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 mean of the squared deviations Square root of variance In Excel =AVERAGE(A1:An)

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.