Statistical Sampling
Sampling
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?
Stratified Random Sampling
Sampling
In Excel
=AVERAGE(A1:An)
mean of the squared
deviations
Square root of variance
Standard Deviation
𝜎=
1
𝑛−1
Use Excel function
=STDEV(A1:An) or =STDEV.S(A1:An)
𝑛
(𝑥1 − 𝑥)2
𝑖=1
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.
Variability
The differences
between
individuals or units
in a population
Standard Error of the mean
• Equals the standard deviation of all possible
sample means around the true population
mean.
Finite Population Correction Factor
The finite population correction factor serves to reduce the standard error when
relatively large samples are drawn from finite populations
Confidence Interval
• specify the precision of the sample mean in
relation to the population mean.
Student’s t distribution
Confidence Interval
Effect of Standard Deviation
The red distribution has a mean of 40 and a standard deviation of 5;
the blue distribution has a mean of 60 and a standard deviation of 10.
For the red distribution, 68% of the distribution is between 45 and 55;
for the blue distribution, 68% is between 40 and 60.
Sampling Error
Rather than work with absolute confidence limits, convert them to a percent of the
sample mean which is called sampling error. The notation in the handbook is an upper
case E. Take the confidence interval quantity and scale it to the sample mean by dividing
by the sample mean. Express this value as a percent by multiplying by 100. By expressing
the confidence interval as a percentage, the mean can be plus or minus the percentage
derived.
For example, at 95% confidence, an estimate of the mean has a confidence interval of
46.4 plus or minus 2.6. When expressed as a sampling error percent, the mean is plus or
minus 5.6% which says the true population mean falls within 95% percent of the
estimate.
Determining Sample Size
For a 95% confidence level, the t value approaches 2 as the sample size gets
large, so a t value of 2 is commonly used when estimating sample size. The CV is
the relative variability in the population being sampled. Use the population CV if
known or use an estimate if it is not known. The E represents the desired
sampling error, for example, 10%
Items with
Possible
Impacts on
Sampling
Intensity
Coefficient of Variation
• The relative variability in the population being
sampled
• A unitless measure usual for comparing sampling
methods
• Sample Standard Deviation divided by sample
mean times 100
sd/mean X 100 = CV
Using CV for Comparison
Because CVs have no associated unit of measure, they can be useful in
comparing sampling methods to determine which is most efficient.
So which method of sampling would require fewer samples?
Effect of CV Change
As the coefficient of variation increases, so does the required sample size.
Sampling Intensity Revisited
• The USFS Way
Sample Selection – from Precruise data
1. Determine the sampling error for the sale as a whole. (set to
10%)
2. Subdivide (or stratify) the sale population into sampling
components as needed to reduce the variability within the
sampling strata.
3. Calculate the coefficient of variation (CV) by stratum and a
weighted CV over all strata. (this will be covered more later in
the statistics lectures)
4. Calculate number of plots for the sale as a whole and then
distribute by stratum.
Number of Plots
Value of t is assumed to be 2
Error is set at 10%
Distribute Plots by Stratum
• For each stratum, the calculation would look like this:
• n1 = (17.6 * 185) / 67.9 = 48 plots
• n2 = (7.7 * 185) / 67.9 = 21 plots
• n3 = (7.2 * 185) / 67.9 = 20 plots
• n4 = (35.4 * 185) / 67.9 = 96 plots
• Which totals to the 185 plots for the entire tract.
Tree Expansion Factor
• 1 divided by the fixed plot size times the
number of plots
• n = number of plots
• SZ = fixed plot size
• Ft = tree factor
Sample Error Example step 1 (Calculate Standard Error)
Plots 1/5 acre in size were used in
this example and acres is equal to
18. So the total number of plots for
the strata would be 5 plots per acre
times 18 acres = 90 potential plots.
Notice the application of the Finite
Correction Factor (FCF) for this method.
Sample Error – Step 2
Recall the Standard Error was calculated as 8.3 ft3
36.2% is a bit larger than the level we set to begin with (10%) – Implications?