Among those with three and four years of experience, the average performance appraisal score of managers differs from that of engineers by ______ points

 Maximum difference(at most):

Total engineers are 8, and managers are 15. To find the maximum difference for average appraisal score, if we take minimum values for managers, it will eventually be less compared to when we take minimum values for engineers. Therefore, we take the minimum possible values for managers and maximum possible values for engineers. (As any number divided by 15 will be very less, compared to 8)

Engineers

For engineers with 3 years of experience from 3 engineers, one will have an appraisal score of 12; the other two will have a score equal to 16.

So, a total score = 12+16+16 = 44

For engineers with 4 years of experience, from 5 engineers, one will have an appraisal score of 9, other four will have a score equal to 15.

So, a total score = 5+15+15+15+15 = 65

Therefore, the average appraisal score is $$\ \frac{44+65}{3+5}$$ = 14.125

Managers

For managers with 3 years of experience from 10 managers, one will have an appraisal score of 12; the other nine will have a score equal to 10.

So, a total score = 12+10*9 = 102

For managers with 4 years of experience from 5 managers, one will have an appraisal score of 13; the other four will have a score equal to 10.

So, a total score = 13+10*4 = 53

Therefore, the average appraisal score is $$\ \frac{102+53}{10+5}$$ = 10.2

The average performance appraisal score of managers differs from that of engineers by at most (14.125-10.2) points

=> 3.79 points