1. Sampling Designs
Avery and Burkhart,
Chapter 3
Source: J. Hollenbeck

2. Complete Enumeration (Census)
Why Not Measure Everything?
Complete Enumeration (Census)
Measure every feature of interest.
The result is a highly accurate description of the population.
Drawbacks:
Only viable with small populations.
Only cost-effective with high-valued features.
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3. Why Not Randomly Select Samples?
We will discuss several sampling designs in this lecture
Most statistical methods assume simple random sampling was used.
Sampling methods, however, will vary depending on the objectives of the survey, the nature of the population being sampled, and prior information about the population being sampled.
FOR 220 Aerial Photo Interpretation and Forest Measurements

4. Sampling Designs Covered Today:
Simple Random Sampling
Systematic Sampling
Stratified Sampling
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5. Assumptions: Sampling Design I: Simple Random Sampling
Every possible combination of sampling units has an equal and independent chance of being selected.
The selection of a particular unit to be sampled is not influenced by the other units that have been selected or will be selected.
Samples are either chosen with replacement or without replacement.
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6. Design: Sampling Design I: Simple Random Sampling
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7. Sampling Design I: Simple Random Sampling
We use the familiar equations for estimating common statistics for the population
Estimate the population mean:
Estimate the variance of individual values:
Compute the coefficient of variation:
FOR 220 Aerial Photo Interpretation and Forest Measurements

8. Sampling Design I: Simple Random Sampling
Compute the standard error: OR
(with replacement, or
infinite population)
(without replacement and
from a finite population)
Compute confidence limits:
t = t statistic from table, determined by degrees of freedom (n-1).
In general, it is safe to use: t = 2.0 for small n (n < 30), or t = 1.96 for large n (n > 30) if you do not have access to a t-table.
FOR 220 Aerial Photo Interpretation and Forest Measurements

9. How many samples to take?
Sampling Design I: Simple Random Sampling
How many samples to take?
Sample size should be statistically and practically efficient.
Enough sample units should be measured to obtain the desired level of precision (no more, no less).
FOR 220 Aerial Photo Interpretation and Forest Measurements

10. Sampling Design I: Simple Random Sampling
Determine sample size (withoout replacement, or finite population):
Knowing the coefficient of variation and that error is expected to be within x % of the value of the mean, we use the following forumula
n =
where, n = required sample size, A = allowable error percent, t = t-value, CV = coefficient of variation, and N = population size

11. Sampling Design I: Simple Random Sampling
Determine sample size (withoout replacement, or finite population):
Example using 0.10 ac fixed radius plots in a ten acre stand
Assume:
We have calculated the CV and found it was 30%
Our allowable error is +/- 10% of the mean
Are using t-value of 2
N is the total number of potential plots that could be placed in the stand (i.e., population).
In this case, N is stand size/plot size or 10/0.10 = 100
n =
= 26
n =

12. Assumptions: Sampling Design II: Systematic Sampling
The initial sampling unit is randomly selected or established on the ground. All other sample units are spaced at uniform intervals throughout the area sampled.
Sampling units are easy to locate.
Sampling units appear to be representative of an area.
FOR 220 Aerial Photo Interpretation and Forest Measurements

13. Design: Sampling Design II: Systematic Sampling
A Grid Scheme is most common
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14. Arguments: For: *** Against: Sampling Design II: Systematic Sampling
Regular spacing of sample units may yield efficient estimates of populations under certain conditions.
*** Against:
Accuracy of population estimates can be low if there is
periodic or cyclic variation inherent in the population.
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15. Arguments: For: Against: Sampling Design II: Systematic Sampling
There is no practical alternative to assuming that populations are distributed in a random order across the landscape.
Against:
Simple random sampling statistical techniques can’t logically be applied to a systematic design unless populations are assumed to be randomly distributed across the landscape.
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16. Summary: Sampling Design II: Systematic Sampling
We can (and often do) use systematic sampling to obtain estimates about the mean of populations.
When an objective, numerical statement of precision is required, however, it should be viewed as an approximation of the precision of the sampling effort. (i.e. 95% confidence intervals)
Use formulas presented for simple random sampling, and where appropriate, use the “without replacement” variations of those equations (if sampling from a small population), otherwise use the normal SRS statistical techniques.
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17. Assumptions: Sampling Design III: Stratified Sampling
A population (i.e., tract) is subdivided into subpopulations (i.e., stands) of known sizes.
A simple random sample (or systematic sample) is completed in each subpopulation (i.e., stand).
Why? To obtain a more precise estimate of the tract mean. If the variation within a stand is small in relation to the tract variance, the estimate of the population mean will be considerably more precise than a simple random sample of the same size.
Why? To obtain an estimate of the resources within each stand.
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18. Design: Sampling Design III: Stratified Sampling
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19. Sampling Design III: Stratified Sampling
Estimate the overall tract mean:
Where:
<< Essentially a weighted average >>
We often use area (acres) for stand units in natural resource applications; total
stand volume may also be used.
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20. Sampling Design III: Stratified Sampling
Estimate the overall tract mean:
Example: 76 acre tract
Stand
Acreage
Volume
(bdft per acre) 1 10
981.6 2 18
2017.9 3 48
3859.7
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21. Sampling Design III: Stratified Sampling
Estimate the overall population standard error of the mean:
1. First compute variance within each strata
Where: nh = total number of [tree] units sampled in strata h
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22. (without replacement)
Sampling Design III: Stratified Random Sampling
Estimate the overall population standard error of the mean:
2. Second, compute population standard error of the mean
(with replacement) OR
(without replacement)
Remember: N = [area] units, n = [tree] units
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23. Sampling Design III: Stratified Random Sampling
Estimate the overall population standard error of the mean:
3. Compute confidence intervals if desired
Estimated confidence intervals:
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24. Sampling Design III: Stratified Sampling
Calculating Sample Size for Stratified Random Sample:
Specify the sampling error objective for tract as a whole
Subdivide (stratify) the tract (i.e., population) into sampling components (i.e., stands). The purpose is to reduce the coefficient of variation within stands.
Calculate coefficient of variation by stand and a weighted CV for the tract using preliminary cruise data.
Calculate number of plots for the tract, then allocate by stand

25. Sampling Design III: Stratified Sampling
Calculating Sample Size for Stratified Random Sample:
An Illustrative Example

26. Sampling Design III: Stratified Sampling

27. Sampling Design III: Stratified Sampling

28. Sampling Design III: Stratified Sampling