The prism tally that made nine pines in ten look full-grown
Showing the misleading chart
A woodlot newsletter we invented charts its 216 tallies from a forester’s prism and finds 88% of them are adults, 30 cm or more across. Every diameter is real, from a classic stem map of 584 longleaf pines in southern Georgia. But a prism counts a tree from farther away the bigger it is, so its tally is weighted by each trunk’s cross-section. Count the trees on the map and 46.4% are adults, a quarter are saplings under 10 cm, and the average is 26.8 cm, not 45.5.
01The claim
A do-it-yourself prism cruise of a 4-hectare longleaf pine tract shows that nearly nine in ten of its pines are already adults, 30 cm or more across, the typical tree is 45 cm, and young pines are so scarce that the owner should plan for replacing an ageing stand.
02The trick
Every diameter is real, and the cruise was run by the book: 36 points on a fixed grid, a metric prism with a basal area factor of 2, every “in” tree tallied. The trouble is what one tally stands for. A prism, or angle gauge, counts a tree when you stand within a distance proportional to its trunk, 0.354 m per centimetre of diameter at this factor, so each tree has its own inclusion circle and a point lands in it with probability proportional to diameter squared. A 50-cm pine is “in” from anywhere in a 982 m² circle; a 10-cm pine only within 39 m², one twenty-fifth the area. The tally is therefore a sample weighted by basal area, which is exactly what the method is for: 6.0 tallies per point times the factor gives 12.0 m² of basal area per hectare, against 12.1 on the stem map. Read the same tallies as a sample of trees, though, and every class is weighted by its cross-section. That is how 88.0% of tallies (190 of 216) come out at 30 cm or more (the cutoff analyses of this stem map use for “adult”), the mean tallied tree is 45.5 cm, and the 154 saplings under 10 cm on the plot, more than a quarter of all its pines, yield one tally. Nor was the grid unlucky: averaged over every possible point in the same interior area, 85% of tallies would be adults and the mean tallied tree 43 cm. (Wiregrass Woodlot News and its advice are our invention, drawn to show the mechanism; the cruise is one we ran on the published stem map.)
03The fix
The redraw counts the 584 pines on the stem map, one tally each, beside the prism tallies on the same 10-cm classes. Of those pines, 271 (46.4%) are 30 cm or more, 154 (26.4%) are under 10 cm, and the mean diameter is 26.8 cm (median 26.2), not 45.5. Far from lacking young pines, the tract’s most crowded class is its smallest. A panel draws the inclusion circles to one scale, a 10-cm pine’s 3.5 m radius against a 70-cm pine’s 24.7 m, so the weighting is visible as area. A table shows the one figure on which the two counts agree, basal area per hectare, because that is the question a prism answers. The cruise can still describe trees if each tally is weighted by the number of trees per hectare it represents, the basal area factor divided by that tree’s own basal area. Done here, it gives 120 trees per hectare and a mean of 31.9 cm, against about 172 and 23.9 cm in the part of the plot the grid can reach. The reweighting is right on average, but its small-tree figure rests on a single tally: that one sapling stands for 9.8 saplings per hectare where the ground holds 48. That is why cruisers pair the prism with a small fixed-radius plot for small trees. (The grid’s interior square holds even more small trees than the plot as a whole, 37.7% adults against 46.4%, so comparing the tallies with the whole plot understates the gap.) When a sample was drawn with probability proportional to size, say so on the chart, and either reweight it or keep its claims to the quantity it was built to measure.