Matemática discreta Ejemplos

$42$ , $42$ , $44$ , $44$ , $45$ , $47$ , $48$ , $49$ , $51$ , $58$ , $60$ , $65$

Paso 1

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

$\overline{x} = \frac{42 + 42 + 44 + 44 + 45 + 47 + 48 + 49 + 51 + 58 + 60 + 65}{12}$

Paso 2

Simplifica el numerador.

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Paso 2.1

Suma $42$ y $42$ .

$\overline{x} = \frac{84 + 44 + 44 + 45 + 47 + 48 + 49 + 51 + 58 + 60 + 65}{12}$

Paso 2.2

Suma $84$ y $44$ .

$\overline{x} = \frac{128 + 44 + 45 + 47 + 48 + 49 + 51 + 58 + 60 + 65}{12}$

Paso 2.3

Suma $128$ y $44$ .

$\overline{x} = \frac{172 + 45 + 47 + 48 + 49 + 51 + 58 + 60 + 65}{12}$

Paso 2.4

Suma $172$ y $45$ .

$\overline{x} = \frac{217 + 47 + 48 + 49 + 51 + 58 + 60 + 65}{12}$

Paso 2.5

Suma $217$ y $47$ .

$\overline{x} = \frac{264 + 48 + 49 + 51 + 58 + 60 + 65}{12}$

Paso 2.6

Suma $264$ y $48$ .

$\overline{x} = \frac{312 + 49 + 51 + 58 + 60 + 65}{12}$

Paso 2.7

Suma $312$ y $49$ .

$\overline{x} = \frac{361 + 51 + 58 + 60 + 65}{12}$

Paso 2.8

Suma $361$ y $51$ .

$\overline{x} = \frac{412 + 58 + 60 + 65}{12}$

Paso 2.9

Suma $412$ y $58$ .

$\overline{x} = \frac{470 + 60 + 65}{12}$

Paso 2.10

Suma $470$ y $60$ .

$\overline{x} = \frac{530 + 65}{12}$

Paso 2.11

Suma $530$ y $65$ .

$\overline{x} = \frac{595}{12}$

Paso 3

Divide.

$\overline{x} = 49.58\overline{3}$

Paso 4

La media debería redondearse a una cifra decimal más que los datos originales. Si los datos originales fueran mixtos, redondea a una cifra decimal más que el dato menos preciso.

$\overline{x} = 49.6$

Paso 5

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 6

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

$s = \frac{{(42 - 49.6)}^{2} + {(42 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7

Simplifica el resultado.

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Paso 7.1

Simplifica el numerador.

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Paso 7.1.1

Resta $49.6$ de $42$ .

$s = \frac{{(- 7.6)}^{2} + {(42 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.2

Eleva $- 7.6$ a la potencia de $2$ .

$s = \frac{57.76 + {(42 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.3

Resta $49.6$ de $42$ .

$s = \frac{57.76 + {(- 7.6)}^{2} + {(44 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.4

Eleva $- 7.6$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + {(44 - 49.6)}^{2} + {(44 - 49.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.5

Resta $49.6$ de $44$ .

$s = \frac{57.76 + 57.76 + {(- 5.6)}^{2} + {(44 - 49.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.6

Eleva $- 5.6$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + {(44 - 49.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.7

Resta $49.6$ de $44$ .

$s = \frac{57.76 + 57.76 + 31.36 + {(- 5.6)}^{2} + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.8

Eleva $- 5.6$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + {(45 - 49.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.9

Resta $49.6$ de $45$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + {(- 4.6)}^{2} + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.10

Eleva $- 4.6$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + {(47 - 49.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.11

Resta $49.6$ de $47$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + {(- 2.6)}^{2} + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.12

Eleva $- 2.6$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + {(48 - 49.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.13

Resta $49.6$ de $48$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + {(- 1.6)}^{2} + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.14

Eleva $- 1.6$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + {(49 - 49.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.15

Resta $49.6$ de $49$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + {(- 0.6)}^{2} + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.16

Eleva $- 0.6$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + {(51 - 49.6)}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.17

Resta $49.6$ de $51$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + {1.4}^{2} + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.18

Eleva $1.4$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + {(58 - 49.6)}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.19

Resta $49.6$ de $58$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + {8.4}^{2} + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.20

Eleva $8.4$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + {(60 - 49.6)}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.21

Resta $49.6$ de $60$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + {10.4}^{2} + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.22

Eleva $10.4$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + {(65 - 49.6)}^{2}}{12 - 1}$

Paso 7.1.23

Resta $49.6$ de $65$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + {15.4}^{2}}{12 - 1}$

Paso 7.1.24

Eleva $15.4$ a la potencia de $2$ .

$s = \frac{57.76 + 57.76 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.25

Suma $57.76$ y $57.76$ .

$s = \frac{115.52 + 31.36 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.26

Suma $115.52$ y $31.36$ .

$s = \frac{146.88 + 31.36 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.27

Suma $146.88$ y $31.36$ .

$s = \frac{178.24 + 21.16 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.28

Suma $178.24$ y $21.16$ .

$s = \frac{199.4 + 6.76 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.29

Suma $199.4$ y $6.76$ .

$s = \frac{206.16 + 2.56 + 0.36 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.30

Suma $206.16$ y $2.56$ .

$s = \frac{208.72 + 0.36 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.31

Suma $208.72$ y $0.36$ .

$s = \frac{209.08 + 1.96 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.32

Suma $209.08$ y $1.96$ .

$s = \frac{211.04 + 70.56 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.33

Suma $211.04$ y $70.56$ .

$s = \frac{281.6 + 108.16 + 237.16}{12 - 1}$

Paso 7.1.34

Suma $281.6$ y $108.16$ .

$s = \frac{389.76 + 237.16}{12 - 1}$

Paso 7.1.35

Suma $389.76$ y $237.16$ .

$s = \frac{626.92}{12 - 1}$

Paso 7.2

Simplifica la expresión.

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Paso 7.2.1

Resta $1$ de $12$ .

$s = \frac{626.92}{11}$

Paso 7.2.2

Divide $626.92$ por $11$ .

$s = 56.99\overline{27}$

Paso 8

Aproxima el resultado.

$s^{2} \approx 56.9927$