Statistical measures such as the mean, [[Variance]], [[Covariance]], and [[Correlation]] can be estimated from sample data $X_1, \dots, X_n$, to approximate the corresponding population parameters.
**Mean:**
$
\begin{align}
\theta_X&=\mathbb{E}[X] && \text{(Population Mean)}\\[4pt]
\hat \Theta_X&=\frac{1}{n} \sum_{i=1}^n X_i &&\text{(Sample Mean)}
\end{align}
$
For a general function $g(X)$, any population parameter can be estimated sample-based.
$
\begin{align}
\theta&=\mathbb{E}[g(X)] && \text{(Population Parameter)}\\[4pt]
\hat \Theta&=\frac{1}{n} \sum_{i=1}^n g(X_i) &&\text{(Sample Estimator)}
\end{align}
$
This allows us to derive sample estimators for variance, covariance and correlation, by substituting specific forms of $g(X)$.
**Variance:**
$
\begin{align}
v_X &= \mathbb{E}\big[(X-\theta_X)^2 \big] && \text{(Population Variance)}\\[4pt]
\hat v_X&=\frac{1}{n} \sum_{i=1}^n(X_i- \hat \Theta_X)^2 && \text{(Sample Variance)}
\end{align}
$
**Covariance:**
$ \begin{align}
\mathrm{Cov}(X,Y) &= \mathbb{E}\big[(X-\theta_X)*(Y-\theta_Y)\big] && \text{(Population Covariance)}\\[4pt]
\widehat{\mathrm{Cov}}(X,Y) &= \frac{1}{n} \sum_{i=1}^n [(X-\hat \Theta_X)*(Y-\hat \Theta_Y)\big] && \text{(Sample Covariance)}
\end{align}
$
**Correlation:**
$
\begin{align}
\rho &= \frac{\mathrm{Cov}(X,Y)}{\sqrt{v_X}*\sqrt{v_Y}} && \text{(Population Correlation)}\\[10pt]
\hat \rho &= \frac{\widehat {\mathrm{Cov}}(X,Y)}{\sqrt{\hat v_X}*\sqrt{\hat v_Y}}&& \text{(Sample Correlation)}
\end{align}
$
By weak [[Law of Large Numbers]], all these sample-based estimators [[Modes of Convergence#Convergence in Probability|Convergence in Probability]] to the corresponding population parameter. Hence, they are [[Properties of an Estimator#Key Properties of an Estimator|consistent estimators]].