RosettaCodeData/Task/Diversity-prediction-theorem/00-TASK.txt

The   ''wisdom of the crowd''   is the collective opinion of a group of individuals rather than that of a single expert.

Wisdom-of-the-crowds research routinely attributes the superiority of crowd averages over individual judgments to the elimination of individual noise,   an explanation that assumes independence of the individual judgments from each other. 

Thus the crowd tends to make its best decisions if it is made up of diverse opinions and ideologies.


Scott E. Page introduced the diversity prediction theorem: 
: <big>''The squared error of the collective prediction equals the average squared error minus the predictive diversity''.</big> 


Therefore, &nbsp; when the diversity in a group is large, &nbsp; the error of the crowd is small.


;Definitions:
::* &nbsp; Average Individual Error: &nbsp; Average of the individual squared errors
::* &nbsp; Collective Error:         &nbsp; Squared error of the collective prediction
::* &nbsp; Prediction Diversity:     &nbsp; Average squared distance from the individual predictions to the collective prediction
::* &nbsp; Diversity Prediction Theorem: &nbsp; ''Given a crowd of predictive models'', &nbsp; &nbsp; then
:::::: &nbsp; Collective Error &nbsp; = &nbsp; Average Individual Error &nbsp; ─ &nbsp; Prediction Diversity

;Task:
For a given &nbsp; true &nbsp; value and a number of number of estimates (from a crowd), &nbsp; show &nbsp; (here on this page):
:::* &nbsp; the true value &nbsp; and &nbsp; the crowd estimates
:::* &nbsp; the average error
:::* &nbsp; the crowd error
:::* &nbsp; the prediction diversity


Use &nbsp; (at least) &nbsp; these two examples:
:::* &nbsp; a true value of &nbsp; '''49''' &nbsp; with crowd estimates of: &nbsp; ''' 48 &nbsp; 47 &nbsp; 51'''
:::* &nbsp; a true value of &nbsp; '''49''' &nbsp; with crowd estimates of: &nbsp; ''' 48 &nbsp; 47 &nbsp; 51 &nbsp; 42'''



;Also see:
:* &nbsp; Wikipedia entry: &nbsp; [https://en.wikipedia.org/wiki/Wisdom_of_the_crowd Wisdom of the crowd]
:* &nbsp; University of Michigan: [https://web.archive.org/web/20060830201235/http://www.cscs.umich.edu/~spage/teaching_files/modeling_lectures/MODEL5/M18predictnotes.pdf PDF paper] &nbsp; &nbsp; &nbsp; &nbsp; (exists on a web archive, &nbsp; the ''Wayback Machine'').
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