leaf, histogram, boxplot, etc.
for categorical data
counts, percentages, barcharts, pie charts, etc.
The ideas are even more important when we look at relationships between variables.
For relationships between two measurement variables
regression and correlation
For relationships between two categorical variables
chi-squared tests on contingency tables
For a measurement variable depending on a categorical variable
analysis of variance
While this is only a first approximation as to what to use when, most textbooks DO NOT GIVE THIS INFORMATION! (For some ideas on why, see A. Toom's article in the August MAA Focus.) But, while it is important to give some guidelines, it is also important to understand that there are many exceptions. Perhaps an extreme one is to use an ordinary t-test with data placed in two categories and coded 0-1. In most cases, you will get the same result as if you had applied the binomial distribution, and there are good underlying reasons why this is so. Yet we are applying a method from one end of the spectrum to data from the other.
_ | | Robert W. Hayden | | Department of Mathematics / | Plymouth State College | | Plymouth, New Hampshire 03264 USA | * | Rural Route 1, Box 10 / | Ashland, NH 03217-9702 | ) (603) 968-9914 (home) L_____/ email@example.com fax (603) 535-2943 (work)