## Stratified Sampling

Stratification for sampled areas is on of the most useful techniques in resource sampling.  Stratification should be applied when a difference in the statistic of interest is expected to vary by the sampled strata.  If this is true stratification can:

• Increase the precision of the sample estimates

• Or decrease the number of sample need to equal unstratified estimates.

For example consider the following area in Figure 1:

Figure 1. Population with potential strata

If this population were sampled for density without considering the differences in the regions of the population the estimate of variance will be higher.  If this area were stratified and then sampled by strata the overall estimate will have lower variance  or few samples may be required.

Consider the following histogram of a population that was generate from two normal populations (Sample 1 mu=4, sigma=2; and Sample 2 mu=10, sigma=1) (see figure 2). The sample means are plotted as solid lines, with the overall mean thicker than the two strata means.  The sample standard deviations (s) are plotted as dashed line with the overall  line thicker that the strata lines.  This graph illustrates the increase in sample standard deviation because of pooling the two strata versus calculating the strata individually.

Figure 2. Example of pooling two strata

Table 1. Simple Statistics for the two populations.

 Statistic n Mean Standard Deviation Overall 650 6.32 3.397 Strata 1 400 3.99 2.023 Strata 2 250 10.03 1.009

Estimate of the mean per stratum:

Estimate of the mean for the population:

Estimate of the total for X for the entire population:

Variance of the mean for the population:

If the strata are sufficiently large you can use this form to estimate variance:

The standard error of the total estimate of x:

Given these definitions:

M = number of strata in the population

n = total number of sampling units measured for all strata

nj = total number of sampling units measured in the jth stratum

N = total number of sampling units in the population

Nj = total number of sampling units in the jth stratum

Xij = quantity X measured on the ith sampling unit of the jth stratum

= mean of X for thejth stratum

= estimated mean of X for the population

Pj = proportion of the total area in the jth stratum

= estimated total of X for the population

= variance of X for the jthstratum

= estimated variance for the mean for the population

= estimated variance of

Also See:

Chapter 12 -  Sampling  in Forest Inventory pages 156-192, in:

Husch, B., T. W. Beers and J. A. Kershaw, Jr.. 2003. Forest Mensuration. Fourth Edition. John Wiley and Sons, Hoboken, New Jersey 443 p.

Chapter 6 - Sample Designs - Random Sampling pages 200-236, in:

Krebs, C. J. 1998. Ecological Methodology. Harper and Row, Publishers. New York. 620 pp.