Statistic
Definition taught in Statistical inference, part of Statistical inference
Statistic
| Type | Definition |
| Taught in | Statistical inference |
| Part | Statistical inference |
| Properties | 2 |
| Worked examples | 1 |
A statistic is a function of the sample that does not depend on the unknown parameter. Its sampling distribution is the distribution of that function when the sample is still random.
Properties
Moments of the sample mean and variance
Main article: Moments of the sample mean and variance
If X_1,\ldots,X_n are a random sample from a distribution with mean \mu and variance \sigma^2, then
\mathbb{E}\overline X=\mu,\qquad \operatorname{Var}(\overline X)=\frac{\sigma^2}{n},\qquad \mathbb{E} s^2=\sigma^2.
No normality is needed.
Normal sample theorem
Main article: Normal sample theorem
If X_1,\ldots,X_n are a random sample from \operatorname{N}(\mu,\sigma^2), then
- \overline X is \operatorname{N}(\mu,\sigma^2/n),
- (n-1)s^2/\sigma^2 is \chi^2_{n-1}, and
- \overline X and s^2 are independent.
Examples
Which of these are statistics?
Question page: Which of these are statistics?
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.
\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.
Used in
- Taught in Statistical inference, where the derivation and examples are.
- All 4 questions on Statistical inference
See also
Random sample · Moments of the sample mean and variance · Chi-squared distribution · Student t distribution · F distribution · Normal sample theorem