2G Eighths, sixteenths, etc. / 2H How far have we come?
Chapter 2 · pages 72–76 · 1489 words
Beyond the five-number summary, you can dig deeper into a batch by continuing the same halving-of-depths process that produced the hinges. Recall that the hinge depth was found by taking the median's depth, dropping any trailing "h", adding 1, and halving. The section extends this: once you have the depth of the hinges, you compute the depth of eighths as (1 + depth of hinges)/2, again dropping any "h" before dividing. Repeating the same rule on the eighths' depth gives the depth of sixteenths, and so on. The general rule is simply: new depth = ½(1 + integer part of old depth). Each round of halving cuts the batch roughly into finer and finer fractions — quarters (hinges), eighths, sixteenths, thirty-seconds — hence the section's title.
These new values need tags the same way M (median) and H (hinges) do. The convention is to call the eighths E, then continue down the alphabet — D for sixteenths, then C, B, A, Z, Y, and so on as needed. Collectively M, H, E, D, ... are called letter values, a term that is often stretched to include the extremes too. Adding E's to a 5-number summary gives a 7-number summary (extremes, hinges, E's, median); adding D's as well gives a 9-number summary; continuing further yields 11- and 13-number summaries.
A worked example makes the pattern concrete. For the 50 state populations (1960, units of 10,000 people), the seven-number summary looks like this:
50 populations
M25h | 246
H13 | 89 432 | 343 (H-spread)
E7 | 63 782 | 719 (E-spread)
1 | 23 1678 | 1655 (range)Here the median depth is 25h, the hinge depth is 13, and the eighth depth is 7 — each obtained from the previous by the halving rule. Adding a ninth-number line (D4, depth 4) with values 39 and 1008 extends this to a nine-number summary, and its own spread (969) is called the D-spread. Just as the H-spread is hinge-to-hinge and the range is extreme-to-extreme, each new pair of letter values defines its own named spread (E-spread, D-spread, etc.), which later chapters (notably chapter 19) put to use.
The chapter then closes with a review of everything covered so far in "How far have we come?": ranks are counted in from either end of an ordered list, the smaller of a value's two ranks is its depth, and half-integer depths point to the mean of the two neighboring values. The median sits at depth (count+1)/2; each hinge sits at half of (1 + integer part of median's depth). The extremes-to-extremes difference is the range, the hinge-to-hinge difference is the H-spread. A schematic summary displays the count and depths outside an enclosure and the values inside, often with spreads noted under an "eave" to the right. The step — hinge to first fence, or first fence to second fence — equals 1.5 times the H-spread; values beyond the second fence are far out, beyond the first fence are outside, and the closest values still within the first fence are adjacent. A schematic plot is a box-and-whisker plot that shows all outside values individually (far-out ones with extra emphasis) and whose whiskers stop, dashed, at the adjacent values. The overriding lesson is that no exploratory analysis is complete without something visual to look at — a stem-and-leaf, a letter-value display, or a schematic/box-and-whisker plot — because words alone often cannot convey what a display shows.
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