40 %
average CV of one tree on its own
What you are looking at. On page 3, the biomass of one tree was uncertain by about ± 40 %. Here we add up 25 example trees, like the ones measured in a Miyawaki plot. Each tree is still ± 40 %. But the total of the 25 trees is uncertain by only about ± 12 %. Why? Because random errors partly cancel when you add many trees: one tree comes out too high, another too low, and the total ends up close to the truth.
But not all errors cancel. Some errors are shared by many trees: the same wood-density value is used for every tree of a species, and the same equation is used for all 25 trees. If the equation is 1 % too high, every tree is 1 % too high, and the total is also 1 % too high. Shared errors do not cancel. This page lets you watch both effects.
Each box gives the CV (coefficient of variation) of that source: the SD as a percentage of the value. For a 6.0 m tree, CV 10 % means SD = 0.6 m. "Per tree" = a different error for every tree. "Per species" = one error shared by all trees of that species. "All trees" = one error shared by all 25 trees. The last box is different: σ is the SD of the scatter on the log scale, not a CV; σ = 0.357 is a CV of about 37 %, which we round to 40 %.
Changing any control restarts the animation and recalculates panels A, C and D.
Where are the buttons? Step, Play, Fast-forward, Reset and New random numbers are in Panel B below, next to the chart they drive.
| # | Species | D (cm) | H (m) | ρ (g/cm³) | AGB point (kg) |
|---|
Here each tree is simulated separately, as on page 3, with 5,000 draws. The trees are sorted from biggest to smallest. Orange dot = point value from the equation. Green whisker = 95 % interval. Right column = CV % (relative error). Every tree has a CV near 40 %, big or small: the relative error of one tree does not depend on its size.
Panel B does for 25 trees what this box does for three. Read this once, slowly, and Panel B will be easy. The three trees are three rows from the example table above, and they are three different species. (The table prints 3.99 kg for tree #1; here it is rounded to 4.0 kg.)
Each tree gets its own weight from the same equation. The total of the plot is the three weights added together. That is all "adding up trees" means.
On page 3 we saw that one tree's weight from the equation has a CV of about 40 %, almost all of it from the scatter die (a real tree is not on the line). Written as an SD in kg, and as a 95 % interval (from ÷ 2 to × 2):
The bigger tree has the bigger SD in kg, although all three have the same 40 %. This is what Panel A showed: on a log scale every whisker is the same length.
A round is one possible world. In a round, every tree gets its own roll of the scatter die. The roll is a multiplying factor: a factor of 1.4 means "this tree is really 40 % heavier than the equation says", a factor of 0.7 means "30 % lighter". We multiply each tree's weight by its own factor, then add the three results. (To keep the arithmetic short, this box rolls only the scatter die. The page rolls all five.)
Round 1: tree #1 went up by 40 %, tree #14 went down by 30 %, tree #8 did not move. One up and one down: they cancelled, and the total is only 1 % away from 20.0 kg. Round 4: the same thing, tree #1 far down and tree #14 far up, total 2 % away.
Round 2: all three went up. Round 3: all three went down. Nothing cancelled in these rounds, so the total moved a lot, by 18 % and 22 %. This can happen, but it is not common: for three independent dice to all go the same way is like three coins all landing on the same side, all heads or all tails: about 1 round in 4. With 25 trees it is like 25 coins all landing on the same side, which almost never happens. That is why the total of many trees stays close to the point total.
Now do this 20,000 times and write down every total. Panel B builds exactly such a pile, for 25 trees. The middle of the pile is a little above 20.0 kg, near 21 kg (the note under Panel B explains why), and its spread is the uncertainty of the plot total.
We do not need 20,000 rounds to guess the answer. When the errors are independent (every tree rolls its own die), the SDs in kg combine by the same rule as on page 3:
So three trees added together are known to 25 %, better than any one of them at 40 %. Not as good as the textbook 40 % ÷ √3 = 23 %, because that rule is for three equal trees. Here tree #8 is half the total, so its die matters most, and the plot behaves like 2.6 equal trees, not 3. Panel C shows the same thing for 25 trees: both curves, blue and green, stay above the dashed √n line.
So far every tree rolled its own die. Some errors are not like that. The equation (a and b) is the same for all three trees. If it is 1 % too high, all three trees are 1 % too high in the same round. They move together, never one up and one down. There is nothing to cancel.
For a shared error the total moves by the same percentage as each tree. It does not shrink by the square-add-root rule, and it does not shrink however many trees you add. Add 1,000 trees that all use the same equation, and the 1 % is still there. That is why the green curve in Panel C never reaches zero: the shared part is a floor.
Wood density is shared too, but only within a species. Our three trees are three different species, so each uses its own density value from the table, and their density errors are independent, like the scatter: they cancel. On the full page it is different: six of the 25 trees are T. bellirica and all six use the same value 0.62. If that value is 10 % too high, all six go up by 10 % together. In Panel B, if you switch off the H, D and scatter dice and leave only ρ on, you can watch the six T. bellirica trees move in the same direction in every round.
One round = one possible "world". In each round, all 25 trees get their random errors at the same time. The green dots move to the new values (the grey tick on each row is the tree's point value). The 25 values are added up, exactly as in Step 3 of the three-tree example: total of the round = AGB₁ + AGB₂ + … + AGB₂₅, each with its own rolled errors. The sum drops into the histogram of plot totals below. Trees of the same species share the same ρ error. With all dice on, the big scatter die hides this; switch off the H, D and scatter dice to see the six T. bellirica trees move together, in the same direction.
Add the trees one at a time, in the order of the table, and watch the CV % of the running total fall. Horizontal axis: how many trees have been added so far. Vertical axis: CV % of their total. The grey dashed line is 40 % ÷ √n, the textbook rule for n trees that are all equal in size and have independent errors.
Field plots are small. Reports give results "per hectare" (10,000 m²). To convert, divide by the plot area and multiply by 10,000. This keeps the same percentage uncertainty. The numbers get bigger, and the interval gets bigger with them.
Area = number of trees ÷ density. Change either the density or the area; the other one is updated. If you change the number of trees away from 25, the total is multiplied by (trees ÷ 25). This assumes the 25 measured trees are typical of the others.
Carbon = 0.47 × biomass (dry wood is about 47 % carbon). CO₂ = 3.67 × carbon (44 ÷ 12, the weight of CO₂ made from one unit of carbon).