Simple random sampling

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{{Content Tree|HEADER=Forest Inventory lecturenotes|NAME=Forest Inventory lecturenotes}}
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==General observations==
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Simple random sampling (SRS) is the basic theoretical [[:Category:Sampling design|sampling technique]].  
 
Simple random sampling (SRS) is the basic theoretical [[:Category:Sampling design|sampling technique]].  
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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.
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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.  
 
Every possible combination of sampling units from the population has an equal and independent chance of being in the sample.  
 
   
 
   
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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 inventory|forest inventories]] because there are various other sampling techniques which are more efficient, given the same sampling effort.
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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 inventory|forest inventories]] because there are various other sampling techniques which are more efficient, given the same sampling effort<ref>Kleinn, C. 2007. Lecture Notes for the Teaching Module  Forest Inventory. Department of Forest Inventory and Remote Sensing. Faculty of Forest Science and Forest Ecology,  Georg-August-Universität Göttingen. 164 S.</ref>.
  
 
For information about how exactly sampling units are choosen see [[Random selection]].
 
For information about how exactly sampling units are choosen see [[Random selection]].
  
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==Notations==
  
  
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{{construction}}
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{| class="wikitable"
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|-
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!''Statistic''
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!''Parametric value''
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!''Sample based estimator''
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|-
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|Mean
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|<math>\mu = \frac{\sum_{i=1}^N y_i}{N}</math>
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|<math>\bar {y} = \frac{\sum_{i=1}^n y_i}{n}</math>
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|-
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|Variance
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|<math>\sigma^2 = \frac{\sum_{i=1}^N (y_i - \mu)^2}{N}</math>
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|<math>S_y^2 = \frac{\sum_{i=1}^n (y_i - \bar {y})^2}{n-1}</math>
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|-
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|Standard deviation
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|<math>\sigma = \sqrt{\frac{\sum_{i=1}^N (y_i - \mu)^2}{N}}</math>
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|<math>S_y = \sqrt{\frac{\sum_{i=1}^n (y_i - \bar {y})^2}{n-1}}</math>
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|-
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|Standard error
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(without replacement or from a finite population)
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|<math>\sigma_{\bar {y}} = \sqrt{\frac{N-n}{N-1}}*\frac {\sigma}{\sqrt{n}}</math>
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|<math>S_{\bar {y}} = \sqrt{\frac{N-n}{N}}*\frac{S_y}{\sqrt{n}}</math>
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|-
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|Standard error
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(with replacement or from an infinite population)
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|<math>\sigma_{\bar {y}} = \frac{\sigma}{\sqrt{n}}</math>
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|<math>S_{\bar {y}} = \frac{S_y}{\sqrt{n}}</math>
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|}
  
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Where,
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{|
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|-
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| <math>N \!</math> || number of sampling elements in the population (= population size);
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|-
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| <math>n \!</math> || number of sampling elements in the sample (= sample size);
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|-
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| <math>y_i \!</math> || observed value of i-th sampling element;
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|-
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| <math>\mu \!</math> || parametric mean of the population;
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|-
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| <math>\bar {y} </math> || estimated mean;
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|-
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| <math>\sigma \!</math> || standard deviation in the population;
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|-
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| <math>S \!</math> || estimated standard deviation in the population;
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|-
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| <math>\sigma^2 \!</math> || parametric variance in the population;
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|-
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| <math>s^2 \!</math> || estimated variance in the population;
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|-
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| <math>\sigma_{\bar {y}} </math> || parametric standard error of the mean;
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|-
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| <math>s_{\bar {y}} </math> || estimated standard error of the mean.
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|}
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{{Exercise
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|message=Simple random sampling examples
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|alttext=test
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|text=2 exercises for this topic
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}}
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=References=
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<references/>
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[[Category:Sampling design]]
 
[[Category:Sampling design]]

Latest revision as of 14:35, 26 October 2013

Forest inventory
Introduction to forest inventory
Forest mensuration
Introduction to sampling
Plot design
Sampling design
Estimation issues
Planning issues
UNFCCC Implications for Forest Monitoring
Forest Inventory Examples
  Forest Inventory Glossary

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[1].

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

[edit] Notations

Statistic Parametric value Sample based estimator
Mean \(\mu = \frac{\sum_{i=1}^N y_i}{N}\) \(\bar {y} = \frac{\sum_{i=1}^n y_i}{n}\)
Variance \(\sigma^2 = \frac{\sum_{i=1}^N (y_i - \mu)^2}{N}\) \(S_y^2 = \frac{\sum_{i=1}^n (y_i - \bar {y})^2}{n-1}\)
Standard deviation \(\sigma = \sqrt{\frac{\sum_{i=1}^N (y_i - \mu)^2}{N}}\) \(S_y = \sqrt{\frac{\sum_{i=1}^n (y_i - \bar {y})^2}{n-1}}\)
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.



Exercise.png Simple random sampling examples: 2 exercises for this topic

[edit] References

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