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,
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(1)
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- For grades = with probabilities , we get:
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(2)
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- For grades = with probabilities , we have:
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(3)
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- If we have all the possible grades with probabilities , we have:
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(4)
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