Yes, that's obviously how the math works out. Both the overall mean and mean-of-means have sensible interpretations though. Suppose you observe a random sampl

Yes, that's obviously how the math works out. Both the overall mean and mean-of-means have sensible interpretations though. Suppose you observe a random sample of stars and find their temperatures to be 6000, 6250, 6750, and 7000C. The best possible guess[0] for the temperature of a new star drawn from that population is 6500C. The average temperature of all the starstuff may be different if one star is larger than another, but that's an answer to a slightly different question. In fact, I would argue that the mean-of-means is often a more appropriate summary statistic than the overall mean of the observations when there are repeated measurements. [0] Under L2 loss, with no prior, etc etc