# Formally speaking, is converting some variable into a per capita measure a transformation?

When I convert some data for a model (e.g., wealth, pollution) into per capita figures (e.g., wealth per inhabitant, tons of pollution per inhabitant) is this a transformation or re-expression of the data? Or is the term transformation or re-expression reserved for logarithmic or exponential transformations (including roots and inverses)?

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To supplement @John's reply, a re-expression is given by a mathematical function $f$ of a single variable. The re-expressed values of a dataset $(x_1,x_2,\ldots,x_n)$ are, by definition, $(f(x_1), f(x_2), \ldots, f(x_n))$. This precludes, for instance, normalization of one quantity by values of another. Formally, such normalization changes bivariate data $((x_1,y_1), \ldots, (x_n,y_n))$ into $(x_1/y_1, \ldots, x_n/y_n)$. Mathematically that can be considered a "transformation," but in the statistical literature, "re-expression" applies only to univariate transformations. –  whuber Nov 20 '11 at 17:16
@whuber +1. This explains why John is right in a way that's both rigorous and intuitively clear. BTW, rabidotter, if you consider John to have answered your question, you should click the check next to his answer. –  gung Nov 20 '11 at 21:42