3E Looking quickly
Chapter 3 · pages 98–102 · 980 words
Rather than building several stem-and-leaf displays just to see which re-expression symmetrizes a batch, this section offers a shortcut based on a property all our standard re-expressions share: order preservation. Roots, logs, and negative reciprocals never swap the relative order of two values — if 57 is bigger than 43, then log 57 is still bigger than log 43. Because order is preserved, so are ranks and depths: the median, hinges, eighths, and extremes computed from re-expressed values are simply the re-expressions of the corresponding raw summaries. This means you don't need to re-sort a whole re-expressed batch — you can just re-express the handful of summary numbers you already have.
This licenses a fast diagnostic: compute midsummaries, the averages of paired summary values at the same depth. The midhinge is (lower hinge + upper hinge)/2, the mideighth is (lower eighth + upper eighth)/2, and the midextreme (elsewhere called the midrange) is (lower extreme + upper extreme)/2. If a batch is symmetric, these midsummaries should all sit close to the median; if they drift steadily away from the median as you move outward toward the extremes, that drift reveals skewness and points toward the re-expression needed to remove it. You just look at a short sequence like median, midhinge, midextreme, or the longer median, midhinge, mideighth, midextreme, and check whether the numbers trend up, trend down, or stay flat.
Applying this to the volcano-heights example, only one stem-and-leaf (of the raw heights) is needed. The 5-, 7-, and 9-number summaries are each re-expressed as roots, logs, and negative reciprocals, and their midsummaries computed. Roughly:
(raw) (root) (log)
M 65 80 181
midH 66h 80h 177h
midE 72h 79h 173
midD 78 79 167h
mid1 100h 77 130
(trend) (up) (close) (down)The raw values trend up, the logs trend down, but the roots stay close to level across the whole sequence from median to midextreme — a strong, clear signal to re-express volcano heights by square root. The midextreme, though, is flagged as the least trustworthy of these: because it comes from only the two most extreme values, a single stray observation can throw it off. For that reason, it's often better to build the diagnosis from a 7-number or 9-number summary rather than relying on the 5-number summary and its midextreme alone. Midsummaries also attach naturally onto the letter-value displays already in use, adding a mid column and a spread column alongside each set of paired letter values.
With small batches the signal is rarely as crisp as it is for the 219 volcano heights, so it helps to combine evidence across several batches. Tippett's warp-break counts illustrate this: six small batches (AL, AM, AH, BL, BM, BH, 9 counts each) each give a 7-number summary. Summed across all six batches (or across the five excluding the somewhat irregular AL batch), the raw-count midsummaries clearly increase outward from the median, while the same midsummaries computed on square roots of the counts stay roughly constant — again pointing to the square root as the re-expression that produces symmetry. Pooling many small batches this way lets a trend emerge that no single small batch could show convincingly on its own.
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