Finite Math Examples

$8$ , $16$ , $20$ , $20$ , $24$ , $32$

Step 1

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

$\overline{x} = \frac{8 + 16 + 20 + 20 + 24 + 32}{6}$

Step 2

Cancel the common factor of $8 + 16 + 20 + 20 + 24 + 32$ and $6$ .

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Step 2.1

Factor $2$ out of $8$ .

$\overline{x} = \frac{2 (4) + 16 + 20 + 20 + 24 + 32}{6}$

Step 2.2

Factor $2$ out of $16$ .

$\overline{x} = \frac{2 \cdot 4 + 2 \cdot 8 + 20 + 20 + 24 + 32}{6}$

Step 2.3

Factor $2$ out of $2 \cdot 4 + 2 \cdot 8$ .

$\overline{x} = \frac{2 \cdot (4 + 8) + 20 + 20 + 24 + 32}{6}$

Step 2.4

Factor $2$ out of $20$ .

$\overline{x} = \frac{2 \cdot (4 + 8) + 2 (10) + 20 + 24 + 32}{6}$

Step 2.5

Factor $2$ out of $2 \cdot (4 + 8) + 2 (10)$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10) + 20 + 24 + 32}{6}$

Step 2.6

Factor $2$ out of $20$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10) + 2 (10) + 24 + 32}{6}$

Step 2.7

Factor $2$ out of $2 \cdot (4 + 8 + 10) + 2 (10)$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10 + 10) + 24 + 32}{6}$

Step 2.8

Factor $2$ out of $24$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10 + 10) + 2 (12) + 32}{6}$

Step 2.9

Factor $2$ out of $2 \cdot (4 + 8 + 10 + 10) + 2 (12)$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10 + 10 + 12) + 32}{6}$

Step 2.10

Factor $2$ out of $32$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10 + 10 + 12) + 2 (16)}{6}$

Step 2.11

Factor $2$ out of $2 \cdot (4 + 8 + 10 + 10 + 12) + 2 (16)$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10 + 10 + 12 + 16)}{6}$

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $2$ out of $6$ .

$\overline{x} = \frac{2 \cdot (4 + 8 + 10 + 10 + 12 + 16)}{2 (3)}$

Step 2.12.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot (4 + 8 + 10 + 10 + 12 + 16)}{2 \cdot 3}$

Step 2.12.3

Rewrite the expression.

$\overline{x} = \frac{4 + 8 + 10 + 10 + 12 + 16}{3}$

Step 3

Simplify the numerator.

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Step 3.1

Add $4$ and $8$ .

$\overline{x} = \frac{12 + 10 + 10 + 12 + 16}{3}$

Step 3.2

Add $12$ and $10$ .

$\overline{x} = \frac{22 + 10 + 12 + 16}{3}$

Step 3.3

Add $22$ and $10$ .

$\overline{x} = \frac{32 + 12 + 16}{3}$

Step 3.4

Add $32$ and $12$ .

$\overline{x} = \frac{44 + 16}{3}$

Step 3.5

Add $44$ and $16$ .

$\overline{x} = \frac{60}{3}$

Step 4

Divide $60$ by $3$ .

$\overline{x} = 20$

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.

$s = \frac{{(8 - 20)}^{2} + {(16 - 20)}^{2} + {(20 - 20)}^{2} + {(20 - 20)}^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $20$ from $8$ .

$s = \frac{{(- 12)}^{2} + {(16 - 20)}^{2} + {(20 - 20)}^{2} + {(20 - 20)}^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.2

Raise $- 12$ to the power of $2$ .

$s = \frac{144 + {(16 - 20)}^{2} + {(20 - 20)}^{2} + {(20 - 20)}^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.3

Subtract $20$ from $16$ .

$s = \frac{144 + {(- 4)}^{2} + {(20 - 20)}^{2} + {(20 - 20)}^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.4

Raise $- 4$ to the power of $2$ .

$s = \frac{144 + 16 + {(20 - 20)}^{2} + {(20 - 20)}^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.5

Subtract $20$ from $20$ .

$s = \frac{144 + 16 + 0^{2} + {(20 - 20)}^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.6

Raising $0$ to any positive power yields $0$ .

$s = \frac{144 + 16 + 0 + {(20 - 20)}^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.7

Subtract $20$ from $20$ .

$s = \frac{144 + 16 + 0 + 0^{2} + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.8

Raising $0$ to any positive power yields $0$ .

$s = \frac{144 + 16 + 0 + 0 + {(24 - 20)}^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.9

Subtract $20$ from $24$ .

$s = \frac{144 + 16 + 0 + 0 + 4^{2} + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.10

Raise $4$ to the power of $2$ .

$s = \frac{144 + 16 + 0 + 0 + 16 + {(32 - 20)}^{2}}{6 - 1}$

Step 7.1.11

Subtract $20$ from $32$ .

$s = \frac{144 + 16 + 0 + 0 + 16 + 12^{2}}{6 - 1}$

Step 7.1.12

Raise $12$ to the power of $2$ .

$s = \frac{144 + 16 + 0 + 0 + 16 + 144}{6 - 1}$

Step 7.1.13

Add $144$ and $16$ .

$s = \frac{160 + 0 + 0 + 16 + 144}{6 - 1}$

Step 7.1.14

Add $160$ and $0$ .

$s = \frac{160 + 0 + 16 + 144}{6 - 1}$

Step 7.1.15

Add $160$ and $0$ .

$s = \frac{160 + 16 + 144}{6 - 1}$

Step 7.1.16

Add $160$ and $16$ .

$s = \frac{176 + 144}{6 - 1}$

Step 7.1.17

Add $176$ and $144$ .

$s = \frac{320}{6 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $6$ .

$s = \frac{320}{5}$

Step 7.2.2

Divide $320$ by $5$ .

$s = 64$

Step 8

Approximate the result.

$s^{2} \approx 64$