Characteristics of the Distribution of Public Libraries in the US, 2001

For this analysis, the same five variables listed on the Public Libraries in the US main page: BKVOL (book and serial volumes held), POPU_LSA (population of the library's legal service area), TOTINCM (total library income), TOTSTAFF (total library staff), and TOTOPEXP (total library expenditures) have been analyzed using all 9,133 PLDF3 libraries reporting in 2001 as well as the "1990" dataset for those cases where span is 'A'. To make the presentation of results more complex, the results in the following tables for PLDF3 and the subset of "1990" data present measures based first on the raw values, then present these same data for the log transforms of those five variables of these two datasets. Transforming values by taking their logarithms is a respected technique used both make the data more symmetrical and to provide more tools for analysis. Distributions with logarithmic transforms are commonly examined with skewed distributions.

PLDF3 Libraries in 2001 (raw values)

The skewness and kurtosis of a Gaussian ("normal") distribution are 0. Skewness is a measure of the shape of the distribution and this distribution is positively skewed meaning there are many small values and fewer very large values. Kurtosis is a measure of how sharp the peak is in a distribution. With the high peak in the small values the distributions for these variables have large values for kurtosis. These statistics, among others, for these values indicate that the distributions are not "normal." In fact, I have never seen a Gaussian, "normal," distribution in any library data. The arithmetic mean and 3rd Quartile figures indicate another important characteristic of these libraries: the large libraries are so large that they pull the arithmetic mean above the 75th percentile figure. See the discussion of "heavy tails" on the main page for another way of looking at this characteristic.

"1990" Libraries for 2001, Span = 'A' (raw values)

These libraries have a similar structure as the PDLF3 group.

What are the characteristics of logarithmic tranformations of these data?

In two of the variables the N changes between the raw and log values. The differences are libraries with 0 (zero) reported for those variables. The logarithm of 0 has no meaning.

The logarithm of a book or of dollar figures is not a notion one encounters every day but we are concerned not with the books or dollars in this kind of analysis but the underlying characteristics of the libraries--of all the libraries whose data we have--and what these data tell us about these libraries. What we see here is that the skewness and kurtosis have been reduced so a logarithmic transform makes the data more symmetrical--they are still not "normal," however. This result is about what is expected and with information from the correlations on the page with summary 2001 data, there are hints about underlying structures that will be dealt with in time.

May 20, 2004