Learn English – Are “nil” and “null” interchangeable

I think you mostly have it right, but for further exemplification, here are the most common immediately following collocates for each these words from the COCA:

Most common collocates for mandatory

1   TESTING      151   
2   MINIMUM      125   
3   RETIREMENT   107   
4   MINIMUMS     100   
5   REPORTING    87    
6   SENTENCES    78    
7   EVACUATION   74    
8   SENTENCING   74    
9   DISCLOSURE   63    
10  ARBITRATION  56    
11  SPENDING     41    
12  DRUG         36    
13  SENTENCE     35    
14  EVACUATIONS  32    
15  ARREST       27    
16  PRISON       27    
17  COUNSELING   26    
18  OVERTIME     24    
19  RECYCLING    23    
20  BUSING       22    
21  SANCTIONS    22    
22  INSURANCE    21    
23  SERVICE      21    
24  HIV          20    
25  RELEASE      20    

Most common collocates for compulsory

1   EDUCATION       86
2   MILITARY        53
3   SCHOOL          31
4   SCHOOLING       21
5   STERILIZATION   19
6   LICENSING       18
7   ATTENDANCE      18
8   NATIONAL        16
9   SERVICE         15
10  MILITIA         14
11  MEMBERSHIP      14
12  JURISDICTION    11
13  HEALTH          11
14  HETEROSEXUALITY 10
15  LABOR           10
16  ARBITRATION     9 
17  LICENSE         9 
18  FIGURES         9 
19  PROCESS         9 
20  PUBLIC          9 
21  RETIREMENT      8 
22  DISPUTE         7 
23  ARMY            7 
24  VOTING          6 
25  INSURANCE       6 

Most common collocates for obligatory

1   SERVICE          10 
2   NATURE           8  
3   STOP             8  
4   REFERENCE        6  
5   STRUCTURE        5  
6   CELIBACY         4  
7   EXCHANGE         4  
8   VISIT            4  
9   READING          4  
10  MILITARY         4  
11  QUESTION         4  
12  SLAP             3  
13  NOD              3  
14  RETURNS          3  
15  BASIC            3  
16  PERIOD           3  
17  UPON             3  
18  PHOTO            3  
19  BOOK             3  
20  EARLY            3  
21  PARTHENOGENESIS  2  
22  PREPOSITION      2  
23  PILGRIMAGES      2  
24  HAGGLING         2  
25  ALTRUISTIC       2  

As you can see, mandatory and compulsory have much stronger collocates. The collocates for mandatory are more things that are simply required by law, policy, or rule, whereas compulsory is more for specifically things that someone must do (e.g. education, military, sterilization). Obligatory, on the other hand, doesn’t have much of a strong affinity for certain things, and can be used generally for anything that is required by social custom.

The Wikipedia article explains the difference in the best way possible:

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual (true) value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results. Although the two words can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.

High accuracy, but low precision:

High precision, but low accuracy: