## 4.2 Universe or Population $$\mathcal{U}$$

Definition 4.4 Universe or population is the set of all elementary units of interest. $$\\$$

Usually the universe has a large $$N$$ size, even infinite, but in some cases it can be relatively small. It is formally denoted by $\mathcal{U} = \lbrace 1,2, \ldots, N \rbrace.$

Example 4.3 (Electoral research III) In 2018, the universe of voters in the municipality of Porto Alegre comprised 1,100,163 voters17, i.e., $$N=1\,100\,163$$. Formally $\mathcal{U} = \lbrace 1, 2, \ldots, 1\,100\,163 \rbrace.$

Definition 4.5 Universal element, population element or elementary unit denotes an element $$i \in \mathcal{U}$$.

Definition 4.6 Characteristic(s) of interest denotes the variable or set of $$k$$ variables associated with each element of the universe, noted by $$\boldsymbol{X} = (\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots, \boldsymbol{x}_N) = \left( \begin{bmatrix} x_{11} \\ x_{12} \\ \vdots \\ x_{1k} \end{bmatrix}, \begin{bmatrix} x_{21} \\ x_{22} \\ \vdots \\ x_{2k} \end{bmatrix}, \cdots , \begin{bmatrix} x_{N1} \\ x_{N2} \\ \vdots \\ x_{Nk} \end{bmatrix} \right) = \left( \begin{array}{cccc} x_{11} & x_{21} & \cdots & x_{N1} \\ x_{12} & x_{22} & \cdots & x_{N2} \\ \vdots & \vdots & \ddots & \vdots \\ x_{1k} & x_{2k} & \cdots & x_{Nk} \end{array} \right).$$ $$\\$$

Example 4.4 Consider that in the universe $$\mathcal{U} = \lbrace 1,2,3 \rbrace$$ of size $$N=3$$, subject 1 is female, 24 years old and 1.66m tall, subject 2 is male aged 32 years and 1.81m tall, and subject 3 male aged 49 years old with a height of 1.73m. Therefore, $\boldsymbol{X} = (\boldsymbol{x}_1,\boldsymbol{x}_2,\boldsymbol{x}_3) = \left( \begin{bmatrix} 24 \\ 1.66 \\ F \end{bmatrix}, \begin{bmatrix} 32 \\ 1.81 \\ M \end{bmatrix}, \begin{bmatrix} 49 \\ 1.73 \\ M \end{bmatrix} \right) = \left( \begin{array}{ccc} 24 & 32 & 49 \\ 1.66 & 1.81 & 1.73 \\ F & M & M \end{array} \right).$

### 4.2.1 Parameter

Definition 4.7 Parameter denotes a function or measurement that depends on all characteristics of interest. $$\\$$

Example 4.5 The total parameter is given by Eq. (2.11). $$\\$$

Example 4.6 The average parameter is given by Eq. (2.8). $$\\$$

Example 4.7 A variable is called dichotomous when it takes only two possible values such as yes/no, true/false, on/off, etc. The characteristic of interest is called success and the other characteristic failure. For convenience, success is associated with the value $$x=1$$ and failure with $$x=0$$. In this way, $$\sum_{i=1}^N x_i$$ is symbolized as the total number of successes observed in the universe. In this situation the parameter proportion is given by $\begin{equation} \pi = \frac{1}{N} \sum_{i=1}^N x_i. \tag{4.1} \end{equation}$

Exercise 4.1 Consider universal proportion and mean, respectively given by Eqs. (4.1) and (2.8).
(a) What makes these two quantities different?
(b) Can the proportion be considered an average? Why? $$\\$$

Example 4.8 The parameter variance is given by the Equations (2.21) and (2.22). $$\\$$

Example 4.9 The parameter standard deviation is the square root of the universal variance, given by Equation (2.26). $$\\$$

Example 4.10 The parameter covariance is given by $\begin{equation} \sigma_{XY} = Cov[X,Y] = \frac{1}{N} \sum_{i=1}^N (x_i - \mu_X)(y_i - \mu_Y). \tag{4.2} \end{equation}$

Example 4.11 The parameter correlation is given by $\begin{equation} \rho_{XY} = Cor[X,Y] = \frac{\sigma_{XY}}{\sigma_X \sigma_Y}. \tag{4.3} \end{equation}$

Exercise 4.2 Using the data from Example 4.4, calculate the parameters of the Examples 4.5 to 4.11. $$\\$$

Exercise 4.3 Show that the Equations (2.21) and (2.22) are equivalent.