3H How far have we come? / 3K How to think about logs (background) / 3P Additional problems
Chapter 3 · pages 111–116 · 1484 words
Section 3H steps back and sorts the data types we re-express into three families. Amounts and counts (tonnage, lake areas, river lengths) usually need re-expression whenever the largest value is much bigger than the smallest — the rule of thumb is that if the ratio of biggest to smallest is close to 1, re-expression barely matters, but once that ratio grows, logs are almost always the right tool, though roots or other powers sometimes help instead. Balances — quantities that are already differences or can go negative — are re-expressed less often, and if they are re-expressed at all it has to happen before the balance is formed, not after; note also that a log of anything is automatically a balance, since it can be negative. Counted fractions, percentages, and grades almost always need re-expression, but the book postpones the actual machinery for them to later chapters.
The break tables introduced earlier are now placed in a pecking order of "commonest" re-expressions: logs first, then square roots, then negative reciprocals, then negative reciprocal square roots. All are meant to be quick, few-digit jobs — "efficient laziness" — with the understanding that we can always come back later and redo a promising re-expression with full precision. Plain counted data (not tiny counts) get treated like amounts: going to roots rarely hurts, and going all the way to logs often pays off. A useful mental shortcut is to think of the log as standing in for the "zeroth power" in the ladder of re-expressions (square, none, root, log, reciprocal root, reciprocal, ...).
Two practical techniques round out the toolkit. Starting means adding the same constant to every count (or to any amount that can be zero) before re-expressing; this is optional when counts are all large, but becomes essential when some counts are zero and logs are going to be taken, since the log of zero is undefined. Midsummaries — averages of paired values equally far from the extremes — can be calculated and their trend examined; a trend in the midsummaries reveals asymmetry more delicately than simply eyeballing a stem-and-leaf display. The section closes by framing re-expression as a tool for the eye: at this stage its main payoff is a more symmetric, easier-to-grasp display, with other benefits still to come, and it reinforces the chapter's theme that taking a firm hand with data before displaying or calculating is often the best or only sound approach.
Section 3K is a reference list of background facts about logarithms, meant to be recalled rather than derived. Logs can be looked up in tables; logs "to" different bases differ only by a multiplicative constant, so for data-analysis purposes any base is essentially as good as any other, though the book writes log₁₀ explicitly when the base 10 matters and will later use logₑ (or ln) for natural logs. Four defining facts follow: the log of 1 is 0; the log of zero is undefined; the log of a negative number exists but is not a real number (the book avoids using it, sometimes writing "L" — read as "lower than any number" — as a placeholder instead); and the log of a product is the sum of the logs of its factors (log uvw = log u + log v + log w), while the log of a ratio is the difference of the logs (log(s/t) = log s − log t). Finally, log₁₀ of 10 is 1, of 100 is 2, of 1000 is 3, of 0.1 is −1, of 0.01 is −2, of 0.001 is −3, and so on — each power-of-ten step in the number shifts the log by exactly one unit.
Section 3P is a problem set (exhibits 20–22) giving practice batches — port tonnages and state lake areas; ocean and sea areas and river lengths; waterfall heights and dam reservoir capacities — each posed as a stem-and-leaf display spanning a huge range of magnitudes. For example, a tiny illustrative batch of tonnages like 8,9,14,18,24,33,58,66,70,127,207,460,771,826 would need a stem-and-leaf with stems marked at successive powers of ten (8, 9, 1, 2*, ...) to keep the display compact, exactly the situation that signals a log re-expression is called for:
8* | 4,9
9* | 6
1** | 63,65,68,71,78
2 | 86
3 | 18,63,64,71,87
4 | 33,58
5** | 66
6 | 24
8 | 14,70
9** | 09The problems ask the reader to decide, for each panel, what re-expression is indicated, build the corresponding stem-and-leaf, and comment — direct practice applying the logarithm-first, then-root, then-reciprocal ordering from 3H to real data whose values span several orders of magnitude.
Exhibits and original pages (5)
