The one challenge in trying to determine an “optimal sample size” (the statistical term for picking the smallest number of plots to get your desired standard error) is that you have to know something about how variable your population is beforehand. This is an especially difficult problem in natural resource projects where natural variability is an inherent part of the resource.

For any methods that I am aware of you have to have some idea of what the variability of your population is before you begin estimating an optimal sample size. In our case this would mean having some idea what the standard error is. This “expected” standard error can come from the following sources:

1. Published literature in a forest similar to yours using the same sized plots and estimating the same results you are interested in.

2. Other work you have done in similar forests.

3. An educated guess based on experience.

4. A “pilot study” or test sample in the area you are interested in measuring.

The pilot study approach which would involve generating a small random sample of something like 8-10 plots in the area you are interested in measuring, to get a first estimate of standard error, has a couple advantages. First, it is quantitative and based directly on the data you are interested in. Second, this small initial sample could then become the first plots in your larger project so you are not wasting time collecting this data just to estimate the number of plots.

No matter how you decide to estimate your initial standard error the formula for the number of plots to use is as follows:

n

_{opt}= (se

_{exp}/se

_{des})

^{2}*n

_{exp}

Where:

n

_{opt}= optimal sample size (the smallest number of plots to use to achieve you desired standard error)

se

_{exp}= expected standard error, estimated from pilot study or other method

se

_{des}= desired standard error, the standard error you would like to achieve in your results

n

_{exp}= the number of plots used to determine your expected standard error (se

_{exp}), e.g. the number of plots in you pilot study.

There are a couple things to keep in mind if you want to start using this approach. 1) Standard error is different for different variables so you may want to pick one key variable or a few variables you are most interested in to use in this formula. For example you may go through this process using the standard errors for the total number of trees in your population however that doesn’t guarantee that you will achieve the standard errors you want for the value of air pollution reduction. 2) Your sample size estimate is only going to be as good as your estimate of expected standard error (se

_{exp}). If the estimate of expected standard error is way off because you made a bad guess or you got unlucky and your pilot sample was not representative of the whole population there isn’t much you can do. This method only allows you to make an estimate of how many plots you should use based on available information and sometimes the information that is available is not that good.

Sometimes it is useful to think of standard errors as a percent of the value you are estimating since you may not know what that value is before you start sampling. So you may not be able to say I want to estimate the total value of benefits with a standard error of $1,000 because you don’t know if you are going to have $5,000 in total benefits or $500,000 in benefits. In this case you may want to say “I want to estimate the total number tree benefits with a standard error of 20%.” The above formula can be adjusted by substituting percents for se

_{des}and se

_{exp}. The percent se

_{exp}would be the standard error from your pilot study divided by the value that was estimated (e.g. If you were using the total leaf biomass as your variable of interest from the pilot study and you got 50 tons of leaf biomass with a standard error of 10, your se

_{exp}as a percent would be 10/50=20%).

n

_{opt}= (se%

_{exp}/se%

_{des})

^{2}*n

_{exp}

Where:

n

_{opt}= optimal sample size (the smallest number of plots to use to achieve you desired standard error)

se%

_{exp}= expected standard error as percent estimated from pilot study or other method by dividing the standard error by value being estimated

se%

_{des}= desired standard error expressed as a percent of the value you are interested in estimating

n

_{exp}= the number of plots used to determine your expected standard error (se

_{exp}), e.g. the number of plots in you pilot study.