Finite Math Examples

$190$ , $260$ , $250$ , $260$ , $240$ , $310$ , $400$ , $240$ , $250$ , $330$

Step 1

The mean of a set of numbers is the sum divided by the number of terms.

$\overline{x} = \frac{190 + 260 + 250 + 260 + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2

Cancel the common factor of $190 + 260 + 250 + 260 + 240 + 310 + 400 + 240 + 250 + 330$ and $10$ .

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Step 2.1

Factor $10$ out of $190$ .

$\overline{x} = \frac{10 \cdot 19 + 260 + 250 + 260 + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.2

Factor $10$ out of $260$ .

$\overline{x} = \frac{10 \cdot 19 + 10 \cdot 26 + 250 + 260 + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.3

Factor $10$ out of $10 \cdot 19 + 10 \cdot 26$ .

$\overline{x} = \frac{10 \cdot (19 + 26) + 250 + 260 + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.4

Factor $10$ out of $250$ .

$\overline{x} = \frac{10 \cdot (19 + 26) + 10 \cdot 25 + 260 + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.5

Factor $10$ out of $10 \cdot (19 + 26) + 10 (25)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25) + 260 + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.6

Factor $10$ out of $260$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25) + 10 \cdot 26 + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.7

Factor $10$ out of $10 \cdot (19 + 26 + 25) + 10 (26)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26) + 240 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.8

Factor $10$ out of $240$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26) + 10 \cdot 24 + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.9

Factor $10$ out of $10 \cdot (19 + 26 + 25 + 26) + 10 (24)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24) + 310 + 400 + 240 + 250 + 330}{10}$

Step 2.10

Factor $10$ out of $310$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24) + 10 \cdot 31 + 400 + 240 + 250 + 330}{10}$

Step 2.11

Factor $10$ out of $10 \cdot (19 + 26 + 25 + 26 + 24) + 10 (31)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31) + 400 + 240 + 250 + 330}{10}$

Step 2.12

Factor $10$ out of $400$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31) + 10 \cdot 40 + 240 + 250 + 330}{10}$

Step 2.13

Factor $10$ out of $10 \cdot (19 + 26 + 25 + 26 + 24 + 31) + 10 (40)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40) + 240 + 250 + 330}{10}$

Step 2.14

Factor $10$ out of $240$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40) + 10 \cdot 24 + 250 + 330}{10}$

Step 2.15

Factor $10$ out of $10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40) + 10 (24)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24) + 250 + 330}{10}$

Step 2.16

Factor $10$ out of $250$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24) + 10 \cdot 25 + 330}{10}$

Step 2.17

Factor $10$ out of $10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24) + 10 (25)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25) + 330}{10}$

Step 2.18

Factor $10$ out of $330$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25) + 10 \cdot 33}{10}$

Step 2.19

Factor $10$ out of $10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25) + 10 (33)$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25 + 33)}{10}$

Step 2.20

Cancel the common factors.

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Step 2.20.1

Factor $10$ out of $10$ .

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25 + 33)}{10 (1)}$

Step 2.20.2

Cancel the common factor.

$\overline{x} = \frac{10 \cdot (19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25 + 33)}{10 \cdot 1}$

Step 2.20.3

Rewrite the expression.

$\overline{x} = \frac{19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25 + 33}{1}$

Step 2.20.4

Divide $19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25 + 33$ by $1$ .

$\overline{x} = 19 + 26 + 25 + 26 + 24 + 31 + 40 + 24 + 25 + 33$

Step 3

Simplify by adding numbers.

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Step 3.1

Add $19$ and $26$ .

$\overline{x} = 45 + 25 + 26 + 24 + 31 + 40 + 24 + 25 + 33$

Step 3.2

Add $45$ and $25$ .

$\overline{x} = 70 + 26 + 24 + 31 + 40 + 24 + 25 + 33$

Step 3.3

Add $70$ and $26$ .

$\overline{x} = 96 + 24 + 31 + 40 + 24 + 25 + 33$

Step 3.4

Add $96$ and $24$ .

$\overline{x} = 120 + 31 + 40 + 24 + 25 + 33$

Step 3.5

Add $120$ and $31$ .

$\overline{x} = 151 + 40 + 24 + 25 + 33$

Step 3.6

Add $151$ and $40$ .

$\overline{x} = 191 + 24 + 25 + 33$

Step 3.7

Add $191$ and $24$ .

$\overline{x} = 215 + 25 + 33$

Step 3.8

Add $215$ and $25$ .

$\overline{x} = 240 + 33$

Step 3.9

Add $240$ and $33$ .

$\overline{x} = 273$

Step 4

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

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

Step 5

Set up the formula for variance for this set of numbers.

$s = \frac{{(190 - 273)}^{2} + {(260 - 273)}^{2} + {(250 - 273)}^{2} + {(260 - 273)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $273$ from $190$ .

$s = \frac{{(- 83)}^{2} + {(260 - 273)}^{2} + {(250 - 273)}^{2} + {(260 - 273)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.2

Raise $- 83$ to the power of $2$ .

$s = \frac{6889 + {(260 - 273)}^{2} + {(250 - 273)}^{2} + {(260 - 273)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.3

Subtract $273$ from $260$ .

$s = \frac{6889 + {(- 13)}^{2} + {(250 - 273)}^{2} + {(260 - 273)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.4

Raise $- 13$ to the power of $2$ .

$s = \frac{6889 + 169 + {(250 - 273)}^{2} + {(260 - 273)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.5

Subtract $273$ from $250$ .

$s = \frac{6889 + 169 + {(- 23)}^{2} + {(260 - 273)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.6

Raise $- 23$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + {(260 - 273)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.7

Subtract $273$ from $260$ .

$s = \frac{6889 + 169 + 529 + {(- 13)}^{2} + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.8

Raise $- 13$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + 169 + {(240 - 273)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.9

Subtract $273$ from $240$ .

$s = \frac{6889 + 169 + 529 + 169 + {(- 33)}^{2} + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.10

Raise $- 33$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + {(310 - 273)}^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.11

Subtract $273$ from $310$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 37^{2} + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.12

Raise $37$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + {(400 - 273)}^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.13

Subtract $273$ from $400$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 127^{2} + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.14

Raise $127$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 16129 + {(240 - 273)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.15

Subtract $273$ from $240$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 16129 + {(- 33)}^{2} + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.16

Raise $- 33$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 16129 + 1089 + {(250 - 273)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.17

Subtract $273$ from $250$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 16129 + 1089 + {(- 23)}^{2} + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.18

Raise $- 23$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 16129 + 1089 + 529 + {(330 - 273)}^{2}}{10 - 1}$

Step 6.1.19

Subtract $273$ from $330$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 16129 + 1089 + 529 + 57^{2}}{10 - 1}$

Step 6.1.20

Raise $57$ to the power of $2$ .

$s = \frac{6889 + 169 + 529 + 169 + 1089 + 1369 + 16129 + 1089 + 529 + 3249}{10 - 1}$

Step 6.1.21

Add $6889$ and $169$ .

$s = \frac{7058 + 529 + 169 + 1089 + 1369 + 16129 + 1089 + 529 + 3249}{10 - 1}$

Step 6.1.22

Add $7058$ and $529$ .

$s = \frac{7587 + 169 + 1089 + 1369 + 16129 + 1089 + 529 + 3249}{10 - 1}$

Step 6.1.23

Add $7587$ and $169$ .

$s = \frac{7756 + 1089 + 1369 + 16129 + 1089 + 529 + 3249}{10 - 1}$

Step 6.1.24

Add $7756$ and $1089$ .

$s = \frac{8845 + 1369 + 16129 + 1089 + 529 + 3249}{10 - 1}$

Step 6.1.25

Add $8845$ and $1369$ .

$s = \frac{10214 + 16129 + 1089 + 529 + 3249}{10 - 1}$

Step 6.1.26

Add $10214$ and $16129$ .

$s = \frac{26343 + 1089 + 529 + 3249}{10 - 1}$

Step 6.1.27

Add $26343$ and $1089$ .

$s = \frac{27432 + 529 + 3249}{10 - 1}$

Step 6.1.28

Add $27432$ and $529$ .

$s = \frac{27961 + 3249}{10 - 1}$

Step 6.1.29

Add $27961$ and $3249$ .

$s = \frac{31210}{10 - 1}$

Step 6.2

Subtract $1$ from $10$ .

$s = \frac{31210}{9}$

Step 7

Approximate the result.

$s^{2} \approx 3467.7778$