Ensino Superior ⇒ Estatística: Média, Mediana e Desvio padrão

[silvia]

Calcule a média, a mediana e o desvio padrão da variável salário da tabela abaixo.

[tex3]\hspace{70px}\begin{array}{|r|c|}\hline\text{Salário}\,\, \text{(R\$)} \,\,\,\,& \text{N. de funcionários}\\
\hline 350,00 \,\,\,\,& 10\\
400,00 \,\,\,\,& 7\\
550,00 \,\,\,\,& 2\\
2640,00 \,\,\,\,& 1\\
\hline \text{Total}\,\,\,\, & \\ \hline \end{array}[/tex3]

a) [tex3]\text{R}\$\, 520,00;\, \text{R}\$\,357,00;\,\text{R}\$\,539,49[/tex3]

b) [tex3]\text{R}\$\,550,00;\,\text{R}\$\,387,00;\,\text{R}\$\,469,49[/tex3]

c) [tex3]\text{R}\$\,600,00;\,\text{R}\$\,437,00;\,\text{R}\$\,519,49[/tex3]

d)[tex3]\text{R}\$\,502,00;\,\text{R}\$\,375,00;\,\text{R}\$\,493,94[/tex3]

e) [tex3]\text{R}\$\,505,00;\,\text{R}\$\,400,00;\,\text{R}\$\,475,00[/tex3]

[Karl Weierstrass]

[tex3]\hspace{70px}\overline{x}=\frac{\small\sum \large x_if_i}{n}[/tex3]

[tex3]\hspace{70px}\begin{array}{|r|cc|cc|cc|cc|}\hline x_i && f_i&&F_i&&x_if_i&&x_i^2f_i\\
\hline 350 && 10 && 10&& 3500 && 1225000\\
400 && 7 && 17 &&2800 && 1120000\\
550 && 2 && 19&& 1100 && 605000\\
2640 && 1 && 20 && 2640 && 6969600\\
\hline \text{Total}&& n=20 && \,&& \small\sum\large x_if_i=10040 && \small\sum\large x_i^2f_i=9919600\\ \hline\end{array}[/tex3]

[tex3]\hspace{70px}\overline{x}=\Large\frac{\small\sum \large x_if_i}{n}\large=\Large\frac{10040}{20}\large=502,00\,\,\text{reais.}[/tex3]

Como o número de observações é par, segue que o elemento mediano é dado por

[tex3]\hspace{70px}E_{Md}=\Large\frac{n}{2}\large\,=\,\Large\frac{20}{2}\large\,=\,10.[/tex3]

Desse modo, o salário mediano é a média aritmética simples entre as observações de ordem [tex3]10[/tex3] e [tex3]11[/tex3]:

[tex3]\hspace{70px}Md=\Large\frac{x_{10}\,+\,x_{11}}{2}\large\,=\,\Large\frac{350\,+\,400}{2}\large\,=\,375,00 \,\text{reais}[/tex3]

Considerando a Variância Populacional, temos

[tex3]\hspace{70px}V\,=\,\Large\frac{1}{n}\,\cdot\,\large\left[\small{\sum}\large(x_i^2f_i)\,-\,\Large\frac{\left(\small{\sum}\large x_if_i\right)\large ^2}{n}\large\right][/tex3]

[tex3]\hspace{70px}V\,=\,\Large\frac{1}{20}\,\cdot\,\large\left[9919600\,-\,\Large\frac{(10040)\large ^2}{20}\large\right]\,=\,243976,00\,\text{reais}^2[/tex3]

[tex3]\hspace{70px}D\,=\,\sqrt V\,=\,\sqrt{243976}\,\approx \,493,94 \,\text{reais}[/tex3]

[tex3]\boxed{\text{D}}[/tex3]