The variance of an unbiased estimator $$\hat{\theta}$$ is defined as: $$Var(\hat{\theta}) = \mathbb{E}((\hat{\theta} - \mathbb{E}(\hat{\theta}))^2)$$ where $$\mathbb{E}$$ is the expected value function. The Standard Error of an unbiased estimator $$\hat{\theta}$$ is simply the positive square root of the Variance: $$SE(\hat{\theta}) = \sqrt{Var(\hat{\theta})}$$

Variance and Standard Error of an Estimator

• Variance: The variance of the data is the average squared distance between the mean and each data value.

• Standard deviation: Standard deviation is calculated as the square root of the variance (the average squared deviation from the mean).



University of California, Davis

• mean

• median

• mode

• percentiles

• range

• variance

• standard deviation

• mean squared error

• consistency

statistical measures

• Mean: The "average" number; found by adding all data points and dividing by the number of data points.

• Median: The middle number; found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers).

• Mode: The most frequent number—that is, the number that occurs the highest number of times.


mean, median and mode

variance and standard deviation

Mean Squared Error is simply the variance of a potentially biased estimator $\hat{\theta}_m$ of the Expected Value or Mean $\theta$
$$MSE(\hat{\theta}_m) = \mathbb{E}((\hat{\theta}_m - \theta)^2)$$
$$ = Bias^2(\hat{\theta}_m) + Var(\hat{\theta}_m)$$

Mean Squared Error

In statistics, consistency is a property of an estimator where its probabilistic limit converges to the true underlying parameter value as the sample size approaches infinity ($$	ext{plim}_{mightarrow\infty}\hat{	heta}_m=	heta$$). When an estimator is consistent, accumulating more data guarantees that empirical estimates will eventually match the true theoretical probabilities.

Learn Before

Related

Learn After