Exploratory Data Analysis
This opening section sets up the guiding metaphor for the whole book: exploratory data analysis (EDA) is detective work. Just as a detective looks for clues, the data analyst looks for numerical, counting, or graphical clues hidden in data. The section insists that doing this well takes two separate ingredients — tools and understanding — and that neither alone is enough.
The detective analogy is spelled out concretely. A detective without fingerprint powder simply cannot find fingerprints, no matter how sharp his instincts; a detective without a sense of where a criminal is likely to have touched things won't know where to dust for prints even if he has the powder. In the same way, an analyst needs concrete techniques (tools) to extract information from numbers, and also a feel for where in the data the interesting information is likely to be hiding (understanding). The book promises to supply some of each, though only for a deliberately narrow selection — a few of the most general and powerful simple tools, not necessarily the objectively "best" ones, since the author doubts a unique best even exists.
The section then draws a distinction about the limits of "understanding." Detective skill needed in a London slum, a Welsh village, among Parisian aristocrats, in the cattle-raising American west, or in the Australian outback are all different — a Scotland Yard officer would not automatically excel at trailing cattle thieves, nor a Texas ranger at working the streets of Birmingham. Likewise, data about earthquakes, chemical manufacturing, firm profits, human hearing, suicide rates, population growth, fossil dinosaurs, fruit-fly genetics, or molecular biology each demand their own specialized, detailed understanding — far more than one book could teach across all fields.
Yet the analogy has an optimistic side too: a Scotland Yard detective, dropped into the wild west or the outback, would still not be useless, because he carries general understandings of conventional detective work that transfer anywhere. The book's ambition is to hand the reader the data-analytic equivalent of that transferable general understanding — a handful of broadly useful habits of mind that work across earthquake data, business data, biological data, and beyond, even without field-specific expertise.
This section sets out the small arithmetic habits that make hand analysis of data manageable. First, precision is a matter of judgment rather than a fixed rule: sometimes you keep extra decimal places, sometimes you round, depending on how the rest of the data behaves. If nearby values are close together, more precision matters; if they are far away, a rounded answer loses nothing.
When rounding is done, ties (numbers ending in exactly .5) are broken by rounding to the nearest even number rather than always up. So 17.5 becomes 18, but 18.5 also becomes 18, while 19.5 becomes 20. This differs from cutting, where you simply drop the extra digits without adjusting the remaining ones — cutting is faster but less accurate. The choice between the two is pragmatic: cut when more detail is available nearby if you need it later, round when the data matters enough that you want a fair, careful value.
For writing numbers down, a single number is clearest when both its raw units and decimal point are kept (like 0.00127 inches). But once several numbers must be compared, decimal points become a nuisance because they disrupt visual alignment and make comparison harder. The fix is to choose convenient units so the numbers become clean integers — expressing population in hundreds of thousands rather than in millions with decimals, for instance — and to label the table or panel with the unit used, such as "unit = 100,000 people," so the reader knows how to interpret the bare digits.
Finally, the section introduces small notational shortcuts used throughout the book: "h" stands for ".5" or "and a half" (so counting by halves goes 4, 4h, 5, 5h, 6...), **"*" is a place-holder digit used when cutting a number to indicate unknown or insignificant trailing digits (always by cutting, not rounding), and "#"** is shorthand for "count" or "number." These conventions let numbers be written compactly without pretending to a precision the data doesn't have.
flowchart LR
A[Raw number] --> B{Need to compare many numbers?}
B -- No, single number --> C[Keep units and decimal point]
B -- Yes --> D[Pick convenient units]
D --> E[Drop decimal points]
E --> F[Label panel with unit]
A batch of numbers is simply a set of similar values, however they were gathered — the weights of 21 students, ten years of winter snowfall, mite counts on 49 rats. Just writing them down as a raw list is a start, but it does not let you feel their shape "as a whole." The first improvement is a tally: for the number of cars listed in 18 used-car ads (6,7,7,3,5,7,3,11,16,4,17,17,6,7,9,4,7,5), you list every possible value from 3 to 17 down the page and mark each occurrence next to its value:
3 | X X
4 | X X
5 | X X
6 | X X
7 | X X X X X
8 |
9 | X
...
16 | X
17 | X XA small refinement replaces each "X" with the actual last digit of the number being tallied — this "digit-reminder" form gives the eye a bit more to grasp and doubles as a check that each mark landed on the right line:
3 | 33
4 | 44
5 | 55
6 | 66
7 | 77777
9 | 9
11 | 1
16 | 6
17 | 77To make bold-face entries (used elsewhere to distinguish kinds of entries) work on paper rather than in print, the book recommends two colors where practical (a four-color ballpoint is handy), or pen-for-bold/pencil-for-not-bold as a fallback, or, as a last resort, circling entries — more work and less clear.
Tallying breaks down once the range of values is too wide relative to the count of values — cutting used-Chevrolet prices from \$150 to \$1895 in single-dollar steps would need 1746 lines to display only 17 prices. The fix is to let each line absorb a range of possible values instead of just one: lines can be grouped by twos, threes, or tens. When you condense ten possible raw values onto each stem, and abbreviate the label, you get what the book calls a simple stem-and-leaf display — for the car-count data, condensing by tens gives two lines that already show the whole batch:
0* | 33445566777779
1* | 1677In this display each line is a stem, and each digit written after the stem's label is a leaf; the label itself (the digits shared by every value on that line) is the starting part. One-digit leaves are preferred when they suffice, though two- or more-digit leaves are sometimes worthwhile. Every stem-and-leaf display is expressed in some unit — for Chevrolet prices, using a \$10 unit turns \$250 into "25", shown as stem 2\* with leaf 5. Values that don't divide evenly (like \$1099) can either be cut (truncated, e.g. to 109) or rounded (e.g. to 110); the book leaves the choice open here, as it did earlier. Comparing the same 17 Chevrolet prices at three unit sizes — \$100 (a plain check-off tally), \$10 (one-digit-leaf stem-and-leaf), and \$1 (two-digit-leaf stem-and-leaf) — shows that all three convey much the same overall picture, detail increases from the coarsest to the finest, and the cost of that extra detail is small going from \$100 to \$10 units but can be more noticeable going from \$10 to \$1 units. That's why the \$10-unit or \$1-unit style (or both together) is usually the sensible choice.
Because errors made while scratching down a stem-and-leaf are easy to make and painful to find later, checking matters — but not too much checking, or you spend your time debugging the checks themselves. The minimum recommended check is simply confirming the number of leaf-entries equals the number of original values, kept track of with a "(check count)" column. To count many leaves on one stem quickly, the book recommends covering the leaf-string in blocks of three digits at a time with your forefinger, counting "3, 6, 9, ..." as you go, then adding any digits left uncovered at the end. If the count comes out wrong, you re-walk the original data placing a dot over each leaf already verified correct, to isolate the mistake.
Choosing how to split a value between stem and leaf is not automatic: sometimes one digit's worth of stem gives too few lines (values crammed together), while moving the split one digit further gives too many lines (values spread too thin). Section 1E works through several ways to adjust the number of stems to fit the batch.
The stretched stem-and-leaf doubles the number of lines per starting part: one line carries leaves 0-4 and the next carries leaves 5-9, with the "*" tag shown only on the line holding leaves 0 and 1. Applied to Mississippi county land areas (rounded to tens of square miles), this produces a much clearer picture than a single line per starting part would, and lets a few distinctive counties (like Hinds, or Bolivar and Yazoo) be labeled right on the display:
3*|8 (1)
4*|0121243121300214202 (19)
4.|597886556569 (12)
5*|142010 (6)
5.|977899958797 (12)The squeezed stem-and-leaf goes the other way, using five lines per starting part instead of one line per digit, tagged * (zero/one), t (two/three), f (four/five), s (six/seven), and . (eight/nine) — a scheme that works because English digit names pair up neatly in fives. A batch running from 118 up through 330 can then be laid out compactly:
1*|1 (1)
t|2333 (4)
f|445555 (6)
s|66677 (5)
.|88 (2)
2*|0000011 (7)A third technique, mixed leaves, is used when the aim is mainly to collect values rather than scrutinize their shape, and especially when a few values trail far from the bulk of the batch. Here the split between starting part and leaf can change partway through the display — typically at a multiple of ten — and every such change is marked both by a change in the number of asterisks and by a blank line. A value can be handled two ways: for looking at, only the first digit or two of a long leaf is kept (1345 becomes 1|3, meaning "somewhere in the 1300s"); for storage, the entire leaf is kept (1345 becomes 1|345). The looking-at form sacrifices exact recovery of the original number in exchange for a picture; the storage form sacrifices compactness in exchange for being able to reconstruct every original value exactly. Multi-digit leaves in the storage form are separated by commas so their length stays unambiguous, and this style is well suited to batches running from many small values up to a few very large ones — values often pile up on and just after each power of ten, sometimes called the "abnormal law of large numbers."
Mixed-leaf displays can also mix leaf lengths within one display and handle negative and positive values together, which requires both a "+0" stem and a "-0" stem so that small positive and small negative values are kept apart. In one worked example, one-digit leaves are used from -59 up through +59, while two-digit leaves take over from -60 downward and +60 upward, letting the whole batch fit into 16 lines rather than the 31 that a uniform digit-width would have required.
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 crossedBecause 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" markThis 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.
This section closes the chapter with a summary, then adds an optional extra technique. The summary reminds us that every tool introduced so far — stem-and-leaf, tallying — exists to do one of two jobs: give a feeling for how a batch behaves as a whole, or store the values conveniently for later use. Sometimes, the author notes, you need one display for insight and a different one for storage, and there is no shame in making both. The chapter also names the whole enterprise: exploratory data analysis is "detective work," finding and revealing clues, while confirmatory data analysis (not covered in this book) goes further and assesses how strong the evidence is.
The summary also collects the practical lessons about stem-and-leaf itself: every display, however made, should at least be checked against the total count (overdotting makes this check easy for stem-and-leaf); leaves can be one digit, two digits, or mixed, chosen for convenience or necessity, and sometimes two styles are worth making side by side; and when you have many values and can sacrifice detail, tallying-by-tens is the efficient choice. The things a detailed display can reveal are named explicitly: separation, asymmetry, irregularities, centering, and width. The chapter then makes the case for why batches of similar numbers need special techniques at all — a single number like 46,930 motor-vehicle deaths, or a short table of death rates (trains 0.09, planes 0.16, buses 0.17, cars-turnpike 1.2, cars-general 2.3 per hundred million person-miles) can be grasped directly, but a whole batch of many similar numbers needs a display to be grasped "firmly or rapidly."
The optional section 1K then shows that stem-and-leaf's basic principle — the stem locates where you write, the leaf says more — can carry richer information than a single digit. Working from newspaper ads for 18 used-car sales, the model year becomes the stem and a coded letter or digit for the car's make becomes the leaf, so each entry is really a pair (year, make). Suggestive letters are used for makes where possible (F for Ford, C for Chevrolet), stepping out to digits only when letters run short, and capital versus small letters are kept distinct only where handwriting would naturally distinguish them (a book convention, not a rule of the technique). Organizing the leaves alphabetically within each stem year then exposes repeated name-year combinations at a glance — most strikingly, fifteen '65 Chevrolets show up as a run of identical leaves:
'63 | b B C C C F F I I I L O I C C b P
'64 | A B B C C C D D g h I I I M O P P r s 1 3 4
'65 | C C C C C C C C C C C C C C C
'66 | f t v h 1 C v 3 B 4 h F I v A C O h m P J F F C p 3Once you see leaves as carrying a code rather than just a digit, the same trick extends to storing pairs of numbers rather than a number and a category. The section illustrates this with 50 pairs — a logarithmic bank-deposit figure and a governor's salary — where the stem is the leading digits of the deposit figure and each leaf is itself a pair of numbers, such as stem 10 carrying leaves (4, 574) and (6, 556), which together decode to the full pairs (104, 574) and (106, 556). This shows that stem-and-leaf is not limited to single batches of plain numbers: the stem-plus-leaf idea generalizes to any situation where you want a compact, checkable way to store linked pieces of information.
This closing section of Chapter 1 gives practice problems and then steps back to explain why the whole chapter's technique matters and what comes next. The three problems ask you to build stem-and-leaf displays from real scientific data — capillary-rheometer viscosity measurements, radial velocities of stars near Alpha Persei, and repeated measurements of a plutonium complex-ion association constant — and then to comment on the shape each display reveals. This is the same skill built up through 1C–1K: turn raw numbers into a stem-and-leaf picture, then read off its pattern.
The General comment that follows the problems reflects on the purpose of stem-and-leaf: it lets us scratch down a batch quickly and, more importantly, lets us see the batch's overall pattern. When a pattern is unusual — such as the three distinct groupings seen in Exhibit 10, or the asymmetric tailing-off seen in Exhibits 4 and 5, where the display stops more abruptly at the small end than the large end — we should describe that unusualness in words rather than expect any standard numerical summary to capture it.
This leads to the book's larger agenda: because we cannot expect a single "standard summary" to capture unusual features, future chapters will build a small set of agreed numbers — with agreed names, symbols, and standard ways of writing and plotting them — that summarize the typical characteristics of a batch's pattern. These summaries deliberately have limited ambition: they are meant to reveal what is more or less expected, not the exceptional. The chapter argues that summaries are useful and often necessary for large amounts of data, but they can never substitute for the full stem-and-leaf detail, which remains the tool for catching the unexpected.
The section closes with review questions asking why we summarize a batch at all, how completely we can expect summaries to succeed, how broad their aims can be, and how important full detail remains — questions that bridge directly into Chapter 2's topics of ranges, hinges, five-number summaries, and schematic plots, previewed here by the reproduced chapter index.
This section picks out the first few "easily-found numbers" that will summarize a whole batch: the two extremes (the highest and lowest values) and a middle value, the median. The point is that all of these should be locatable just by counting through a stem-and-leaf display, no matter whether the batch has 8, 82, or 208 values — you never need to add anything up, just count stems and leaves.
Counting is done by assigning a rank to every value: starting at one end, call the first value's rank "1", the next "2", and so on. You can count from either end. Counting from the top down (so the largest value is rank 1) is ranking down — the natural sense of "he was first in his class." Counting from the bottom up (so the smallest value is rank 1) is ranking up. Whichever direction gives a value its smaller rank is called that value's depth; an extreme value always has depth 1, since it is rank 1 from its own end no matter how you count. Here is the Chevrolet-price example worked both ways:
Upward Downward
rank Prices rank Depth
1 150 17 1
2 250 16 2
3 688 15 3
4 695 14 4
5 795 13 5
6 795 12 6
7 895 11 7
8 895 10 8
9 895 9 9
10 1099 8 8
11 1166 7 7
12 1333 6 6
13 1499 5 5
14 1693 4 4
15 1699 3 3
16 1775 2 2
17 1895 1 1Notice the general property visible in every row: upward rank plus downward rank always equals 1 plus the total count (here 1 + 17 = 18). This is not a coincidence of this example — it holds for any batch, and it is exactly what lets you find the middle of a batch without knowing which end you started from.
The median, marked "M", is the single middle value when the count is odd, or the mean of the two middle values when the count is even. Since upward rank + downward rank = 1 + count for every entry, the median — the value equally far from both ends — must sit at rank ½(1 + count) counted from either direction. For the 17-price batch that is rank 9 = ½(1+17), a whole number, so the median is simply the 9th value. But for a six-value batch like 7, 12, 18, 20, 27, 54, ½(1+6) = 3.5, which falls between the values ranked 3 and 4 (18 and 20); the median is their mean, ½(18+20) = 19. To handle this smoothly the book allows interpolated ranks (and correspondingly interpolated depths) in half-integers — writing "3h" for "three and a half," using "h" to mean "and a half" — so that the single formula depth of median = (1 + count)/2 works whether the count is odd or even. Occasionally ranks or depths in quarters are used for other quantities, but these are treated as rare exceptions; halves are the normal unit needed for medians.
Having found the extremes and the median (2A), we now add two more landmark values by repeating the same halving trick one level down: instead of folding the whole batch in half to find the median, we fold each half again, counting halfway from each extreme in to the median. The two values this produces are called hinges, because they sit at the points where the batch would fold if you creased it first at the median and then again at each half. Together the two extremes, the two hinges, and the median make up a 5-number summary.
Finding a hinge's depth follows the same discard-half-then-add-one rule used for the median, but applied to the half-batch. If the median's depth is a whole number, treat it (ignore any fraction) and find the hinge depth as half of (that whole number plus one). For a batch of 13, the median depth is 7 (since ½(1+13)=7); the hinge depth is then ½(1+7)=4, so each hinge is the 4th value in from its end. For a batch of 20, the median depth is 10h (a half-integer, ½(1+20)=10.5); to get the hinge depth we drop the "h", getting 10, then compute ½(1+10)=5h — so each hinge sits at depth 5h, interpolated between the 5th and 6th values in.
These five numbers are recorded compactly in a letter-value display, an arrangement that lists, to the left of an inverted-U bracket, the depths (marked # for the total count, M for the median's depth, H for the hinges' depth, and an unmarked 1 for the extremes, since their depth is always one), and inside the bracket the corresponding values, median centered, one side's values above the other's — which side is "upper" and which "lower" is arbitrary. For example, a batch of 13 values might fold into:
#13
M7 1.5
H4 0.1 3.0
1 -3.2 9.8Here the count is 13, the median (depth 7) is 1.5, the hinges (depth 4) are 0.1 and 3.0, and the extremes (depth 1) are -3.2 and 9.8.
The hinges are useful because comparing hinge-to-median distances against extreme-to-hinge distances reveals asymmetry in a batch's trail-off. In one worked example the hinges were 45 and 69 — each exactly 12 away from the median of 57, perfectly symmetric — while the extremes 38 and 94 sat only 7 units past the lower hinge but 25 units past the upper hinge, exposing a much longer trail toward high values that the hinges alone had hidden. This is also why a stem-and-leaf display benefits from cumulative counts written in from each end (and a separate count for the line holding the median): once you know, say, that 38 values lie at or below a certain line and that the median's rank is 41h, you can count inward from the nearest known cumulative total instead of tallying the whole batch, using the display's own digits to read off the median and hinges directly.
A box-and-whisker plot is the picture version of the 5-number summary from section 2B. You draw a long thin box running from the lower hinge to the upper hinge, cross it with a bar marking the median, and then extend a "whisker" from each end of the box out to the corresponding extreme. In one glance you see spread (the box), center (the bar), and range (the whisker tips), laid out vertically along a measurement scale — for example, comparing the heights of the 50 states' highest points against the heights of 219 volcanoes side by side, both scaled in feet from 0 to 20,000.
Height
(feet)
20,000 - o <- extreme (e.g. Alaska)
|
15,000 - |
|
10,000 - [ ] [====] <- box: hinge to hinge
[===] <- median [====]
5,000 - [ ] [====]
| |
0 - o o <- extreme
States VolcanoesBecause a box-and-whisker plot spends so much ink on empty space compared to just listing the five numbers, that empty space is not wasted — it is room for identification. Rather than leaving the extremes as bare dots, you can label them with the case they belong to (Alaska at the top, Delaware and Florida at the bottom, for instance), and you can identify a handful of other unusual points near the ends too. When you do this, the recommended practice is to stop the whisker at the innermost point you've bothered to identify, rather than running it all the way to the single most extreme value — the whisker then marks the identified range, not just the numerical extreme.
This section also uses box-and-whisker plots to launch a broader lesson on plotting technique. The key distinction is between what makes plotting easy and what makes a plot effective: a grid of ruled lines on ordinary graph paper helps you place points accurately but clutters the final picture and hides the pattern you're trying to see. The fix is to plot on tracing paper (or acetate) laid over a sheet of well-printed graph paper slipped underneath — you plot using the grid showing through, add your reference lines and a few scale ticks, then lift away the graph sheet so the finished plot is uncluttered. Good graph paper for this purpose has clean lines with every 10th line heavy and every 5th line medium; cheap "five and dime" or quadrille paper makes the job harder. On a finished plot only three or four scale values should be shown per axis, though while actively plotting it helps to have many more marked, and to put the horizontal scale above the plot rather than below so your hand doesn't block it. In a pinch, with no graph paper at all, you can improvise a grid by placing one sheet of ruled paper sideways beneath another sheet of ruled paper, letting the lines of the bottom sheet show through the top one.
Hinges do more than build the 5-number summary: they let us define a rule of thumb for flagging values that stray unusually far from the bulk of the batch. The key quantity is the H-spread, the difference between the two hinges. Multiplying the H-spread by 1.5 gives the step. Measuring one step outward from each hinge gives the inner fences; measuring two steps outward (equivalently, one step beyond the inner fences) gives the outer fences. Any actual data value that sits between a hinge and its inner fence is simply part of the ordinary batch. The most extreme such value on each side — the one closest to, but still inside, its inner fence — is called adjacent. A value that lies between the inner and outer fence is outside, and a value beyond the outer fence altogether is far out.
These fences are not drawn from the data as separate marks — they are computed numbers used only to classify the extremes and any suspicious values. A convenient bookkeeping layout, the fenced letter display, writes the H-spread in a small box just outside (above) the letter-value display, and the step in a second small box ("penthouse") above the fence calculations that follow. For example, for a batch of 34 "ultimate power" values with hinges 33 and 162 (H-spread 129, step 194), the fences work out to inner fences at about -161 and 356, and outer fences at about -355 and 550. Sorting the raw values against these fences produces a report like this:
34 ultimate powers
M17h | 73 |
H9 |33 162|129
1 |15 1974|
adj: 15(Boysen), 285(Hungry Horse)
out: 379(Shasta), 424(San Luis)
far: 900(Glenn Canyon), 1345(Hoover)
far: 1974(Grand Coulee)Alongside fences, the box-and-whisker plot also suggests two further numbers worth recording. The range is just the top extreme minus the bottom extreme — trivial to compute since both extremes are already in the 5-number summary, and worth writing down routinely, typically to the right of the box on the extremes line. The H-spread is usually the better single measure of spread to quote, since the range is heavily influenced by the two most extreme (possibly stray) values; but quoting both together is often informative. Separately, when we want a single "typical value" that reflects the visual impression of centering given by both the median and the hinges together, we use the trimean: (lower hinge + 2×median + upper hinge)/4. The trimean can substitute for the median almost anywhere the median would be used.
Section 2E turns these fence classifications into a plotting convention: the schematic plot. In a schematic plot, outside and far-out values are always shown as individually plotted, identified points rather than absorbed into a whisker. The whiskers themselves are drawn dashed, ending in dashed crossbars exactly at the adjacent values (not at the true extremes, unless an extreme happens to be adjacent). Far-out values get an emphatic mark and are labeled in CAPITAL letters; outside and adjacent values are labeled in small letters, dropped only if there are so many (more than six at one end) that labels would clutter the picture. Abbreviated labels are fine when they will still be understood. Because schematic plots always use dashed whiskers by this rule, a solid-line whisker signals instead a "free-form" box-and-whisker plot not built by these fence rules. A worked example uses the 83 Michigan county areas: hinges 54.5 and 78.5 give adjacent values at Benzie (32) and Gogebic (111), outside values at Delta, Iron, Schoolcraft and Ontonagon, and far-out values at Chippewa (158) and Marquette (184) — plotted individually and capitalized, with dashed whiskers running from the box to the Benzie/Gogebic crossbars:
Area
(sq mi)
1800 | o MARQUETTE
|
1500 | o CHIPPEWA
|
| o Ontonagon
1000 | Iron o o Schoolcraft
| Gogebic-o Delta
| : [box]
500 | : [box]
| :
| Benzie- - (dashed whisker/crossbar)Lord Rayleigh's 1893-94 measurements of the weight of a standard volume of "nitrogen" prepared from different sources became a famous test case for schematic plots, because they show both what these plots hide and what they reveal. When all 15 weights are lumped into a single batch, the values actually split into two tightly clustered subgroups — a low cluster near 2.299-2.300 and a high cluster near 2.310 — with nothing in between. A dot plot of the batch shows this gap immediately, since every individual value is visible:
Dot plot (single batch of 15)
2.310 | xxx
| x
|
2.305 |
|
2.300 | x
| x
| xxx
| xA schematic plot of that same single batch, by contrast, only draws a box from hinge to hinge and whiskers out to the adjacent values, so the two-cluster structure is almost entirely swallowed up — the median line and box sit in the empty middle, and only unusually short whiskers relative to the box give an alert viewer a hint that something is being hidden. The lesson is that schematic plots are not built to reveal structure near the center of a batch, such as gaps or clusters; for that job a dot plot or a stem-and-leaf display is needed instead.
Where schematic plots do their real work is comparing batches. When Rayleigh's 15 weights are split by their true grouping — 7 weights "from air" versus 8 weights "from other sources" — and each subgroup gets its own schematic plot side by side, the enormous separation between the two batches jumps out, along with secondary features: a slight upward straggle in the "other sources" batch, and a hint that this batch is also more variable. This comparison is exactly what let Rayleigh recognize that air-derived nitrogen differed from chemically-derived nitrogen — a discrepancy that led directly to the discovery of the new element argon. The general principle this illustrates is that aims must drive the choice of plot: a plot can show general behavior (as the two-batch schematic comparison does beautifully) or fine detail near the center (as dot plots or stem-and-leaf do), but usually not both at once, so the analyst must decide which one a given picture needs to deliver.
The Rayleigh example also warns against a common piece of graphical advice: many books on graphical presentation insist that a vertical scale must start at zero, so that bar heights faithfully represent relative size, and treat any convention of exaggerating a scale as a "journalistic" or "political" trick. But if Rayleigh's two batches had been drawn as bar graphs starting at zero, the tiny but real difference between them would have been squashed into invisibility, making the two bars look almost identical — a picture that is neither evidence for a new chemical element nor a basis for a Nobel prize, even though it might be perfectly adequate for someone only interested in freight charges for compressed gas. Zooming the scale to enlarge small but real differences is therefore not dishonest but a vital scientific tool, and schematic plots — used well, with an appropriate scale — earn their place as the standard comparison display precisely because they focus attention where it matters and cut down visual "busyness" enough to let many batches be compared side by side, something stem-and-leaf displays cannot do once more than two or three batches are involved.
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.
This section names the single most useful re-expression tool: the logarithm, shortened throughout to log. Building on the earlier observation (from the power-plant stem-and-leaf in chapter 1) that batches of always-positive numbers whose values are not too close together tend to have their leaves crowded into the low digits, the book argues that taking logs is the first re-expression to try, with square roots as the second choice. The key property that makes logs useful is that equal differences in logs correspond to equal ratios of the raw values — so taking logs turns products into sums and ratios into differences, which is why logs help whenever people work with ratios or products (price indexes, growth rates, and so on). Whenever a kind of log is needed other than base 10 (common logarithms), the text promises to flag it specially; otherwise "log" always means base-10.
The worked example re-does the Bureau of Reclamation hydroelectric-plant data. In raw megawatts the leaves were washed into the corner where leading digits are 1 or 2; taking logs spreads the batch out and makes it far more symmetric. A compressed version of that log stem-and-leaf, unit 0.1, looks like this (leaves are the second digit after the decimal point of the log):
depth | stem | leaf
1 | 1* | 1 (smallest plant)
5 | t | 2333
11 | f | 445555
16 | s | 66677
(2) | . | 88
16 | 2* | 0000011
9 | t | 23
7 | f | 445
4 | s | 6 San Luis
3 | . | 9 Glenn Canyon
2 | 3* | 1 Hoover
1 | t | 3 Grand CouleeBoth the extremes, hinges, and median are reported in both logs and the original megawatts side by side, because when working with a re-expressed batch it is good practice to show letter values in both scales — this keeps the "feel" for the raw units while doing the analysis in the more manageable, more symmetric log scale.
Section 3B turns to the practical question of how to get logs fast without a full-precision table or a calculator's worth of decimals. Two-decimal logs — and often even one-decimal logs — are precise enough for almost any re-expression job; the guiding rule is to use two-decimal logs first, and only fall back to many-decimal logs in the rare case where more precision is truly needed. To make two-decimal logs fast, the book introduces the break table (what others might call a critical table): a table that tells you which of a small number of possible log values a raw number's leading digits falls between, so you just find where your number lies between two "breaks" and read off the log's fractional part directly, with no interpolation. When a value lands exactly on a break, the convention is to round to an even answer.
An example applies the method: for the Corps of Engineers hydroelectric data, a value beginning "14" falls between the breaks 1396 and 1429, so its two-decimal log ends in .15; a value beginning "18" falls between 1799 and 1841, giving a log ending in .26. Applying this systematically to the whole batch turns a raw stem-and-leaf whose leaves pile up at low leading digits into a log stem-and-leaf that is longer and more "gangling" but gives, as the text puts it, a far more honest picture of how the values are spread out — confirming that the raw display, useful mainly for storage, gave little help for visualization compared to the log version.
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. | 55667889Re-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.
Square roots can be pulled from a break table much as logs were in section 3B, but the geometry is less forgiving. A log table can be reused every time x changes by a factor of 10; a root table only repeats itself when x changes by a factor of 100, since sqrt(100·x) = 10·sqrt(x). Worse, a plain two-decimal root table has wildly uneven step sizes: going from sqrt(100)=10 to sqrt(121)=11 is a jump of 21 in the argument for one unit of root, while going from sqrt(9604)=98 to sqrt(9801)=99 needs a jump of about 197 — roughly ten times as many raw numbers — to move the root by the same one unit. A well-built break table compensates by using closer-together breaks near the low end (near 11) and more widely spaced breaks near the high end (near 98), so that each row of the table advances the root by a controlled amount; that is exactly the design behind the root break table (its "exhibit 7").
Quick reciprocals raise a different problem: ordinary reciprocals reverse order. Since 3 > 2 but 1/3 < 1/2, working directly with 1/x scrambles a batch's ranking. The fix is to use negative reciprocals: -1/3 is indeed more positive than -1/2, so -1/x preserves the original ordering. Because raw values like 57 are inconvenient to invert directly, it's handy to work with -1000/number instead, and a break table (called "exhibit 8" in the book) is built for exactly that quantity — you read off the nonzero digits from the main table and fix the decimal point from a companion panel, the same two-step lookup style used for logs and roots.
A worked example: to find -1000/43.7, look at the digits 437 in the break table; it falls between two adjacent break entries (4347 and 4425), giving nonzero digits -228, and the decimal-point panel converts that to -22.8.
The volcano heights example shows why you'd bother with roots at all. The raw heights of 219 volcanoes (in hundreds of feet) trail off strongly toward large values:
STEM-and-LEAF, unit = 100 feet (rounded), heights of 219 volcanoes
0 | many low leaves (short stem, few counts)
...
5*| many leaves (heavily populated middle)
...
19 | 39730 <- long climb up to the tallest volcanoesTaking logs (as in section 3A/3B) overcorrects and now trails the other way, toward low values, so logs are too strong a pull. Taking square roots instead lands in between and produces a nicely symmetric stem-and-leaf, roughly:
STEM-and-LEAF, units 0.1 of sqrt(height in hundreds of feet)
1*| 4
2 | 65525
12 | 44488
13*| 2266
14 | 0000with a median near 6500 feet and extremes near 200 and 19,900 feet — a batch that finally looks even-handed rather than skewed. Comparing the three schematic plots (raw, log, root) side by side makes the improvement even more obvious than reading the three stem-and-leaf displays does, because a schematic plot shows symmetry or skew at a glance through the position of the median and the lengths of the whisker-like fences.
The last topic is reciprocal times. When timing an event — a rat running a maze, a fish stopping swimming, a house collapsing in an earthquake — some trials never finish. A rat that never finishes is really just an extreme case of slowness, much closer to a rat that takes 1000 seconds than to one that finishes in 4 seconds; yet on the raw time scale "never" sits infinitely far from everything. Working with 1000/time (or its negative, -1000/time, to preserve order as "more positive means slower") turns "never" into a perfectly ordinary value near zero, and turns very long times into unremarkable small speeds. Data from a rat-running study illustrate this: raw times for four rats show erratic, seemingly very different amounts of spread, with some rats looking wildly variable and one looking oddly compressed; converting to reciprocal times (1000/seconds) produces a much more consistent pattern of roughly symmetric, similarly sized spreads across the rats, with only one rat genuinely standing out as unusually spread out. That changed impression — driven purely by choice of scale — is the section's point: how we look at numbers can matter enormously. When data include both "never" and literal "zero" values, the recommended fix for the zero is to replace it with half the smallest nonzero value that could have been recorded, while also complaining about how the data were collected.
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.
Counted data — things like numbers of deaths, warp breaks, or state populations — get treated with the same casual willingness to re-express as measured data such as heights or dollars. Many analysts feel free to take logs or roots of a length or a dollar amount, but balk at doing the same to a count of "how many times something happened." That hesitation is not principled; it is only a habit, and giving in to it risks missing patterns that re-expression would reveal. The two re-expressions that work best for counts, just as for measured batches, are logs and square roots. Special refinements come later for tricky cases — very small counts (especially when zero occurs), "how many out of N" survival/success counts, and counts laid out in structured patterns — but for ordinarily-sized counts, treat them exactly like amounts.
The state-population example makes the case vividly. Fifty raw population counts (in tens of thousands) sprawl out so unevenly that a stem-and-leaf of the raw values is nearly useless — a few huge states (California, New York) sit far off from the rest. Once you take logs, the display compresses into something readable, and a further "squeezed" version (using coarser leaf digits, keyed by t/f/s marks under each stem) makes the shape even easier to see:
LOGS (unit 0.01) SQUEEZED (unit 0.1, for viewing)
1 13* 6 1*
2 14 6 t 3 Alaska
4 15 29 f 455
5 16 5 s 67
6 17* 9 1· 888888999
12 18 003338 2* 11 Ariz, Nebr
15 19 589 t 22223333
20 f 444445555
17 21* 15 s 6666666677
21 22 4557 2· 899
25 23 4478 3* 00 Ill, Pa
25 24 04589 t 22 Cal, NY
20 25* 1135
16 26 0004679
9 27 18
7 28 9
6 29* 89
4 30 05
2 32 02The deaths-by-cause data pushes the point further: fifty-nine causes of death, ranging from 17 deaths (polio) to hundreds of thousands (heart disease). Raw counts are unreadable as a batch; logs (in a condensed stem-and-leaf) finally give a "first clear picture," showing a pronounced trailing-off toward low values, a weaker trailing-off toward high values, and persistent asymmetry (lower values stretched out farther) even after trimming the extremes. Trying square roots instead only sharpens the impression that three causes — malignant neoplasm of the digestive organs, vascular lesions of the central nervous system, and arteriosclerotic heart disease — behave like disconnected "catch-all" categories, separate from a main mass of medium counts and a cluster of unusually small, mostly infectious-disease counts. So even an "unpromising" set of counts yields real structure once properly re-expressed.
The optional section on powers and logs steps back to ask how re-expressions relate to each other in general. Multiplying or dividing every value in a batch by the same constant — feet to inches, kilowatts to megawatts — is called a trivial re-expression: it changes nothing about how easy the data are to analyze or what conclusions follow, as long as you track what you did. Adding or subtracting a constant is also usually trivial, with one key exception: if the original numbers have a natural zero, shifting that zero means you should no longer take a power, root, or log of the result — except for small counts, where nudging "none counted" up to a small positive value specifically so a root or log can be taken afterward is a standard and useful move. Beyond scaling and shifting, the next simplest change is raising every value to a common power. Because instruments and habits already tend to spread out large values, re-expression usually moves the other way — toward compressing them — via square roots (the 1/2 power), reciprocals (the −1 power), and reciprocals of square roots (the −1/2 power). Lined up on a ladder of powers −1, −1/2, +1/2, +1, there's a conspicuous gap at the zero power, since anything to the zero power is trivially 1. That gap is filled, not by a power at all, but by the logarithm — logs behave as the natural "zero power" member of this ladder. Moving up the ladder (from x to x² to x³, or equivalently from −1/x² toward log x) stretches out differences among larger x values relative to smaller ones; moving down the ladder does the opposite, spreading out differences among smaller x values. In practice, most re-expression work stays within a narrower stretch of this ladder, from about +1 down to −1, rather than using the full range from +3 to −3.
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.