Abstract
Stratum boundaries at equal steps of cumulative root frequency come close to optimal, while equal counts do worst (Paper 957). Subsampling within plots and differences in measuring cost were not treated there. Splitting the timber-volume distribution of Paper 957 (log-normal with log mean 4.5 and log standard deviation 0.8) into a plot effect and a tree effect, this paper takes $n$ plots and $m$ trees per plot and finds, by formula, exhaustive search and random sampling, the $m$ that minimises the variance of the mean under a cost per plot and a cost per tree. No new theorem or law is claimed.
First, the best number does not depend on the budget: with plot-effect share 0.1 and plot cost 20, the formula gives $\mathbf{15.8518}$ trees, and exhaustive search gave 16 trees for budgets of 1000, 2000 and 5000 alike. In all four combinations the best was the same for the three budgets (Section 2).
Second, measuring twice or half as many costs the same: at twice and at half the best number the variance was the same $\mathbf{1.123327}$ times the best (share 0.1, cost 20), and $1.064604$ to $1.123327$ times across the four combinations (Section 3).
Third, the size of the loss is set by one share: with $a$ the share of the plot cost taken by the trees’ total cost at the best, the loss agreed within $10^{-12}$ with a formula in $a$ and the factor missed alone. Measuring one tree gave $\mathbf{4.432163}$ times and measuring all 50 gave $1.362887$ times (share 0.1, cost 20) (Section 3).
Fourth, random sampling follows the formula: the variance of the mean over ten thousand fresh draws from the population was $0.983340$ to $\mathbf{1.033172}$ times the formula in six settings (Section 4).
Fifth, the separator is whether the variation lies between plots or within them: raising the plot-effect share from 0.1 to 0.3 lowered the best number at cost 20 from $15.8518$ to $\mathbf{8.0440}$ trees. The sum of the two variances stayed at $13776.529821$ (Section 5).
Keywords: two-stage sampling, survey sampling, optimal subsample size, survey cost, variance decomposition, forest inventory.
Scope (what is not claimed)
No new theorem or law is claimed: that in two-stage sampling the variance of the mean is the between-plot variance over the number of plots plus the within-plot variance over the total number of trees, and that under costs the best number is the square root of the product of the cost ratio and the variance ratio and does not depend on the budget, are known in sampling theory. These are set values: the volume distribution is the same log-normal as Paper 957; plot-effect shares of 0.1 and 0.3, plot costs of 20 and 100, a tree cost of 1 and budgets of 1000, 2000 and 5000 are set here. They correspond to no forest or species. Sampling is simplified: the population is taken as infinite; finite-population corrections, combination with strata, costs depending on travel distance between plots, and varying the number per plot are not included. No measuring plan is recommended: how many trees a real survey should measure is not treated. Relation to earlier papers: Paper 938, in its scope note, did not treat two-stage sampling that further subsamples within plots, and Paper 957, in its scope note, also did not treat subsampling within plots or different measuring costs by stratum; this paper sits in the seat of subsampling within plots (no earlier occupant; Paper 957 sat in the stratum-boundary seat of the same scope note of Paper 938). Paper 957 showed that stratum boundaries at equal steps of cumulative root frequency come close to optimal while equal counts do worst. Here, with the same distribution, the relation between how many trees to measure per plot and cost is counted. What is added: finding the best number by formula and exhaustive search for four combinations of costs and variances and comparing three budgets, confirming that the loss from missing it matches a formula in one share, comparing random sampling with the formula, seeing how far the best number moves when the variances are estimated from one forest, and placing the separator on “variation between plots, or within”.
1. Notation and the Question
Setting: tree volume is $V=\exp(4.5+b+e)$, with plot effect $b$ and tree effect $e$ normal with mean 0 and variances $0.64s$ and $0.64(1-s)$ ($s$ is the plot-effect share, 0.1 or 0.3). The variance of the true plot means (between-plot variance) and the mean within-plot variance are
$$S_b^2=e^{9+0.64(1-s)}\,e^{0.64s}\bigl(e^{0.64s}-1\bigr),\qquad S_w^2=e^{9+1.28s}\,e^{0.64(1-s)}\bigl(e^{0.64(1-s)}-1\bigr)$$With $n$ plots and $m$ trees per plot, the variance of the mean and the number of plots under budget $B$, plot cost $c_1$ and tree cost $c_2$ are
$$\mathrm{Var}(m)=\frac{S_b^2}{n}+\frac{S_w^2}{n\,m},\qquad n=\frac{B}{c_1+m\,c_2},\qquad m^*=\sqrt{\frac{c_1\,S_w^2}{c_2\,S_b^2}}$$where $m^*$ minimises the variance. $c_1=20$ and $100$, $c_2=1$, and exhaustive search covered $m=1$ to $200$.
Question: how many trees per plot is best, how does that change with the budget, and how much is lost by missing it?
2. The Best Number
Computation 2.1: true variances and the best number.
| share $s$ | between-plot variance | within-plot variance | plot cost | formula best | exhaustive (budgets 1000, 2000, 5000) |
|---|---|---|---|---|---|
| $0.1$ | $1015.664610$ | $12760.865212$ | $20$ | $\mathbf{15.8518}$ | 16, 16, 16 |
| $0.1$ | $1015.664610$ | $12760.865212$ | $100$ | $35.4458$ | 35, 35, 35 |
| $0.3$ | $3252.813592$ | $10523.716229$ | $20$ | $\mathbf{8.0440}$ | 8, 8, 8 |
| $0.3$ | $3252.813592$ | $10523.716229$ | $100$ | $17.9868$ | 18, 18, 18 |
The ratio of within- to between-plot variance was $12.564054$ for share 0.1 and $3.235266$ for 0.3. In all four combinations the exhaustive best was the same for the three budgets and was the formula’s best rounded down or up.
Observation 2.2 the best number does not depend on the budget. $\mathrm{Var}(m)$ can be written $(S_b^2+S_w^2/m)(c_1+m c_2)/B$, and the budget $B$ only divides the whole: raising the plot cost from 20 to 100 raised the best number for share 0.1 from $15.8518$ to $35.4458$ trees.
3. The Loss from Missing It
Computation 3.1: dividing the variance at $k$ times the best number by the best variance gives, with $a=m^*c_2/c_1$,
$$\frac{\mathrm{Var}(k\,m^*)}{\mathrm{Var}(m^*)}=\frac{1+a\,(k+1/k)+a^2}{(1+a)^2}$$which takes the same value for $k$ and $1/k$.
| share $s$ | plot cost | $a$ | twice | half | one tree | all 50 |
|---|---|---|---|---|---|---|
| $0.1$ | $20$ | $0.792592$ | $\mathbf{1.123327}$ | $1.123327$ | $\mathbf{4.432163}$ | $1.362887$ |
| $0.1$ | $100$ | $0.354458$ | $1.096606$ | $1.096606$ | $7.467578$ | $1.023093$ |
| $0.3$ | $20$ | $0.402198$ | $1.102280$ | $1.102280$ | $2.261785$ | $1.895303$ |
| $0.3$ | $100$ | $0.179868$ | $\mathbf{1.064604}$ | $1.064604$ | $3.072804$ | $1.147238$ |
In all four combinations the values for twice and half agreed with each other and with the formula within $10^{-12}$.
Observation 3.2 the loss is symmetric in the logarithm of the factor by which the number is missed. The larger $a$, the larger the loss for the same factor (at twice, from $1.064604$ for $a=0.179868$ to $1.123327$ for $a=0.792592$): the losses from measuring one tree and from measuring all 50 are set by how far each is from the best as a factor and by $a$; for share 0.1 and cost 100, one tree gave $7.467578$ and all 50 gave $1.023093$.
4. Checking with Random Sampling
Computation 4.1: with plot cost 20 and budget 2000, the number of plots was $n=\lfloor 2000/(20+m)\rfloor$, and plot and tree effects were drawn fresh from the population to form the mean ten thousand times.
| share $s$ | trees per plot | plots | sampled variance | formula | ratio |
|---|---|---|---|---|---|
| $0.1$ | $1$ | $95$ | $148.239353$ | $145.016103$ | $1.022227$ |
| $0.1$ | $16$ | $55$ | $32.920832$ | $32.967612$ | $0.998581$ |
| $0.1$ | $50$ | $28$ | $45.067825$ | $45.388640$ | $0.992932$ |
| $0.3$ | $1$ | $95$ | $142.843287$ | $145.016103$ | $0.985017$ |
| $0.3$ | $8$ | $71$ | $66.476279$ | $64.341945$ | $\mathbf{1.033172}$ |
| $0.3$ | $50$ | $28$ | $121.628233$ | $123.688854$ | $0.983340$ |
In six settings the sampled variance was $0.983340$ to $1.033172$ times the formula. For both shares the variance at the best number was smaller than with one tree and with all 50.
5. The Separator
Observation 5.1 the separator is whether the variation lies between plots or within them. Raising the plot-effect share from 0.1 to 0.3 lowered the best number from $15.8518$ to $8.0440$ trees at cost 20 and from $35.4458$ to $17.9868$ at cost 100: the sum of the two variances was $13776.529821$ for both shares, and the formula variance with one tree was the same $145.016103$ for both. What changed was only how the variation divides between and within plots.
Observation 5.2 estimated from one forest, the best number moves. Estimating the variances from one forest of 400 plots with 50 trees each gave, for share 0.1, a between-plot variance of $925.245704$ and a within-plot variance of $12609.537335$, and a best of $16.5096$ trees at cost 20 (exhaustive 17): one tree away from the true $15.8518$ (16). For share 0.3 it was $8.3518$ (8).
Observation 5.3 what is not said. Finite-population corrections, combination with strata, costs of travel between plots, varying the number per plot, and how many trees a real survey should measure are not treated. What is shown is that, for the set distribution and costs, the best number does not depend on the budget and the loss from missing it follows a formula in one share.
6. Roll Call and Closing
- variance formula for two-stage sampling (Section 1): between- and within-plot variance.
- best $\mathbf{15.8518}$ trees (share 0.1, cost 20) (Computation 2.1): exhaustive 16 for all three budgets.
- the same $\mathbf{1.123327}$ at twice and half (Computation 3.1): the loss is a formula in $a$ and the factor.
- $\mathbf{4.432163}$ with one tree (Computation 3.1): far from the best as a factor.
- 0.983340 to $\mathbf{1.033172}$ of the formula by random sampling (Computation 4.1): as the formula says.
- between plots, or within (Observation 5.1): the separator; the scope notes of Paper 957 and Paper 938.
Closing: The best number of trees to measure per plot ignores the budget, and doubling or halving it costs the same. Splitting the volume distribution of Paper 957 into plot and tree effects, with plot-effect share 0.1 and plot cost 20 the best was $15.8518$ trees, and exhaustive search gave 16 for budgets of 1000, 2000 and 5000: twice and half the best gave the same $1.123327$ times the variance, and the loss matched a formula in the share of the plot cost taken by the trees’ total cost and the factor alone. Random sampling gave $0.983340$ to $1.033172$ times the formula. The separator is whether the variation lies between plots or within them. Placed among the earlier papers: to the forest samples Paper 957 and Paper 938 counted by stratum boundaries and allocation, the subsampling within plots and costs that neither treated were added. To be honest. This is one distribution with an infinite population, comparing a variance formula with random sampling. What sets the best number and how much is lost by missing it, and no further.
References
- Cochran, W. G. (1977). Sampling Techniques. Wiley.
- Yamagishi, Y. (2026). Stratifying a Forest Removes Between-Stratum Differences from Sampling Error, and Allocating More Plots to Variable Strata Reduces It Further [Paper 938]. YOI: https://www.yamagishiyuuki.com/yoi/938
- Yamagishi, Y. (2026). Stratum Boundaries at Equal Steps of Cumulative Root Frequency Come Close to Optimal, While Equal Counts Do Worst [Paper 957]. YOI: https://www.yamagishiyuuki.com/yoi/957
Reference
Yamagishi, Y. (2026). The Best Number of Trees to Measure per Plot Ignores the Budget, and Doubling or Halving It Costs the Same [Paper 980]. YOI: https://www.yamagishiyuuki.com/yoi/980
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 980. This paper is a preprint and has not been peer reviewed.