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A song has a total of 1,000 plays, and on average only 50% of the song was played (say a song is 4 mins long and on average listeners only listen to half the song).

What other data can we interpret from the numbers above? I'm trying to estimate how many of the 1,000 plays actually listened to the whole song, but it doesn't seem possible, does it?

Thank you.

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No, with this single observation, and no statistical model behind, you cannot estimate the proportion of listeners who listened to the whole song. You would need to define the possible durations (e.g., 0, 1/2, 1) or else a continuous distribution for those who do not listen the whole duration (e.g., a uniform(0,1)). For instance, if the duration was a random variable $x$ with a distribution $$ p U(0,1)(x) + (1-p) \mathbb{I}_1(x) $$ your data $\bar x$ and a first moment estimator would imply $p=1$...

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    $\begingroup$ It is wonderful(!) to see you on this site Xi'an :) $\endgroup$ – Tal Galili Nov 5 '11 at 10:30

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