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Distributions of Statistics

Definition: A statistic is any value that can be computed from the data that is contained in a sample.

Examples include the sample mean, the sample S.D., the sample median, or any arbitrary function of the sample data. Clearly a statistic is also a random variable.

A Sampling Distribution for a statistic is the probability distribution of that statistic.

That is, if from some population

then the sampling distribution is the probability distribution of these values (which are r.v.’s).

The sampling distribution also depends on how the sampling procedure is conducted. We will concentrate only on Random Sampling - a sample of n observations from a population of size N is said to be random if it is selected by a procedure that ensures that any sample of size n has an equal chance of being picked. The individual observations in a random sample are said to be independent, and identically distributed (iid).

Distribution of the Sample Mean: Suppose we have a random sample X1, X2,..., Xn from some underlying population with mean m and variance s2 and we compute =(åiXi)/n. Then

E[] = = m and V[] == s2/n
If the underlying population is N(m ,s2) then ~N(m, s2/n).
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Jayant Rajgopal