Finite Math Examples

$1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

Add $1$ and $2$ .

$\overline{x} = \frac{3 + 3 + 4 + 5 + 6 + 7 + 8}{8}$

Step 2.2

Add $3$ and $3$ .

$\overline{x} = \frac{6 + 4 + 5 + 6 + 7 + 8}{8}$

Step 2.3

Add $6$ and $4$ .

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

Step 2.4

Add $10$ and $5$ .

$\overline{x} = \frac{15 + 6 + 7 + 8}{8}$

Step 2.5

Add $15$ and $6$ .

$\overline{x} = \frac{21 + 7 + 8}{8}$

Step 2.6

Add $21$ and $7$ .

$\overline{x} = \frac{28 + 8}{8}$

Step 2.7

Add $28$ and $8$ .

$\overline{x} = \frac{36}{8}$

Step 3

Cancel the common factor of $36$ and $8$ .

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Step 3.1

Factor $4$ out of $36$ .

$\overline{x} = \frac{4 (9)}{8}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4$ out of $8$ .

$\overline{x} = \frac{4 \cdot 9}{4 \cdot 2}$

Step 3.2.2

Cancel the common factor.

$\overline{x} = \frac{4 \cdot 9}{4 \cdot 2}$

Step 3.2.3

Rewrite the expression.

$\overline{x} = \frac{9}{2}$

Step 4

Divide.

$\overline{x} = 4.5$

Step 5

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 6

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

$s = \frac{{(1 - 4.5)}^{2} + {(2 - 4.5)}^{2} + {(3 - 4.5)}^{2} + {(4 - 4.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $4.5$ from $1$ .

$s = \frac{{(- 3.5)}^{2} + {(2 - 4.5)}^{2} + {(3 - 4.5)}^{2} + {(4 - 4.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.2

Raise $- 3.5$ to the power of $2$ .

$s = \frac{12.25 + {(2 - 4.5)}^{2} + {(3 - 4.5)}^{2} + {(4 - 4.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.3

Subtract $4.5$ from $2$ .

$s = \frac{12.25 + {(- 2.5)}^{2} + {(3 - 4.5)}^{2} + {(4 - 4.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.4

Raise $- 2.5$ to the power of $2$ .

$s = \frac{12.25 + 6.25 + {(3 - 4.5)}^{2} + {(4 - 4.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.5

Subtract $4.5$ from $3$ .

$s = \frac{12.25 + 6.25 + {(- 1.5)}^{2} + {(4 - 4.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.6

Raise $- 1.5$ to the power of $2$ .

$s = \frac{12.25 + 6.25 + 2.25 + {(4 - 4.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.7

Subtract $4.5$ from $4$ .

$s = \frac{12.25 + 6.25 + 2.25 + {(- 0.5)}^{2} + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.8

Raise $- 0.5$ to the power of $2$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + {(5 - 4.5)}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.9

Subtract $4.5$ from $5$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + {0.5}^{2} + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.10

Raise $0.5$ to the power of $2$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + {(6 - 4.5)}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.11

Subtract $4.5$ from $6$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + {1.5}^{2} + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.12

Raise $1.5$ to the power of $2$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + {(7 - 4.5)}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.13

Subtract $4.5$ from $7$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + {2.5}^{2} + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.14

Raise $2.5$ to the power of $2$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + {(8 - 4.5)}^{2}}{8 - 1}$

Step 7.1.15

Subtract $4.5$ from $8$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + {3.5}^{2}}{8 - 1}$

Step 7.1.16

Raise $3.5$ to the power of $2$ .

$s = \frac{12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25}{8 - 1}$

Step 7.1.17

Add $12.25$ and $6.25$ .

$s = \frac{18.5 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25}{8 - 1}$

Step 7.1.18

Add $18.5$ and $2.25$ .

$s = \frac{20.75 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25}{8 - 1}$

Step 7.1.19

Add $20.75$ and $0.25$ .

$s = \frac{21 + 0.25 + 2.25 + 6.25 + 12.25}{8 - 1}$

Step 7.1.20

Add $21$ and $0.25$ .

$s = \frac{21.25 + 2.25 + 6.25 + 12.25}{8 - 1}$

Step 7.1.21

Add $21.25$ and $2.25$ .

$s = \frac{23.5 + 6.25 + 12.25}{8 - 1}$

Step 7.1.22

Add $23.5$ and $6.25$ .

$s = \frac{29.75 + 12.25}{8 - 1}$

Step 7.1.23

Add $29.75$ and $12.25$ .

$s = \frac{42}{8 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $8$ .

$s = \frac{42}{7}$

Step 7.2.2

Divide $42$ by $7$ .

$s = 6$

Step 8

Approximate the result.

$s^{2} \approx 6$