Revision as of 14:51, 12 January 2011

General observations

This article is based upon the lecturenotes for the teaching modul "Forest Inventory" by Kleinn et al. (2007^[1]).

Simple random sampling (SRS) is the basic theoretical sampling technique. The sampling elements are selected as an independent random sample from the population. Each element of the population has the same probability of being selected. And, likewise, each combination of n sampling elements has the same probability of being eventually selected.

Every possible combination of sampling units from the population has an equal and independent chance of being in the sample.

Simple random sampling is introduced and dealt with here and in sampling textbooks mainly because it is a very instructive way to learn about sampling; many of the underlying concepts can excellently be explained with simple random sampling. However, it is hardly applied in forest inventories because there are various other sampling techniques which are more efficient, given the same sampling effort.

For information about how exactly sampling units are choosen see Random selection.

Notations used

Estimators	Parametric	Sample
Mean	\(\mu = \frac{\sum_{i=1}^N y_i}{N}\)	\(\bar {y} = \frac{\sum_{i=1}^n y_i}{n}\)
Variance	\(\sigma^2 = \frac{\sum_{i=1}^N (y_i - \mu)^2}{N}\)	\(s^2 = \frac{\sum_{i=1}^n (y_i - \bar {y})^2}{n-1}\)
Standard deviation	\(\sigma = \sqrt{\frac{\sum_{i=1}^N (y_i - \mu)^2}{N}}\)	\(s = \sqrt{\frac{\sum_{i=1}^n (y_i - \bar {y})^2}{n-1}}\)
Standard error (without replacement or from a finite population)	\(\sigma_{\bar {y}} = \sqrt{\frac{N-n}{N-1}}*\frac {\sigma}{\sqrt{n}}\)	\(S_{\bar {y}} = \sqrt{\frac{N-n}{N}}*\frac{S_y}{\sqrt{n}}\)
Standard error (with replacement or from an infinite population)	\(\sigma_{\bar {y}} = \frac{\sigma}{\sqrt{n}}\)	\(S_{\bar {y}} = \frac{S_y}{\sqrt{n}}\)

Where,

\(N \!\)	number of sampling elements in the population (= population size);
\(n \!\)	number of sampling elements in the sample (= sample size);
\(y_i \!\)	observed value of i-th sampling element;
\(\mu \!\)	parametric mean of the population;
\(\bar {y} \)	estimated mean;
\(\sigma \!\)	standard deviation in the population;
\(S \!\)	estimated standard deviation in the population;
\(\sigma^2 \!\)	parametric variance in the population;
\(s^2 \!\)	estimated variance in the population;
\(\sigma_{\bar {y}} \)	parametric standard error of the mean;
\(s_{\bar {y}} \)	estimated standard error of the mean.

Simple random sampling examples: 2 exercises for this topic

References

1. ↑ Kleinn, C. 2007. Lecture Notes for the Teaching Module ForestInventory. Department of Forest Inventory and Remote Sensing. Facultyof Forest Science and Forest Ecology, Georg-August-UniversitätGöttingen. 164 S.