Q.For any set of positive, unequal observations, which of the following relationships always holds?

The three means always stand in one fixed order for unequal positive data. …

Examining each option:

1. AM > GM > HM: this is exactly the standard relationship among the three means for any set of POSITIVE, UNEQUAL observations — arithmetic mean is always the largest, harmonic mean the smallest, and geometric mean always lies strictly between them. Correct.
2. HM > GM > AM: this reverses the true order entirely and is never true for positive data. Incorrect.
3. AM = GM = HM: equality of all three means holds ONLY when every observation in the data set is identical — the question specifies UNEQUAL observations, so this cannot hold here. Incorrect. …

Assuming the three means are always equal, or assuming there is no fixed order among them — the AM ≥ GM ≥ HM relationship is a locked mathe …