<Table> <Tr> <Th> Population </Th> <Th> Statistic </Th> <Th> Sampling distribution </Th> </Tr> <Tr> <Td> Normal: N (μ, σ 2) (\ displaystyle (\ mathcal (N)) (\ mu, \ sigma ^ (2))) </Td> <Td> Sample mean X _̄ (\ displaystyle (\ bar (X))) from samples of size n </Td> <Td> X _̄ ∼ N (μ, σ 2 n) (\ displaystyle (\ bar (X)) \ sim (\ mathcal (N)) (\ Big () \ mu, \, (\ frac (\ sigma ^ (2)) (n)) (\ Big))) or (In case sigma is not known): X _̄ ∼ T (μ, S 2 n) (\ displaystyle (\ bar (X)) \ sim (\ mathcal (T)) (\ Big () \ mu, \, (\ frac (S ^ (2)) (n)) (\ Big))) Where S (\ displaystyle S) is the standard deviation of the sample and T (\ displaystyle (\ mathcal (T))) is the Student's t - distribution . </Td> </Tr> <Tr> <Td> Bernoulli: Bernoulli ⁡ (p) (\ displaystyle \ operatorname (Bernoulli) (p)) </Td> <Td> Sample proportion of "successful trials" X _̄ (\ displaystyle (\ bar (X))) </Td> <Td> n X _̄ ∼ Binomial ⁡ (n, p) (\ displaystyle n (\ bar (X)) \ sim \ operatorname (Binomial) (n, p)) </Td> </Tr> <Tr> <Td> Two independent normal populations: <P> N (μ 1, σ 1 2) (\ displaystyle (\ mathcal (N)) (\ mu _ (1), \ sigma _ (1) ^ (2))) and N (μ 2, σ 2 2) (\ displaystyle (\ mathcal (N)) (\ mu _ (2), \ sigma _ (2) ^ (2))) </P> </Td> <Td> Difference between sample means, X _̄ 1 − X _̄ 2 (\ displaystyle (\ bar (X)) _ (1) - (\ bar (X)) _ (2)) </Td> <Td> X _̄ 1 − X _̄ 2 ∼ N (μ 1 − μ 2, σ 1 2 n 1 + σ 2 2 n 2) (\ displaystyle (\ bar (X)) _ (1) - (\ bar (X)) _ (2) \ sim (\ mathcal (N)) \! \ left (\ mu _ (1) - \ mu _ (2), \, (\ frac (\ sigma _ (1) ^ (2)) (n_ (1))) + (\ frac (\ sigma _ (2) ^ (2)) (n_ (2))) \ right)) </Td> </Tr> <Tr> <Td> Any absolutely continuous distribution F with density ƒ </Td> <Td> Median X (k) (\ displaystyle X_ ((k))) from a sample of size n = 2k − 1, where sample is ordered X (1) (\ displaystyle X_ ((1))) to X (n) (\ displaystyle X_ ((n))) </Td> <Td> f X (k) (x) = (2 k − 1)! (k − 1)! 2 f (x) (F (x) (1 − F (x))) k − 1 (\ displaystyle f_ (X_ ((k))) (x) = (\ frac ((2k - 1)!) ((k - 1)! ^ (2))) f (x) (\ Big () F (x) (1 - F (x)) (\ Big)) ^ (k - 1)) </Td> </Tr> <Tr> <Td> Any distribution with distribution function F </Td> <Td> Maximum M = max X k (\ displaystyle M = \ max \ X_ (k)) from a random sample of size n </Td> <Td> F M (x) = P (M ≤ x) = ∏ P (X k ≤ x) = (F (x)) n (\ displaystyle F_ (M) (x) = P (M \ leq x) = \ prod P (X_ (k) \ leq x) = \ left (F (x) \ right) ^ (n)) </Td> </Tr> </Table> <Tr> <Th> Population </Th> <Th> Statistic </Th> <Th> Sampling distribution </Th> </Tr> <Tr> <Td> Normal: N (μ, σ 2) (\ displaystyle (\ mathcal (N)) (\ mu, \ sigma ^ (2))) </Td> <Td> Sample mean X _̄ (\ displaystyle (\ bar (X))) from samples of size n </Td> <Td> X _̄ ∼ N (μ, σ 2 n) (\ displaystyle (\ bar (X)) \ sim (\ mathcal (N)) (\ Big () \ mu, \, (\ frac (\ sigma ^ (2)) (n)) (\ Big))) or (In case sigma is not known): X _̄ ∼ T (μ, S 2 n) (\ displaystyle (\ bar (X)) \ sim (\ mathcal (T)) (\ Big () \ mu, \, (\ frac (S ^ (2)) (n)) (\ Big))) Where S (\ displaystyle S) is the standard deviation of the sample and T (\ displaystyle (\ mathcal (T))) is the Student's t - distribution . </Td> </Tr> <Tr> <Td> Bernoulli: Bernoulli ⁡ (p) (\ displaystyle \ operatorname (Bernoulli) (p)) </Td> <Td> Sample proportion of "successful trials" X _̄ (\ displaystyle (\ bar (X))) </Td> <Td> n X _̄ ∼ Binomial ⁡ (n, p) (\ displaystyle n (\ bar (X)) \ sim \ operatorname (Binomial) (n, p)) </Td> </Tr>

The probability distribution of a statistic is called a sampling distribution