Which of these are statistics?

Topic: Statistical inference · more questions on this topic
‹ Summaries of ten cartons The expectation of the sample variance ›

Let X_1,\ldots,X_n be a random sample from \operatorname{N}(\mu,\sigma^2) with both parameters unknown. Which of the following are statistics: \overline X; \overline X-\mu; \max_i X_i-\min_i X_i; (\overline X-\mu)/(s/\sqrt n); \sum_i X_i^2? For each one that is not a statistic, say whether its distribution still depends on unknown parameters.

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\overline X, \max_iX_i-\min_iX_i and \sum_iX_i^2 are statistics: each can be computed from the data alone.

\overline X-\mu is not a statistic, because it contains \mu. Its distribution, \operatorname{N}(0,\sigma^2/n), still depends on the unknown \sigma^2.

(\overline X-\mu)/(s/\sqrt n) is not a statistic either, for the same reason. Its distribution, however, is t_{n-1} whatever \mu and \sigma^2 are. That combination, a function of data and parameter with a known distribution, is a pivot, and it is what makes the t interval possible.