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 4.9 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 \[\begin{equation} \rho_{XY} = Cor[X,Y] = \frac{\sigma_{XY}}{\sigma_X \sigma_Y}. \tag{4.3} \end{equation}\]