2D Fences, and outside values / 2E Schematic plots
Chapter 2 · pages 62–68 · 1193 words
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)Exhibits and original pages (6)
