There was a comment on one of my answers that stated that all instances of average or dice rolling are treated the same by the rules. Specifically mentioned are players gaining HP, Monster Damage and Passive checks.
For Monster Damage in the DM Basic rules (page 6 v0.1) it reads:
Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the “Hit” notation. You have the option of taking average damage or rolling the damage; for this reason, both the average damage and the die expression are presented.
For player's gaining hp when they go up a level in the Basic rules (page 10 v.01) it reads:
Each time you gain a level, you gain 1 additional Hit Die. Roll that Hit Die, add your Constitution modifier to the roll, and add the total to your hit point maximum. Alternatively, you can use the fixed value shown in your class entry, which is the average result of the die roll (rounded up).
Passive checks explicitly states that you do not ever involve any die rolls. (Basic Rules V.02 page 59)
A passive check is a special kind of ability check that doesn’t involve any die rolls. Such a check can represent done repeatedly, such as the average result for a task searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.
The one other place the concept of taking an average vs rolling dice comes up is with Rolling Stats for PCs there it makes it very clear (and is arguably not really the average)
You generate your character’s six ability scores randomly. Roll four 6-sided dice and record the total of the highest three dice on a piece of scratch paper. Do this five more times, so that you have six numbers. If you want to save time or don’t like the idea of randomly determining ability scores, you can use the following scores instead: 15, 14, 13, 12, 10, 8.
Is there a general rule that can be derived here, (Are there enough data points in the rules to create a general principle?) or is the different use of language in all the instances make them all unique situations which need to be interpreted independently?
Additionally, if there is a general rule, can it be used anytime you want to "save time" or "reduce randomness"? (And if not, why not?)