Difference between revisions of "161-A1.1"

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== Student Grading ==
 
== Student Grading ==
 
How much information can we get from a single grade? Note that the maximum information occurs when all the grades have equal probability.
 
How much information can we get from a single grade? Note that the maximum information occurs when all the grades have equal probability.
* For Pass/Fail grades, the possible outcomes are: <math>\{P, F\}</math> with probabilities <math>\{\tfrac{1}{2}, \tfrac{1}{2}\}</math>. Thus,  
+
* For Pass/Fail grades, the possible outcomes are: <math>\{\mathrm{P}, \mathrm{F}\}</math> with probabilities <math>\{\tfrac{1}{2}, \tfrac{1}{2}\}</math>. Thus,  
  
 
{{NumBlk|::|<math>H\left(P\right)=\sum_{i=1}^n p_i\cdot \log_2\left(\frac{1}{p_i}\right) = \frac{1}{2}\cdot \log_2\left(2\right) + \frac{1}{2}\cdot \log_2\left(2\right) = 1\,\mathrm{bit}</math>|{{EquationRef|1}}}}
 
{{NumBlk|::|<math>H\left(P\right)=\sum_{i=1}^n p_i\cdot \log_2\left(\frac{1}{p_i}\right) = \frac{1}{2}\cdot \log_2\left(2\right) + \frac{1}{2}\cdot \log_2\left(2\right) = 1\,\mathrm{bit}</math>|{{EquationRef|1}}}}
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{{NumBlk|::|<math>H\left(P\right)=\sum_{i=1}^n p_i\cdot \log_2\left(\frac{1}{p_i}\right) = 7\cdot \frac{1}{7}\cdot \log_2\left(7\right) = 2.81\,\mathrm{bits}</math>|{{EquationRef|3}}}}
 
{{NumBlk|::|<math>H\left(P\right)=\sum_{i=1}^n p_i\cdot \log_2\left(\frac{1}{p_i}\right) = 7\cdot \frac{1}{7}\cdot \log_2\left(7\right) = 2.81\,\mathrm{bits}</math>|{{EquationRef|3}}}}
  
* If we have all the possible grades <math>\{1.00, 1.25, 1.50, 1.75, 2.00, 2.25, 2.50, 2.75, 3.00, 4.00, 5.00, INC, DRP, LOA\}</math> with probabilities <math>\{\tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}\}</math>, we have:
+
* If we have all the possible grades <math>\{1.00, 1.25, 1.50, 1.75, 2.00, 2.25, 2.50, 2.75, 3.00, 4.00, 5.00, \mathrm{INC}, \mathrm{DRP}, \mathrm{LOA}\}</math> with probabilities <math>\{\tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}, \tfrac{1}{14}\}</math>, we have:
  
 
{{NumBlk|::|<math>H\left(P\right)=\sum_{i=1}^n p_i\cdot \log_2\left(\frac{1}{p_i}\right) = 14\cdot \frac{1}{14}\cdot \log_2\left(14\right) = 3.81\,\mathrm{bits}</math>|{{EquationRef|4}}}}
 
{{NumBlk|::|<math>H\left(P\right)=\sum_{i=1}^n p_i\cdot \log_2\left(\frac{1}{p_i}\right) = 14\cdot \frac{1}{14}\cdot \log_2\left(14\right) = 3.81\,\mathrm{bits}</math>|{{EquationRef|4}}}}

Revision as of 23:18, 13 September 2020

Let's look at a few applications of the concept of information and entropy.

Student Grading

How much information can we get from a single grade? Note that the maximum information occurs when all the grades have equal probability.

  • For Pass/Fail grades, the possible outcomes are: with probabilities . Thus,

 

 

 

 

(1)

  • For grades = with probabilities , we get:

 

 

 

 

(2)

  • For grades = with probabilities , we have:

 

 

 

 

(3)

  • If we have all the possible grades with probabilities , we have:

 

 

 

 

(4)