Sample variance

Builds on Sample mean Variance Statistic Bias of an estimator

Definition

Let $n \ge 2$ and let $X_1, \dots, X_n$ be real random variables with sample mean $\bar{X}_n$ . The sample variance is the statistic $S_n^2 = \frac{1}{n - 1} \sum_{i=1}^{n} \left(X_i - \bar{X}_n\right)^2.$ The sample standard deviation is $S_n = \sqrt{S_n^2}$ .

Remarks

If the $X_i$ are i.i.d. with variance $\sigma^2 < \infty$ , then $E[S_n^2] = \sigma^2$ . So $S_n^2$ is an unbiased estimator of $\sigma^2$ . The divisor $n - 1$ instead of $n$ is Bessel's correction. The version with divisor $n$ has expectation $\frac{n-1}{n} \sigma^2$ , so it has the bias $-\sigma^2/n$ . The reason is that $\sum_i (X_i - \bar{X}_n)^2 \le \sum_i (X_i - \mu)^2$ : the deviations from $\bar{X}_n$ are on average smaller than those from the true mean $\mu$ . For i.i.d. $\mathcal{N}(\mu, \sigma^2)$ variables, $(n - 1) S_n^2 / \sigma^2$ has the chi-squared distribution with $n - 1$ degrees of freedom, and it is independent of $\bar{X}_n$ .

Reading path

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Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Discrete random variable definition
Law of a random variable definition
Independence of two events definition
Random variable with a density definition
Joint probability distribution definition
Identically distributed random variables definition
Expected value definition
Mutual independence of events definition
Independence of two random variables definition
Marginal distribution definition
Variance definition
Linearity of expectation statement
Independent sigma-algebras definition
Independence means the joint law is the product of the marginals statement
Covariance definition
Independence of random variables definition
Variance of a sum statement
Independent and identically distributed random variables definition
Expectation of a product of independent random variables statement
Statistic definition
Independent random variables are uncorrelated statement
Estimator definition
Variance of a sum of independent random variables statement
Bias of an estimator definition
Sample mean definition
Sample variance definition

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17 pages build on this one.

Tag	`0JN9P4A944`
URI	`m2://0JN9P4A944`
Kind	definition
Writing	W1 stated
MSC	62F10
Defines	Sample variance
Uses	Sample mean, Variance, Statistic, Bias of an estimator

Sample variance

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