3C Comparisons of two batches

Chapter 3 · pages 83–88 · 1070 words

When you want to compare two batches side by side, stem-and-leaf displays can be arranged with a shared set of stems and a leaf column on either side. Two layouts are common: a side-by-side stem-and-leaf, where each batch keeps its own stem column, and a back-to-back stem-and-leaf, where a single stem column sits in the middle and the leaves of one batch grow leftward while the leaves of the other grow rightward — with the leaves optionally sorted so each row reads like an ordinary ordered stem-and-leaf reversed. Comparing county land areas of Michigan (83 counties) and Mississippi (82 counties) this way shows that Michigan's areas tail out much further toward large values, while a big cluster of Mississippi counties bunches tightly together. A tiny illustration of the back-to-back idea, leaves sorted:

   Michigan          Mississippi
        2 |  3* |
       75 |  3. | 8
     9865 |  4* | 0122334
      432 |  4. | 55667889

Re-expressing the same land areas as logs and redoing the back-to-back display makes the comparison sharper rather than just prettier: the Michigan pattern becomes a "dinosaur" shape (a blunt cluster with a long tail), while Mississippi's pattern becomes roughly rectangular. Crucially, the gain here isn't only that each single batch becomes more symmetric — the two batches' spreads become much more similar once logged, so that any remaining difference in shape between the two states can be described more simply. This illustrates a general goal: when smoothing stem-and-leafs, it's reasonable to aim for symmetry, but symmetry is a nice-to-have whose main value often comes from side benefits, not from being the real point.

A sharper case is the 1927 atomic-weight data for hydrogen and bromine. The raw values live on totally different scales (hydrogen near 1.008, bromine near 79.9), so a stem-and-leaf with one common unit is useless for comparing them directly — one batch's digits swamp the display. Taking logs helps, but here even five-decimal logs are not fine enough to show the batches' internal spread; six-decimal logs are needed. Once that finer log unit is used, a single comparative stem-and-leaf with a common unit (the sixth decimal place) works well for both elements simultaneously, revealing that the bulk of each batch is spread about equally, while also flagging one or two straying values in each. This works because measurement error for a physical quantity tends to be a roughly constant fraction of the measured amount (matching the idea of a fixed number of significant figures), and working in logs is a better match to "constant fractional error" than working with a fixed number of decimal places in the raw units.

The bigger lesson is that re-expression can help a two-batch comparison in (at least) two distinct ways: it can make each batch's own pattern more symmetric, or — often more importantly — it can bring two batches' spreads onto a matching scale so a single common stem-and-leaf unit works for both. The symmetry gain is the one whose real value tends to come from its byproducts; the scale-matching gain is the one that most directly rescues an otherwise-impossible comparison.

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