Matemática básica Ejemplos

$1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $9$ , $10$ , $11$ , $12$

Paso 1

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

$\overline{x} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Paso 2

Simplifica el numerador.

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Paso 2.1

Suma $1$ y $2$ .

$\overline{x} = \frac{3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Paso 2.2

Suma $3$ y $3$ .

$\overline{x} = \frac{6 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Paso 2.3

Suma $6$ y $4$ .

$\overline{x} = \frac{10 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Paso 2.4

Suma $10$ y $5$ .

$\overline{x} = \frac{15 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Paso 2.5

Suma $15$ y $6$ .

$\overline{x} = \frac{21 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Paso 2.6

Suma $21$ y $7$ .

$\overline{x} = \frac{28 + 8 + 9 + 10 + 11 + 12}{12}$

Paso 2.7

Suma $28$ y $8$ .

$\overline{x} = \frac{36 + 9 + 10 + 11 + 12}{12}$

Paso 2.8

Suma $36$ y $9$ .

$\overline{x} = \frac{45 + 10 + 11 + 12}{12}$

Paso 2.9

Suma $45$ y $10$ .

$\overline{x} = \frac{55 + 11 + 12}{12}$

Paso 2.10

Suma $55$ y $11$ .

$\overline{x} = \frac{66 + 12}{12}$

Paso 2.11

Suma $66$ y $12$ .

$\overline{x} = \frac{78}{12}$

Paso 3

Cancela el factor común de $78$ y $12$ .

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Paso 3.1

Factoriza $6$ de $78$ .

$\overline{x} = \frac{6 (13)}{12}$

Paso 3.2

Cancela los factores comunes.

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Paso 3.2.1

Factoriza $6$ de $12$ .

$\overline{x} = \frac{6 \cdot 13}{6 \cdot 2}$

Paso 3.2.2

Cancela el factor común.

$\overline{x} = \frac{6 \cdot 13}{6 \cdot 2}$

Paso 3.2.3

Reescribe la expresión.

$\overline{x} = \frac{13}{2}$

Paso 4

Divide.

$\overline{x} = 6.5$

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.

$s = \frac{{(1 - 6.5)}^{2} + {(2 - 6.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7

Simplifica el resultado.

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Paso 7.1

Simplifica el numerador.

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Paso 7.1.1

Resta $6.5$ de $1$ .

$s = \frac{{(- 5.5)}^{2} + {(2 - 6.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.2

Eleva $- 5.5$ a la potencia de $2$ .

$s = \frac{30.25 + {(2 - 6.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.3

Resta $6.5$ de $2$ .

$s = \frac{30.25 + {(- 4.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.4

Eleva $- 4.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.5

Resta $6.5$ de $3$ .

$s = \frac{30.25 + 20.25 + {(- 3.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.6

Eleva $- 3.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.7

Resta $6.5$ de $4$ .

$s = \frac{30.25 + 20.25 + 12.25 + {(- 2.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.8

Eleva $- 2.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.9

Resta $6.5$ de $5$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + {(- 1.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.10

Eleva $- 1.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.11

Resta $6.5$ de $6$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + {(- 0.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.12

Eleva $- 0.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.13

Resta $6.5$ de $7$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + {0.5}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.14

Eleva $0.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.15

Resta $6.5$ de $8$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + {1.5}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.16

Eleva $1.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.17

Resta $6.5$ de $9$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + {2.5}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.18

Eleva $2.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.19

Resta $6.5$ de $10$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + {3.5}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.20

Eleva $3.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.21

Resta $6.5$ de $11$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + {4.5}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.22

Eleva $4.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + {(12 - 6.5)}^{2}}{12 - 1}$

Paso 7.1.23

Resta $6.5$ de $12$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + {5.5}^{2}}{12 - 1}$

Paso 7.1.24

Eleva $5.5$ a la potencia de $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.25

Suma $30.25$ y $20.25$ .

$s = \frac{50.5 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.26

Suma $50.5$ y $12.25$ .

$s = \frac{62.75 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.27

Suma $62.75$ y $6.25$ .

$s = \frac{69 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.28

Suma $69$ y $2.25$ .

$s = \frac{71.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.29

Suma $71.25$ y $0.25$ .

$s = \frac{71.5 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.30

Suma $71.5$ y $0.25$ .

$s = \frac{71.75 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.31

Suma $71.75$ y $2.25$ .

$s = \frac{74 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.32

Suma $74$ y $6.25$ .

$s = \frac{80.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.33

Suma $80.25$ y $12.25$ .

$s = \frac{92.5 + 20.25 + 30.25}{12 - 1}$

Paso 7.1.34

Suma $92.5$ y $20.25$ .

$s = \frac{112.75 + 30.25}{12 - 1}$

Paso 7.1.35

Suma $112.75$ y $30.25$ .

$s = \frac{143}{12 - 1}$

Paso 7.2

Simplifica la expresión.

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Paso 7.2.1

Resta $1$ de $12$ .

$s = \frac{143}{11}$

Paso 7.2.2

Divide $143$ por $11$ .

$s = 13$

Paso 8

Aproxima el resultado.

$s^{2} \approx 13$