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Statistical Sampling
Sample
A subset of units selected from
the population to represent it.
Hopefully it is representative
Reasons to Sample - review
 Reduced Cost
 For populations having an extremely large number of individuals (like forest trees),
measuring each individual would be impractical. Neither the time nor the money is
available to do such an inventory. Additionally, the inventory costs of doing such an
inventory would exceed the value of the sale.
 Greater Speed
 Sampling reduces the magnitude of the job, allowing the task to be completed in a
shorter period of time. Not many timber sales would make it to market if the volume is
measured on all of the trees and it takes ten months to do the sale prep work.
 Greater Scope
 Sampling provides the ability to study a larger area and include diverse information
about the population. Additional information such as species present in the population,
type of defect, or other conditions in the population can be included when sampling is
used.
 Greater Accuracy
 Often overlooked is the quality of work suffers when budgets and resources are
stretched too thin. Good measurements on a sample of individuals provide more
reliable information than bad measurements on the entire population.
Sampling Types
Simple Random Sampling
 Every possible combination of sample units has an equal and
independent chance of being selected.
 However…
Systemic Sampling
 Beware coincidental bias of sample interval and natural area.
 Ridges
 River bends
 Etc.
Stratified Random Sampling
The point is to reduce variability within strata.
Example: if you were measuring average
estrogen levels in humans, you would stratify
male versus female.
Can you think of some forest examples?
Population sampling
Involves studying a portion of the population to
gain information about the entire population.
When applying sample information to the
population, it is called making an inference.
Inferences include estimation, prediction,
hypothesis testing and determining relationships.
Populations and Samples
Sample Size
The number of units selected randomly from
the population for observation.
Sample size is designated by the lower case n
Sample Total
The sum of the values for all units in the sample
Sample Mean
 The sum of the individual unit values divided by the sample size.
Variability
The differences
between individuals
or units in a
population
Taking
measurements on
these two stands
should result in
values that reflect
their differences.
Sample Variance
 Estimates of the variability of the population. This is a measure of
how closely the observations are clustered around the mean. The
notation for the variance is usually shown as lower case sigma
squared with a caret or as lower case s squared or a capital V. All
squared differences are summed and then divided by the sample
size minus one. Both equations below give the same answer.
Sample Standard Deviation
 Equals the square root of the
variance.
 Returns the variability index to the
units of the mean.
 The notation can be lower case
sigma with a caret, lower case s or
SD.
In Microsoft Excel the function is…
=STDEV(A1:An) or =STDEV.S(A1:An)
Sample Coefficient of Variation
 Because populations with large means tend to have larger
standard deviations than those with small means, the
coefficient of variation permits a comparison of relative
variability about different-sized means. The sample
coefficient of variation is an expression of the standard
deviation as a percentage of the mean. It is usually
represented by upper case CV. It is calculated as the
standard deviation divided by the mean multiplied by 100 to
convert to a percentage.
Sampling Summary
In Excel
=AVERAGE(A1:An)
mean of the
squared deviations
Square root of
variance
Exercise in Random Sampling
 Student heights equals population
 Calculate population mean, etc.
 Take a systemic 20% sample compare estimates of population.
 Take a 50% sample (systemic or random) and compare results.
 Calculate mean, variance, SD and CV of both population and samples.