1F How to count by tallying / 1G What does it mean to "feel what the data are like"?

Chapter 1 · pages 35–39 · 981 words

When a stem in a stem-and-leaf display starts collecting more than about 20 leaves, the rows get cramped and hard to count at a glance. At that point it pays to switch, for that overloaded stem, to a tallying scheme instead of writing every leaf digit out.

The obvious tally-by-fives (single strokes bundled in fives, with a diagonal slash closing each group of five) is widely used but the author calls it treacherous for fast work: it is easy to miscount and end up with four or six strokes in a bundle instead of five. The recommended tally-by-tens scheme builds up a small square in stages so that each stage is visually unmistakable:

 4 dots :  : .        (four dots placed anywhere around a square, order doesn't matter)
 8       : a completed box (all four sides drawn in, any order)
10       : the box with both diagonals crossed

Because closing the four dots into a square, and later closing the four sides, are visually crisp events, there's no tendency to drift into "5" after 3 dots or "11" after the box is done — unlike five-bar-gate tallying. A few beginners forget to cross the diagonals, leaving an "8" where a "10" was meant, but the shapes stay distinguishable when the tally is later converted into digits.

A worked example of small counts turned into this tally notation, then displayed as a miniature stem-and-leaf, looks like this:

count   tally
  1       .
  2       : (two dots)
  3       three dots
  4       four dots (square outline of dots)
  5       one side of the box drawn in
  6,7     more sides filled
  8       box complete (all four sides)
  9       box plus one diagonal
 10       box with both diagonals crossed
 27       two full tens-boxes plus a box (10+10+7)
 42       four full tens-boxes plus a "2" mark

This scheme also works two-dimensionally: counts of, say, boys and girls in each of four class years can be tallied directly into a small grid of cells, each cell filled in with its own dots/box/diagonals, giving a compact two-way tally table without ever writing a two-digit number until the very end.

The used-car example revisited: model years of cars advertised, split into "ads for one or two cars" versus "ads for three or more cars," were tallied by tens next to their digit counts. Comparing the two forms side by side, tallies are read faster than the equivalent printed numbers — enough that it can be worth converting an already-digited count back into tally form just to look at it. Reading the tallied display revealed two things a table of raw counts would have hidden less obviously: small ads tend to advertise somewhat older cars, and 1955 model cars are unexpectedly popular in these ads — later traced partly to two of the three '55s being advertised specifically as "classic Thunderbirds."

Tallying-by-tens, though effective, gives less detail than stem-and-leaf, since it only records a count, not the individual leaf digits. The author's advice is to treat it as an overflow procedure — used only for the specific stem or subset that gets too crowded for stem-and-leaf, not as a general replacement. Often overflow can be avoided altogether by collecting data in separate parts (for instance splitting by "1-2 cars" vs "3+ cars") that stay small enough individually for full stem-and-leaf treatment, while still being combinable later if desired.

Section 1G turns to a broader question: what should a stem-and-leaf display show us? Most of the batches seen so far were "well-behaved" — nothing dramatic. But the heights of the highest point in each of the 50 U.S. states, when put into a stem-and-leaf, visibly split into three separate clusters: Alaska standing alone at the extreme high end, a middle cluster of the Rocky Mountain and Pacific states (California through Washington) plus Hawaii, and then all the remaining, generally lower-elevation states. This is offered as an example of "feeling what the data are like" — letting the shape of the display itself suggest structure, rather than starting from a preconceived expectation.

The section lists the kinds of features a stem-and-leaf display typically thrusts on your attention without effort: apparent separation into distinct groups, trailing off unevenly in one direction more than the other (asymmetry), unexpectedly popular or unpopular individual values, roughly where the values are centered, and roughly how spread out they are. The display's shape alone reveals the first three directly; the last two — center and spread — need a bit more deliberate technique, which the book promises to cover soon. One practical tip: when a display shows an apparent "break" or gap between groups, it's worth explicitly noting the specific values that sit right at the edges of that break, not just the extreme values overall.

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