= What is the expected total discrepancy score in a R choice task? =

Suppose we have R possible choices and each of these is equally likely to be the true one. 

If we consider a discrepancy as the difference between the true choice and the one given by a subject then

The expected total discrepancy of the ''absolute value'' of discrepancies equals 

$$\sum_{k=1}^R (k=1)k $$, $$1 \leq k \leq R $$

with the average sum of the absolute values of discrepancies per rating equal to

$$\frac{\sum_{k=1}^R (k=1)k}{R}$$. 

For example the table below gives all the abs(discrepancies) for the case where R = 4.

||||||||<style="TEXT-ALIGN: center"> '''k=1''' || '''k=2''' || '''k=3''' || '''k=4''' || '''True Rank''' ||
||||||||<style="TEXT-ALIGN: center"> '''0''' || '''1''' || '''2''' || '''3''' || '''1''' ||
||||||||<style="TEXT-ALIGN: center"> '''1''' || '''0''' || '''1''' || '''2''' || '''2''' ||
||||||||<style="TEXT-ALIGN: center"> '''2''' || '''1''' || '''0''' || '''1''' || '''3''' ||
||||||||<style="TEXT-ALIGN: center"> '''3''' || '''2''' || '''1''' || '''0''' || '''4''' ||


Expected total score assuming random guesses at true rank 

= 2(1+2+3)+2(1+1+2)= 20 

= 1x2 + 2x3 + 3x4 

= $$\sum_{k=1}^4 (k=1)k $$. 


The average sum of abs(discrepancies) per rating = 20/4 = 5.