Lecture 3Forestry 3218

Avery and Burkhart, Chapter 3Shiver and Borders, Chapter 2

Forest Mensuration II

Lecture 3Elementary Sampling Methods: Selective, Simple Random, and Systematic

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Sampling vs. Complete Enumeration

Why sampling?• Measuring all units

(trees, birds, etc.) is sometimes impractical, if not impossible– Some measurements are

destructive

• Sampling saves money and time

Complete Enumeration• Measure every feature

of interest; a highly accurate description of the population.

• Drawbacks: only viable with small populations; only cost-effective with high-valued features.

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Sampling Design

• The method of selecting non-overlapping sample units to be included in a sample

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Sampling Frame

• The list of all possible sampling units that might be drawn in a sample

• Developing a reliable frame may be difficult – Jack pine trees in Crown forest (infinite population)– In most field situation, differences between the

sampling frame and the population are inconsequential

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Elementary Sampling Methods

• Selective• Simple Random

Sampling• Systematic Sampling

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Selective Sampling

• The method involved selecting areas that appeared to be reprehensive of the average stand condition to the sampler (cruiser)

• Was widely used in forestry, is still…• Depends on skill of the cruiser, biased• No valid variance, and therefore no confidence

interval, could be calculated• Because sampled areas appeared to be

average, their variability would be smaller than the true variability

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Simple Random Sampling (SRS)

• Sampling units are chosen completely at random

• Every possible combination of sampling units has an equal and independent chance of being selected

• SRS is the fundamental method for other sampling procedures

• Other procedures are simply modifications to achieve better precision or greater economy

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SRS Procedure

• Requires the development of a frame, implying the need of aerial photographs, or maps

• Select random numbers between one and the total number of sampling units in the population

• Samples are either chosen with replacement or without replacement, the latter means that once a sampling unit is chosen it may not been chosen again

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SRS Estimators

xsCV

Mean

Variance

Coefficient of variation

1

2)(2

nxxs

nxx

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SRS Estimators

• Standard error of the mean– With replacement or

infinite population– without replacement

from a finite population

• Confidence limit

)(2

N

nNs

n

sx

n

sxs

2

xstx

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Sampling Intensity

• How many samples to take? Depends on: – The variability of the population– Desired confidence interval– Acceptable level of error

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Sampling Intensity

• With replacement or infinite population

• Without replacement from a finite population

2

E

stn

NstE

n1

21

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Calculating sample size

2

E

stn

Standard deviation (120 m3/ha)

95% confidence (t=2)

Acceptable level of error

±40 m3/ha

36

2

40

1202

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Calculating sample size from CV and A

2

E

stn

22

ACVt

AxxCVt

xCVsxsCV

AxE

1445302

2

)(nExample:

Allowable percent error of mean

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Relationship between sample size and allowable error for different CVs

n

5 205 405 605 805

Allo

wab

le e

rror

(%

)

CV=100

CV=20

0

20

10

30

40

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Can we use SRS all the time? - problems

• Locating some sample units on the ground may be very time-consuming– Reference point to sample units– Access

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Systematic Sampling

The initial sampling unit is randomly selected. All other sample units are spaced at uniform intervals throughout the area sampled

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Systematic Sampling

Pros:• Sampling units are easy

to locate• Sampling units appear

to be “representative”• Generally acceptable

estimates for the population mean

Cons:• Impossible to estimate the

variance of one sample• Accuracy can be poor

(i.e., bias) if a periodic or cyclic variation inherent in the population

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Arguments of systematic sampling

Against– SRS statistical techniques can’t logically be

applied to a systematic design unless populations are assumed to be randomly distributed

For– There is no practical alternative to assuming that

populations are distributed in a random order

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Summary for Systematic Sampling

• Use systematic sampling to obtain estimates about the mean of populations

• Numerical statement of precision should be viewed as an approximation

• Use SRS formulas

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Summary

• Selective sampling• SRS• Systematic sampling