3P Sampling

Point-3P Sampling

Extending 3P Sampling To Large Areas

Published July 1995

3P sampling was developed by Lew Grosenbaugh in the early 60’s. It has proven to be a very efficient method, and the central idea of first making an estimate (this greatly helps in improving the statistics) then measuring some of the trees carefully (which eliminates any bias in the estimation phase) follows right down the centerline of modern efficient sampling methods.

"The problem" with 3P sampling has always been the myth that you had to go to every tree, and therefore it was too much trouble after the inventory reached a certain size. Going to every tree on a marked sale is not a problem, of course, but what about using 3P sampling for large unmarked areas?

One way to do this is "Point-3P sampling", suggested by Grosenbaugh in the early 70’s. Lew saw immediately that a method to expand the system to large areas was desirable and, as usual, saw how to do it well. To extend the 3P method to large areas the first step is to do a number of point samples with your usual methods, but for each tree "in" with the prism you estimate only the height.

You compare this estimated height to a random number to select some of the trees for detailed measurements. The random numbers are generated using the same software that you would use for any 3P cruise. If you expect to select about 120 trees during the day, with an average height of 100 feet, then the sum of the estimates would be:


Let’s assume that you want to select about 20 trees during the cruise for careful measurement. You ask the 3P software to give you random numbers between 0 and . You compare the height of each tree to the random number. If the random number is less than (or equal to) the tree height, then carefully measure that tree for DBH, Height, value, etc. In this way you will select a tree every time you work through about 600 lineal feet of height on the "in" trees. By the time you get through 12,000 feet you will have selected about 20 trees for sampling.

So much for the mechanics of what you do. How do you compute results? Here is how you would work up the final answer and the statistics. It is very similar to the computations for ordinary Variable Plot sampling.

The basal area of the stand (or any species or grouping within the data) is calculated as it would always be:

As always, make sure you have at least one measurement on each species so you can do the computations by species.

The different process is in calculating the VBAR (Volume/ Basal Area Ratio). If you already had the VBAR you would use it in the usual way:

[Basal Area of the stand * average VBAR] = Volume per acre (or per hectare in Canada using a metric angle guage).

 The measured trees are usually the source for computing the VBAR. As we all know, the VBAR is very proportional to tree height. Just for a moment, let’s assume that the VBAR is equal to the tree height itself. This makes the situation quite simple. At each sample point you can compute the volume per acre. The tree count times the BAF gives you the basal area, and the average tree height is the average VBAR of that sample point. No other computations are necessary. Let’s say that comes out to 50,000 BF/acre (because tree height in feet as the rough estimate of VBAR in BF, is about what it should be).


The statistics of the cruise would be based on the Volume per acre at each sample point, just as they normally are when all the trees are "measured". Let’s say that this comes out to a sampling error of 14% on a sample of 30 variable plots. The computations here could be done right in the field, they are so simple. If the height really was the VBAR we would already be through.

On the other hand, we do not have the correct VBAR simply by using the tree height. No problem – we can fix that. The 3P sample trees we selected with our random number list will give us that information. To get the final answer we only need to multiply our crude volume per acre times the correction ratio and we will have a valid computation.

 For each of the sample trees, compute the ratio between the actual VBAR and the tree height (which we were using as a rough VBAR). Take the average of these. Let’s say that comes to 2.3 as an average of 18 trees. In addition, we can compute the sampling error of that ratio. Let’s say that it is 11%.Next Column


Our final answer uses the formula:


 so our example yields:


How good it this final answer? Well, we are as sure of the final answer as we are of the two parts that make it up (the rough estimate & the correction ratio).

We combine them in the usual way, with "Bruce’s formula".



As always, we want to balance the sampling errors in these two portions of the process. The payoff is being able to put in a large number of plots quickly, since you only need to estimate tree height on each tree. Since the tree height is a consistent estimate of tree VBAR you only have to measure a few trees to calculate the correction ratio.

This process allows the power of 3P sampling to be applied to large areas.

Statistical Niceties

There has been some controversy about how to best calculate the sampling error for the SE%ratio . There are some small corrections which you might want to apply here, but in my opinion they are not really necessary.

These are covered in: ("Approximate Sampling Variance of Adjusted 3P estimates" by L.R. Grosenbaugh, Forest Science, June, 1976, pp. 173-176, volume 22, number 2). The corrections are very minor, except for very small sample sizes which would not occur in practical work.

Other Options

The USFS is currently implementing a type of 3P sampling where the volume per hectare is directly estimated, then entire sample plots are chosen for careful measurement (rather than individual trees). This process is essentially the same, with much the same mathematics. This and other applications of 3P sampling are a continuing benefit to the profession from the genius of Lew Grosenbaugh.

3P software

John Bell and Associates markets a small program for producing the 3P random number list.

John Bell & Associates, Inc.
P.O. Box 1538
Corvallis, Oregon 97339

Holmboe Enterprises produces an excellent 3P program using the Hewlett Packard palmtop computer in a waterproof case, which is probably the best field 3P program available anywhere in North America. It can be used to gather all of the information necessary for Point-3P sampling as well, then it can be processed on a PC.

 Rich Holmboe, Holmboe Enterprises
Suite #11, 2253 Wilgress Road
Nanaimo, BC, Canada
V9S 4N3

Phone: (250) 756-6199
FAX: (250) 756-2290

Further information

For those who are interested in an expanded version of this discussion I have one available, and you can contact me at:

Kim Iles & Associates
412 Valley Place
Nanaimo, BC, Canada V9R 6A6

If possible, I would prefer to send it to you via e-mail, so include your e-mail address if you have one.

[Editor’s note: You can get the basic ideas of this method from this discussion, and this is probably enough to wade through for this issue, but we have asked Kim to produce a numerical example of all of the calculations for the next newsletter. This may make the specific process more clear to many readers.]

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