3F Counted data / 3G Relation among powers and logs (optional)

Chapter 3 · pages 102–111 · 1976 words

Counted data — things like numbers of deaths, warp breaks, or state populations — get treated with the same casual willingness to re-express as measured data such as heights or dollars. Many analysts feel free to take logs or roots of a length or a dollar amount, but balk at doing the same to a count of "how many times something happened." That hesitation is not principled; it is only a habit, and giving in to it risks missing patterns that re-expression would reveal. The two re-expressions that work best for counts, just as for measured batches, are logs and square roots. Special refinements come later for tricky cases — very small counts (especially when zero occurs), "how many out of N" survival/success counts, and counts laid out in structured patterns — but for ordinarily-sized counts, treat them exactly like amounts.

The state-population example makes the case vividly. Fifty raw population counts (in tens of thousands) sprawl out so unevenly that a stem-and-leaf of the raw values is nearly useless — a few huge states (California, New York) sit far off from the rest. Once you take logs, the display compresses into something readable, and a further "squeezed" version (using coarser leaf digits, keyed by t/f/s marks under each stem) makes the shape even easier to see:

LOGS (unit 0.01)         SQUEEZED (unit 0.1, for viewing)
 1  13* 6                 1*
 2  14  6                 t 3      Alaska
 4  15  29                f 455
 5  16  5                 s 67
 6  17* 9                1· 888888999
12  18  003338            2* 11         Ariz, Nebr
15  19  589               t 22223333
    20                    f 444445555
17  21* 15                s 6666666677
21  22  4557              2· 899
25  23  4478              3* 00         Ill, Pa
25  24  04589             t 22          Cal, NY
20  25* 1135
16  26  0004679
 9  27  18
 7  28  9
 6  29* 89
 4  30  05
 2  32  02

The deaths-by-cause data pushes the point further: fifty-nine causes of death, ranging from 17 deaths (polio) to hundreds of thousands (heart disease). Raw counts are unreadable as a batch; logs (in a condensed stem-and-leaf) finally give a "first clear picture," showing a pronounced trailing-off toward low values, a weaker trailing-off toward high values, and persistent asymmetry (lower values stretched out farther) even after trimming the extremes. Trying square roots instead only sharpens the impression that three causes — malignant neoplasm of the digestive organs, vascular lesions of the central nervous system, and arteriosclerotic heart disease — behave like disconnected "catch-all" categories, separate from a main mass of medium counts and a cluster of unusually small, mostly infectious-disease counts. So even an "unpromising" set of counts yields real structure once properly re-expressed.

The optional section on powers and logs steps back to ask how re-expressions relate to each other in general. Multiplying or dividing every value in a batch by the same constant — feet to inches, kilowatts to megawatts — is called a trivial re-expression: it changes nothing about how easy the data are to analyze or what conclusions follow, as long as you track what you did. Adding or subtracting a constant is also usually trivial, with one key exception: if the original numbers have a natural zero, shifting that zero means you should no longer take a power, root, or log of the result — except for small counts, where nudging "none counted" up to a small positive value specifically so a root or log can be taken afterward is a standard and useful move. Beyond scaling and shifting, the next simplest change is raising every value to a common power. Because instruments and habits already tend to spread out large values, re-expression usually moves the other way — toward compressing them — via square roots (the 1/2 power), reciprocals (the −1 power), and reciprocals of square roots (the −1/2 power). Lined up on a ladder of powers −1, −1/2, +1/2, +1, there's a conspicuous gap at the zero power, since anything to the zero power is trivially 1. That gap is filled, not by a power at all, but by the logarithm — logs behave as the natural "zero power" member of this ladder. Moving up the ladder (from x to x² to x³, or equivalently from −1/x² toward log x) stretches out differences among larger x values relative to smaller ones; moving down the ladder does the opposite, spreading out differences among smaller x values. In practice, most re-expression work stays within a narrower stretch of this ladder, from about +1 down to −1, rather than using the full range from +3 to −3.

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