Matemática discreta Ejemplos

$4$ , $5$ , $6$ , $7$ , $8$ , $9$ , $9$ , $10$ , $12$ , $12$ , $14$ , $16$ , $17$ , $17$ , $19$

Paso 1

La media de un conjunto de números es la suma dividida por la cantidad de términos.

$\overline{x} = \frac{4 + 5 + 6 + 7 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2

Simplifica el numerador.

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Paso 2.1

Suma $4$ y $5$ .

$\overline{x} = \frac{9 + 6 + 7 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.2

Suma $9$ y $6$ .

$\overline{x} = \frac{15 + 7 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.3

Suma $15$ y $7$ .

$\overline{x} = \frac{22 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.4

Suma $22$ y $8$ .

$\overline{x} = \frac{30 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.5

Suma $30$ y $9$ .

$\overline{x} = \frac{39 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.6

Suma $39$ y $9$ .

$\overline{x} = \frac{48 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.7

Suma $48$ y $10$ .

$\overline{x} = \frac{58 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.8

Suma $58$ y $12$ .

$\overline{x} = \frac{70 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.9

Suma $70$ y $12$ .

$\overline{x} = \frac{82 + 14 + 16 + 17 + 17 + 19}{15}$

Paso 2.10

Suma $82$ y $14$ .

$\overline{x} = \frac{96 + 16 + 17 + 17 + 19}{15}$

Paso 2.11

Suma $96$ y $16$ .

$\overline{x} = \frac{112 + 17 + 17 + 19}{15}$

Paso 2.12

Suma $112$ y $17$ .

$\overline{x} = \frac{129 + 17 + 19}{15}$

Paso 2.13

Suma $129$ y $17$ .

$\overline{x} = \frac{146 + 19}{15}$

Paso 2.14

Suma $146$ y $19$ .

$\overline{x} = \frac{165}{15}$

Paso 3

Divide $165$ por $15$ .

$\overline{x} = 11$

Paso 4

Establece la fórmula para la varianza. La varianza de un conjunto de valores es una medida de la propagación de sus valores.

$s^{2} = \sum_{i = 1}^{n} \frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}$

Paso 5

Establece la fórmula para la varianza de este conjunto de números.

$s = \frac{{(4 - 11)}^{2} + {(5 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6

Simplifica el resultado.

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Paso 6.1

Simplifica el numerador.

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Paso 6.1.1

Resta $11$ de $4$ .

$s = \frac{{(- 7)}^{2} + {(5 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.2

Eleva $- 7$ a la potencia de $2$ .

$s = \frac{49 + {(5 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.3

Resta $11$ de $5$ .

$s = \frac{49 + {(- 6)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.4

Eleva $- 6$ a la potencia de $2$ .

$s = \frac{49 + 36 + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.5

Resta $11$ de $6$ .

$s = \frac{49 + 36 + {(- 5)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.6

Eleva $- 5$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.7

Resta $11$ de $7$ .

$s = \frac{49 + 36 + 25 + {(- 4)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.8

Eleva $- 4$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.9

Resta $11$ de $8$ .

$s = \frac{49 + 36 + 25 + 16 + {(- 3)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.10

Eleva $- 3$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.11

Resta $11$ de $9$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + {(- 2)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.12

Eleva $- 2$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.13

Resta $11$ de $9$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + {(- 2)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.14

Eleva $- 2$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.15

Resta $11$ de $10$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + {(- 1)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.16

Eleva $- 1$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.17

Resta $11$ de $12$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.18

Uno elevado a cualquier potencia es uno.

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.19

Resta $11$ de $12$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.20

Uno elevado a cualquier potencia es uno.

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.21

Resta $11$ de $14$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 3^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.22

Eleva $3$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.23

Resta $11$ de $16$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 5^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.24

Eleva $5$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.25

Resta $11$ de $17$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 6^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.26

Eleva $6$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.27

Resta $11$ de $17$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 6^{2} + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.28

Eleva $6$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + {(19 - 11)}^{2}}{15 - 1}$

Paso 6.1.29

Resta $11$ de $19$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 8^{2}}{15 - 1}$

Paso 6.1.30

Eleva $8$ a la potencia de $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.31

Suma $49$ y $36$ .

$s = \frac{85 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.32

Suma $85$ y $25$ .

$s = \frac{110 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.33

Suma $110$ y $16$ .

$s = \frac{126 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.34

Suma $126$ y $9$ .

$s = \frac{135 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.35

Suma $135$ y $4$ .

$s = \frac{139 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.36

Suma $139$ y $4$ .

$s = \frac{143 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.37

Suma $143$ y $1$ .

$s = \frac{144 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.38

Suma $144$ y $1$ .

$s = \frac{145 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.39

Suma $145$ y $1$ .

$s = \frac{146 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.40

Suma $146$ y $9$ .

$s = \frac{155 + 25 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.41

Suma $155$ y $25$ .

$s = \frac{180 + 36 + 36 + 64}{15 - 1}$

Paso 6.1.42

Suma $180$ y $36$ .

$s = \frac{216 + 36 + 64}{15 - 1}$

Paso 6.1.43

Suma $216$ y $36$ .

$s = \frac{252 + 64}{15 - 1}$

Paso 6.1.44

Suma $252$ y $64$ .

$s = \frac{316}{15 - 1}$

Paso 6.2

Reduce la expresión mediante la cancelación de los factores comunes.

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Paso 6.2.1

Resta $1$ de $15$ .

$s = \frac{316}{14}$

Paso 6.2.2

Cancela el factor común de $316$ y $14$ .

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Paso 6.2.2.1

Factoriza $2$ de $316$ .

$s = \frac{2 (158)}{14}$

Paso 6.2.2.2

Cancela los factores comunes.

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Paso 6.2.2.2.1

Factoriza $2$ de $14$ .

$s = \frac{2 \cdot 158}{2 \cdot 7}$

Paso 6.2.2.2.2

Cancela el factor común.

$s = \frac{2 \cdot 158}{2 \cdot 7}$

Paso 6.2.2.2.3

Reescribe la expresión.

$s = \frac{158}{7}$

Paso 7

Aproxima el resultado.

$s^{2} \approx 22.5714$