Measurements of Scale: Security Metrics

A fun discussion arose today around security data analysis and the value of data by statistical scale (nominal, ordinal, interval and ratio). One of the funny thing we discussed about applying real metrics to our space was that people often think of certain familiar pieces of data, like ip address or mac address, as higher scale data-points, because they are superficially numerical, when they are only nominal in scale. Yes, that means they're just strings! You generally can't even imply the ordinal scale of an ip address, because it could very well be non-unique (at least in our type of dataset it is).

As these discussion tend to go, we then verged from stats to set theory as we attempted to clear up the usage of "ordinality" from "cardinality" as found in a recent book on security metrics. For those without the desire to follow the links, I will grossly simplify. Ordinality has a special purpose in describing infinite sets as "well ordered" (like the set of all natural number), though it typically can refer to a set of numbers or sets that are in order. Cardinality refers to the size or number of elements in a set.

Let's go back to Cantor, shall we? If you're still reading, checkout Cantor's Theorem and Cantor's Paradox. Here you can read about Aleph and it's use in talking about the cardinality of infinite sets.