Matemática discreta Exemplos

\begin{matrix} Class & Frequency 90 - 99 & 4 80 - 89 & 6 70 - 79 & 4 60 - 69 & 3 50 - 59 & 2 40 - 49 & 1 \end{matrix}

$\begin{array}{c} Class & Frequency \\ 90 - 99 & 4 \\ 80 - 89 & 6 \\ 70 - 79 & 4 \\ 60 - 69 & 3 \\ 50 - 59 & 2 \\ 40 - 49 & 1 \end{array}$

Etapa 1

Reordene as classes com suas frequências relacionadas em ordem crescente (do menor número ao maior), que é a mais comum.

\begin{matrix} Class & Frequency (f) 40 - 49 & 1 50 - 59 & 2 60 - 69 & 3 70 - 79 & 4 80 - 89 & 6 90 - 99 & 4 \end{matrix}

$\begin{array}{c} Class & Frequency (f) \\ 40 - 49 & 1 \\ 50 - 59 & 2 \\ 60 - 69 & 3 \\ 70 - 79 & 4 \\ 80 - 89 & 6 \\ 90 - 99 & 4 \end{array}$

Etapa 2

Encontre o ponto médio

M

$M$ de cada grupo.

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Etapa 2.1

O limite inferior de cada classe é o menor valor dessa classe. Por outro lado, o limite superior de todas as classes é o maior valor dessa classe.

\begin{matrix} Class & Frequency (f) & Lower Limits & Upper Limits 40 - 49 & 1 & 40 & 49 50 - 59 & 2 & 50 & 59 60 - 69 & 3 & 60 & 69 70 - 79 & 4 & 70 & 79 80 - 89 & 6 & 80 & 89 90 - 99 & 4 & 90 & 99 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits \\ 40 - 49 & 1 & 40 & 49 \\ 50 - 59 & 2 & 50 & 59 \\ 60 - 69 & 3 & 60 & 69 \\ 70 - 79 & 4 & 70 & 79 \\ 80 - 89 & 6 & 80 & 89 \\ 90 - 99 & 4 & 90 & 99 \end{array}$

Etapa 2.2

O ponto médio da classe é o limite inferior da classe mais o limite superior da classe dividido por

2

$2$ .

\begin{matrix} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) 40 - 49 & 1 & 40 & 49 & \frac{40 + 49}{2} 50 - 59 & 2 & 50 & 59 & \frac{50 + 59}{2} 60 - 69 & 3 & 60 & 69 & \frac{60 + 69}{2} 70 - 79 & 4 & 70 & 79 & \frac{70 + 79}{2} 80 - 89 & 6 & 80 & 89 & \frac{80 + 89}{2} 90 - 99 & 4 & 90 & 99 & \frac{90 + 99}{2} \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 40 - 49 & 1 & 40 & 49 & \frac{40 + 49}{2} \\ 50 - 59 & 2 & 50 & 59 & \frac{50 + 59}{2} \\ 60 - 69 & 3 & 60 & 69 & \frac{60 + 69}{2} \\ 70 - 79 & 4 & 70 & 79 & \frac{70 + 79}{2} \\ 80 - 89 & 6 & 80 & 89 & \frac{80 + 89}{2} \\ 90 - 99 & 4 & 90 & 99 & \frac{90 + 99}{2} \end{array}$

Etapa 2.3

Simplifique toda a coluna do ponto médio.

\begin{matrix} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) 40 - 49 & 1 & 40 & 49 & 44.5 50 - 59 & 2 & 50 & 59 & 54.5 60 - 69 & 3 & 60 & 69 & 64.5 70 - 79 & 4 & 70 & 79 & 74.5 80 - 89 & 6 & 80 & 89 & 84.5 90 - 99 & 4 & 90 & 99 & 94.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Lower Limits & Upper Limits & Midpoint (M) \\ 40 - 49 & 1 & 40 & 49 & 44.5 \\ 50 - 59 & 2 & 50 & 59 & 54.5 \\ 60 - 69 & 3 & 60 & 69 & 64.5 \\ 70 - 79 & 4 & 70 & 79 & 74.5 \\ 80 - 89 & 6 & 80 & 89 & 84.5 \\ 90 - 99 & 4 & 90 & 99 & 94.5 \end{array}$

Etapa 2.4

Adicione a coluna de pontos médios à tabela original.

\begin{matrix} Class & Frequency (f) & Midpoint (M) 40 - 49 & 1 & 44.5 50 - 59 & 2 & 54.5 60 - 69 & 3 & 64.5 70 - 79 & 4 & 74.5 80 - 89 & 6 & 84.5 90 - 99 & 4 & 94.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 40 - 49 & 1 & 44.5 \\ 50 - 59 & 2 & 54.5 \\ 60 - 69 & 3 & 64.5 \\ 70 - 79 & 4 & 74.5 \\ 80 - 89 & 6 & 84.5 \\ 90 - 99 & 4 & 94.5 \end{array}$

\begin{matrix} Class & Frequency (f) & Midpoint (M) 40 - 49 & 1 & 44.5 50 - 59 & 2 & 54.5 60 - 69 & 3 & 64.5 70 - 79 & 4 & 74.5 80 - 89 & 6 & 84.5 90 - 99 & 4 & 94.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 40 - 49 & 1 & 44.5 \\ 50 - 59 & 2 & 54.5 \\ 60 - 69 & 3 & 64.5 \\ 70 - 79 & 4 & 74.5 \\ 80 - 89 & 6 & 84.5 \\ 90 - 99 & 4 & 94.5 \end{array}$

Etapa 3

Calcule o quadrado do ponto médio de cada grupo

M^{2}

$M^{2}$ .

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} 40 - 49 & 1 & 44.5 & {44.5}^{2} 50 - 59 & 2 & 54.5 & {54.5}^{2} 60 - 69 & 3 & 64.5 & {64.5}^{2} 70 - 79 & 4 & 74.5 & {74.5}^{2} 80 - 89 & 6 & 84.5 & {84.5}^{2} 90 - 99 & 4 & 94.5 & {94.5}^{2} \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 40 - 49 & 1 & 44.5 & {44.5}^{2} \\ 50 - 59 & 2 & 54.5 & {54.5}^{2} \\ 60 - 69 & 3 & 64.5 & {64.5}^{2} \\ 70 - 79 & 4 & 74.5 & {74.5}^{2} \\ 80 - 89 & 6 & 84.5 & {84.5}^{2} \\ 90 - 99 & 4 & 94.5 & {94.5}^{2} \end{array}$

Etapa 4

Simplifique a coluna

M^{2}

$M^{2}$ .

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} 40 - 49 & 1 & 44.5 & 1980.25 50 - 59 & 2 & 54.5 & 2970.25 60 - 69 & 3 & 64.5 & 4160.25 70 - 79 & 4 & 74.5 & 5550.25 80 - 89 & 6 & 84.5 & 7140.25 90 - 99 & 4 & 94.5 & 8930.25 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} \\ 40 - 49 & 1 & 44.5 & 1980.25 \\ 50 - 59 & 2 & 54.5 & 2970.25 \\ 60 - 69 & 3 & 64.5 & 4160.25 \\ 70 - 79 & 4 & 74.5 & 5550.25 \\ 80 - 89 & 6 & 84.5 & 7140.25 \\ 90 - 99 & 4 & 94.5 & 8930.25 \end{array}$

Etapa 5

Multiplique o quadrado de cada ponto médio por sua frequência

f

$f$ .

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} 40 - 49 & 1 & 44.5 & 1980.25 & 1 \cdot 1980.25 50 - 59 & 2 & 54.5 & 2970.25 & 2 \cdot 2970.25 60 - 69 & 3 & 64.5 & 4160.25 & 3 \cdot 4160.25 70 - 79 & 4 & 74.5 & 5550.25 & 4 \cdot 5550.25 80 - 89 & 6 & 84.5 & 7140.25 & 6 \cdot 7140.25 90 - 99 & 4 & 94.5 & 8930.25 & 4 \cdot 8930.25 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 40 - 49 & 1 & 44.5 & 1980.25 & 1 \cdot 1980.25 \\ 50 - 59 & 2 & 54.5 & 2970.25 & 2 \cdot 2970.25 \\ 60 - 69 & 3 & 64.5 & 4160.25 & 3 \cdot 4160.25 \\ 70 - 79 & 4 & 74.5 & 5550.25 & 4 \cdot 5550.25 \\ 80 - 89 & 6 & 84.5 & 7140.25 & 6 \cdot 7140.25 \\ 90 - 99 & 4 & 94.5 & 8930.25 & 4 \cdot 8930.25 \end{array}$

Etapa 6

Simplifique a coluna

f \cdot M^{2}

$f \cdot M^{2}$ .

\begin{matrix} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} 40 - 49 & 1 & 44.5 & 1980.25 & 1980.25 50 - 59 & 2 & 54.5 & 2970.25 & 5940.5 60 - 69 & 3 & 64.5 & 4160.25 & 12480.75 70 - 79 & 4 & 74.5 & 5550.25 & 22201 80 - 89 & 6 & 84.5 & 7140.25 & 42841.5 90 - 99 & 4 & 94.5 & 8930.25 & 35721 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & M^{2} & f \cdot M^{2} \\ 40 - 49 & 1 & 44.5 & 1980.25 & 1980.25 \\ 50 - 59 & 2 & 54.5 & 2970.25 & 5940.5 \\ 60 - 69 & 3 & 64.5 & 4160.25 & 12480.75 \\ 70 - 79 & 4 & 74.5 & 5550.25 & 22201 \\ 80 - 89 & 6 & 84.5 & 7140.25 & 42841.5 \\ 90 - 99 & 4 & 94.5 & 8930.25 & 35721 \end{array}$

Etapa 7

Encontre a soma de todas as frequências. Neste caso, a soma de todas as frequências é

n = 1, 2, 3, 4, 6, 4 = 20

$n = 1, 2, 3, 4, 6, 4 = 20$ .

\sum f = n = 20

$\sum_{}^{} f = n = 20$

Etapa 8

Encontre a soma da coluna

f \cdot M^{2}

$f \cdot M^{2}$ . Neste caso,

1980.25 + 5940.5 + 12480.75 + 22201 + 42841.5 + 35721 = 121165

$1980.25 + 5940.5 + 12480.75 + 22201 + 42841.5 + 35721 = 121165$ .

\sum f \cdot M^{2} = 121165

$\sum_{}^{} f \cdot M^{2} = 121165$

Etapa 9

Encontre a média de

μ

$μ$ .

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Etapa 9.1

Reorder the classes with their related frequencies (ƒ) in an ascending order (lowest number to highest), which is the most common.

\begin{matrix} Class & Frequency (f) 40 - 49 & 1 50 - 59 & 2 60 - 69 & 3 70 - 79 & 4 80 - 89 & 6 90 - 99 & 4 \end{matrix}

$\begin{array}{c} Class & Frequency (f) \\ 40 - 49 & 1 \\ 50 - 59 & 2 \\ 60 - 69 & 3 \\ 70 - 79 & 4 \\ 80 - 89 & 6 \\ 90 - 99 & 4 \end{array}$

Etapa 9.2

Encontre o ponto médio

M

$M$ de cada classe.

\begin{matrix} Class & Frequency (f) & Midpoint (M) 40 - 49 & 1 & 44.5 50 - 59 & 2 & 54.5 60 - 69 & 3 & 64.5 70 - 79 & 4 & 74.5 80 - 89 & 6 & 84.5 90 - 99 & 4 & 94.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) \\ 40 - 49 & 1 & 44.5 \\ 50 - 59 & 2 & 54.5 \\ 60 - 69 & 3 & 64.5 \\ 70 - 79 & 4 & 74.5 \\ 80 - 89 & 6 & 84.5 \\ 90 - 99 & 4 & 94.5 \end{array}$

Etapa 9.3

Multiplique a frequência de cada classe pelo ponto médio da classe.

\begin{matrix} Class & Frequency (f) & Midpoint (M) & f \cdot M 40 - 49 & 1 & 44.5 & 1 \cdot 44.5 50 - 59 & 2 & 54.5 & 2 \cdot 54.5 60 - 69 & 3 & 64.5 & 3 \cdot 64.5 70 - 79 & 4 & 74.5 & 4 \cdot 74.5 80 - 89 & 6 & 84.5 & 6 \cdot 84.5 90 - 99 & 4 & 94.5 & 4 \cdot 94.5 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 40 - 49 & 1 & 44.5 & 1 \cdot 44.5 \\ 50 - 59 & 2 & 54.5 & 2 \cdot 54.5 \\ 60 - 69 & 3 & 64.5 & 3 \cdot 64.5 \\ 70 - 79 & 4 & 74.5 & 4 \cdot 74.5 \\ 80 - 89 & 6 & 84.5 & 6 \cdot 84.5 \\ 90 - 99 & 4 & 94.5 & 4 \cdot 94.5 \end{array}$

Etapa 9.4

Simplifique a coluna

f \cdot M

$f \cdot M$ .

\begin{matrix} Class & Frequency (f) & Midpoint (M) & f \cdot M 40 - 49 & 1 & 44.5 & 44.5 50 - 59 & 2 & 54.5 & 109 60 - 69 & 3 & 64.5 & 193.5 70 - 79 & 4 & 74.5 & 298 80 - 89 & 6 & 84.5 & 507 90 - 99 & 4 & 94.5 & 378 \end{matrix}

$\begin{array}{c} Class & Frequency (f) & Midpoint (M) & f \cdot M \\ 40 - 49 & 1 & 44.5 & 44.5 \\ 50 - 59 & 2 & 54.5 & 109 \\ 60 - 69 & 3 & 64.5 & 193.5 \\ 70 - 79 & 4 & 74.5 & 298 \\ 80 - 89 & 6 & 84.5 & 507 \\ 90 - 99 & 4 & 94.5 & 378 \end{array}$

Etapa 9.5

Some os valores na coluna

f \cdot M

$f \cdot M$ .

44.5 + 109 + 193.5 + 298 + 507 + 378 = 1530

$44.5 + 109 + 193.5 + 298 + 507 + 378 = 1530$

Etapa 9.6

Some os valores na coluna de frequência.

n = 1 + 2 + 3 + 4 + 6 + 4 = 20

$n = 1 + 2 + 3 + 4 + 6 + 4 = 20$

Etapa 9.7

A média ( mu ) é a soma de

f \cdot M

$f \cdot M$ dividida por

n

$n$ , que é a soma das frequências.

μ = \frac{\sum f \cdot M}{\sum f}

$μ = \frac{\sum_{}^{} f \cdot M}{\sum_{}^{} f}$

Etapa 9.8

A média é a soma do produto dos pontos médios e das frequências dividida pelo total de frequências.

μ = \frac{1530}{20}

$μ = \frac{1530}{20}$

Etapa 9.9

Simplifique o lado direito de

μ = \frac{1530}{20}

$μ = \frac{1530}{20}$ .

76.5

$76.5$

76.5

$76.5$

Etapa 10

A equação do desvio padrão é

S^{2} = \frac{\sum f \cdot M^{2} - n {(μ)}^{2}}{n - 1}

$S^{2} = \frac{\sum_{}^{} f \cdot M^{2} - n {(μ)}^{2}}{n - 1}$ .

S^{2} = \frac{\sum f \cdot M^{2} - n {(μ)}^{2}}{n - 1}

$S^{2} = \frac{\sum_{}^{} f \cdot M^{2} - n {(μ)}^{2}}{n - 1}$

Etapa 11

Substitua os valores calculados em

S^{2} = \frac{\sum f \cdot M^{2} - n {(μ)}^{2}}{n - 1}

$S^{2} = \frac{\sum_{}^{} f \cdot M^{2} - n {(μ)}^{2}}{n - 1}$ .

S^{2} = \frac{121165 - 20 {(76.5)}^{2}}{20 - 1}

$S^{2} = \frac{121165 - 20 {(76.5)}^{2}}{20 - 1}$

Etapa 12

Simplifique o lado direito de

S^{2} = \frac{121165 - 20 {(76.5)}^{2}}{20 - 1}

$S^{2} = \frac{121165 - 20 {(76.5)}^{2}}{20 - 1}$ para obter a variância

S^{2} = 216.84210526

$S^{2} = 216.84210526$ .

216.84210526

$216.84210526$

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