Matemática discreta Exemplos

$54556$ , $1545$ , $35385$ , $46055$ , $10555$ , $9614$ , $59425$ , $69898$ , $12407$ , $32020$ , $24230$ , $42143$

Etapa 1

A média de um conjunto de números é a soma dividida pelo número de termos.

$\overline{x} = \frac{54556 + 1545 + 35385 + 46055 + 10555 + 9614 + 59425 + 69898 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2

Simplifique o numerador.

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Etapa 2.1

Some $54556$ e $1545$ .

$\overline{x} = \frac{56101 + 35385 + 46055 + 10555 + 9614 + 59425 + 69898 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2.2

Some $56101$ e $35385$ .

$\overline{x} = \frac{91486 + 46055 + 10555 + 9614 + 59425 + 69898 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2.3

Some $91486$ e $46055$ .

$\overline{x} = \frac{137541 + 10555 + 9614 + 59425 + 69898 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2.4

Some $137541$ e $10555$ .

$\overline{x} = \frac{148096 + 9614 + 59425 + 69898 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2.5

Some $148096$ e $9614$ .

$\overline{x} = \frac{157710 + 59425 + 69898 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2.6

Some $157710$ e $59425$ .

$\overline{x} = \frac{217135 + 69898 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2.7

Some $217135$ e $69898$ .

$\overline{x} = \frac{287033 + 12407 + 32020 + 24230 + 42143}{12}$

Etapa 2.8

Some $287033$ e $12407$ .

$\overline{x} = \frac{299440 + 32020 + 24230 + 42143}{12}$

Etapa 2.9

Some $299440$ e $32020$ .

$\overline{x} = \frac{331460 + 24230 + 42143}{12}$

Etapa 2.10

Some $331460$ e $24230$ .

$\overline{x} = \frac{355690 + 42143}{12}$

Etapa 2.11

Some $355690$ e $42143$ .

$\overline{x} = \frac{397833}{12}$

Etapa 3

Cancele o fator comum de $397833$ e $12$ .

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Etapa 3.1

Fatore $3$ de $397833$ .

$\overline{x} = \frac{3 (132611)}{12}$

Etapa 3.2

Cancele os fatores comuns.

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Etapa 3.2.1

Fatore $3$ de $12$ .

$\overline{x} = \frac{3 \cdot 132611}{3 \cdot 4}$

Etapa 3.2.2

Cancele o fator comum.

$\overline{x} = \frac{3 \cdot 132611}{3 \cdot 4}$

Etapa 3.2.3

Reescreva a expressão.

$\overline{x} = \frac{132611}{4}$

Etapa 4

Divida.

$\overline{x} = 33152.75$

Etapa 5

A média deve ser arredondada para uma casa decimal a mais do que os dados originais. Se os dados originais forem misturados, arredonde para uma casa decimal a mais do que a menos precisa.

$\overline{x} = 33152.8$

Etapa 6

Estabeleça a fórmula da variância. A variância de um conjunto de valores é uma medida da propagação de seus valores.

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

Etapa 7

Estabeleça a fórmula da variância para este conjunto de números.

$s = \frac{{(54556 - 33152.8)}^{2} + {(1545 - 33152.8)}^{2} + {(35385 - 33152.8)}^{2} + {(46055 - 33152.8)}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8

Simplifique o resultado.

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Etapa 8.1

Simplifique o numerador.

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Etapa 8.1.1

Subtraia $33152.8$ de $54556$ .

$s = \frac{{21403.2}^{2} + {(1545 - 33152.8)}^{2} + {(35385 - 33152.8)}^{2} + {(46055 - 33152.8)}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.2

Eleve $21403.2$ à potência de $2$ .

$s = \frac{458096970.24 + {(1545 - 33152.8)}^{2} + {(35385 - 33152.8)}^{2} + {(46055 - 33152.8)}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.3

Subtraia $33152.8$ de $1545$ .

$s = \frac{458096970.24 + {(- 31607.8)}^{2} + {(35385 - 33152.8)}^{2} + {(46055 - 33152.8)}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.4

Eleve $- 31607.8$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + {(35385 - 33152.8)}^{2} + {(46055 - 33152.8)}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.5

Subtraia $33152.8$ de $35385$ .

$s = \frac{458096970.24 + 999053020.84 + {2232.2}^{2} + {(46055 - 33152.8)}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.6

Eleve $2232.2$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + {(46055 - 33152.8)}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.7

Subtraia $33152.8$ de $46055$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + {12902.2}^{2} + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.8

Eleve $12902.2$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + {(10555 - 33152.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.9

Subtraia $33152.8$ de $10555$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + {(- 22597.8)}^{2} + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.10

Eleve $- 22597.8$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + {(9614 - 33152.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.11

Subtraia $33152.8$ de $9614$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + {(- 23538.8)}^{2} + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.12

Eleve $- 23538.8$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + {(59425 - 33152.8)}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.13

Subtraia $33152.8$ de $59425$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + {26272.2}^{2} + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.14

Eleve $26272.2$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + {(69898 - 33152.8)}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.15

Subtraia $33152.8$ de $69898$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + {36745.2}^{2} + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.16

Eleve $36745.2$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + {(12407 - 33152.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.17

Subtraia $33152.8$ de $12407$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + {(- 20745.8)}^{2} + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.18

Eleve $- 20745.8$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + {(32020 - 33152.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.19

Subtraia $33152.8$ de $32020$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + {(- 1132.8)}^{2} + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.20

Eleve $- 1132.8$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + {(24230 - 33152.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.21

Subtraia $33152.8$ de $24230$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + {(- 8922.8)}^{2} + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.22

Eleve $- 8922.8$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + {(42143 - 33152.8)}^{2}}{12 - 1}$

Etapa 8.1.23

Subtraia $33152.8$ de $42143$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + {8990.2}^{2}}{12 - 1}$

Etapa 8.1.24

Eleve $8990.2$ à potência de $2$ .

$s = \frac{458096970.24 + 999053020.84 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.25

Some $458096970.24$ e $999053020.84$ .

$s = \frac{1457149991.08 + 4982716.84 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.26

Some $1457149991.08$ e $4982716.84$ .

$s = \frac{1462132707.92 + 166466764.84 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.27

Some $1462132707.92$ e $166466764.84$ .

$s = \frac{1628599472.76 + 510660564.84 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.28

Some $1628599472.76$ e $510660564.84$ .

$s = \frac{2139260037.6 + 554075105.44 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.29

Some $2139260037.6$ e $554075105.44$ .

$s = \frac{2693335143.04 + 690228492.84 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.30

Some $2693335143.04$ e $690228492.84$ .

$s = \frac{3383563635.88 + 1350209723.04 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.31

Some $3383563635.88$ e $1350209723.04$ .

$s = \frac{4733773358.92 + 430388217.64 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.32

Some $4733773358.92$ e $430388217.64$ .

$s = \frac{5164161576.56 + 1283235.84 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.33

Some $5164161576.56$ e $1283235.84$ .

$s = \frac{5165444812.4 + 79616359.84 + 80823696.04}{12 - 1}$

Etapa 8.1.34

Some $5165444812.4$ e $79616359.84$ .

$s = \frac{5245061172.24 + 80823696.04}{12 - 1}$

Etapa 8.1.35

Some $5245061172.24$ e $80823696.04$ .

$s = \frac{5325884868.28}{12 - 1}$

Etapa 8.2

Simplifique a expressão.

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Etapa 8.2.1

Subtraia $1$ de $12$ .

$s = \frac{5325884868.28}{11}$

Etapa 8.2.2

Divida $5325884868.28$ por $11$ .

$s = 484171351.66\overline{18}$

Etapa 9

Aproxime o resultado.

$s^{2} \approx 484171351.6618$