Cfrtjryk

I am confused about compound-complex sentences that begin with if and because. In general, if a compound-complex sentence begins with an introductory phrase and contains two dependent clauses, all three are separated by commas. For example, I might say, "I would have gone to the park yesterday, but it was raining, and I dislike going to the park in the rain." In this case, the placement of the commas does not change the meaning of the sentence (I think). But I am not sure whether this is the case for sentences that begin with if and because

For example, suppose I were to say, "If every observation is the same number, the mean equals the observation, and there is no spread about the mean." I am not sure whether there should be a comma between observation and and. It seems that there should be a comma since "the mean equals the observation," and "there is no spread about the mean" are independent clauses, but I also want it to be clear that both statements--that the mean equals the observation and that there is no spread about the mean--are conditional on every observation being the same number. I feel like the comma could make it seem as though there are two statements: (1) "if every observation is the same number, the mean equals the observation," and separately, (2) "there is no spread about the mean." Essentially, I am concerned that the comma makes the sentence read like this: "If every observation is the same number, the mean equals the observation. And there is no spread about the mean."

The same situation occurs if I say, "if the observations are all one constant, then the probability that X equals c is one, and the expected value of X is c." Again, both statements are conditional on the observations being one constant, and I do not want to make it seem as though "the expected value of X is c" can stand on its own, apart from the if statement.

Lastly, suppose I were to say, "because the average cost curve is above the average variable cost curve everywhere and the marginal cost curve is rising where it crosses both average curves, the minimum of the average variable cost curve, b, is at a lower output level than the minimum of the average cost curve, a." Should there be a comma between everywhere and and in this example? I feel like there should be one since "the average cost curve is above the average variable cost curve everywhere," and "the marginal cost curve is rising where it crosses both average curves" are both full sentences. But I am concerned that adding the comma might confuse readers since the comma might make it seem as though the two are two separate conditions.

I hope this isn't too confusing. Thank you.

"If every observation is the same number, the mean equals the observation, and there is no spread about the mean."

It would reduce ambiguity somewhat to have a "then" before "the mean". Removing the comma between "observation" and "and" does indeed promote the "(If every observation is the same number), the (mean equals the observation, and there is no spread about the mean)." rather than "(If every observation is the same number, the mean equals the observation), and there is no spread about the mean." parsing. Without the comma, the meaning can still be understood, though. If you want to enforce the former parsing even further, you can say "If every every observation is the same number, then it follows that both the mean equals the observation, and there is no spread about the mean." Once you have the word "both" in there, the alternative interpretation that you're worried about is no longer an option: "If every observation is the same number, both the mean equals the observation. And there is no spread about the mean." is not a grammatically correct possibility.

It seems that there should be a comma since "the mean equals the observation," and "there is no spread about the mean" are independent clauses

It is not grammatically necessary to separate independent clauses, especially if, as in the case here, the two clauses together form a unit that is conjoined to another clause.

I also recommend saying "is equal to that number", rather than "equals the observation": "If every observation is the same number, then the mean is equal to that number".

For your more mathematical statements, an option would be repeating the statement in mathematical language:

If the observations are all one constant, then the probability that X equals c is one, and the expected value of X is c.
i.e.
(∃c:∀o:o=c)->((P(X=c)=1)∧(E(X)=c))

Also note that although it is common to represent constants by c, you should explicitly state that c represents the constant.

Lastly, suppose I were to say, "because the average cost curve is above the average variable cost curve everywhere and the marginal cost curve is rising where it crosses both average curves, the minimum of the average variable cost curve, b, is at a lower output level than the minimum of the average cost curve, a." Should there be a comma between everywhere and and in this example? I feel like there should be one since "the average cost curve is above the average variable cost curve everywhere," and "the marginal cost curve is rising where it crosses both average curves" are both full sentences.

An option would be to split it up into two sentences or more sentences (also, your wording implies that there is a point where the marginal cost curve crosses both average curves):

"The average cost curve is above the average variable cost curve everywhere. Also, for both average curves, the marginal cost curve is rising when it intersects the average curve. Because of these two facts, we can conclude that the minimum of the average variable cost curve, b, is at a lower output level than the minimum of the average cost curve, a."