Finite Math Examples

$6$ , $12$ , $15$ , $22$ , $18$ , $14$ , $8$ , $17$

Step 1

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

$\overline{x} = \frac{6 + 12 + 15 + 22 + 18 + 14 + 8 + 17}{8}$

Step 2

Simplify the numerator.

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Step 2.1

Add $6$ and $12$ .

$\overline{x} = \frac{18 + 15 + 22 + 18 + 14 + 8 + 17}{8}$

Step 2.2

Add $18$ and $15$ .

$\overline{x} = \frac{33 + 22 + 18 + 14 + 8 + 17}{8}$

Step 2.3

Add $33$ and $22$ .

$\overline{x} = \frac{55 + 18 + 14 + 8 + 17}{8}$

Step 2.4

Add $55$ and $18$ .

$\overline{x} = \frac{73 + 14 + 8 + 17}{8}$

Step 2.5

Add $73$ and $14$ .

$\overline{x} = \frac{87 + 8 + 17}{8}$

Step 2.6

Add $87$ and $8$ .

$\overline{x} = \frac{95 + 17}{8}$

Step 2.7

Add $95$ and $17$ .

$\overline{x} = \frac{112}{8}$

Step 3

Divide $112$ by $8$ .

$\overline{x} = 14$

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.

$s = \frac{{(6 - 14)}^{2} + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $14$ from $6$ .

$s = \frac{{(- 8)}^{2} + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.2

Raise $- 8$ to the power of $2$ .

$s = \frac{64 + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.3

Subtract $14$ from $12$ .

$s = \frac{64 + {(- 2)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.4

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

$s = \frac{64 + 4 + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.5

Subtract $14$ from $15$ .

$s = \frac{64 + 4 + 1^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.6

One to any power is one.

$s = \frac{64 + 4 + 1 + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.7

Subtract $14$ from $22$ .

$s = \frac{64 + 4 + 1 + 8^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.8

Raise $8$ to the power of $2$ .

$s = \frac{64 + 4 + 1 + 64 + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.9

Subtract $14$ from $18$ .

$s = \frac{64 + 4 + 1 + 64 + 4^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.10

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

$s = \frac{64 + 4 + 1 + 64 + 16 + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.11

Subtract $14$ from $14$ .

$s = \frac{64 + 4 + 1 + 64 + 16 + 0^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.12

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

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.13

Subtract $14$ from $8$ .

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + {(- 6)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.14

Raise $- 6$ to the power of $2$ .

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.15

Subtract $14$ from $17$ .

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + 3^{2}}{8 - 1}$

Step 6.1.16

Raise $3$ to the power of $2$ .

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.17

Add $64$ and $4$ .

$s = \frac{68 + 1 + 64 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.18

Add $68$ and $1$ .

$s = \frac{69 + 64 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.19

Add $69$ and $64$ .

$s = \frac{133 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.20

Add $133$ and $16$ .

$s = \frac{149 + 0 + 36 + 9}{8 - 1}$

Step 6.1.21

Add $149$ and $0$ .

$s = \frac{149 + 36 + 9}{8 - 1}$

Step 6.1.22

Add $149$ and $36$ .

$s = \frac{185 + 9}{8 - 1}$

Step 6.1.23

Add $185$ and $9$ .

$s = \frac{194}{8 - 1}$

Step 6.2

Subtract $1$ from $8$ .

$s = \frac{194}{7}$

Step 7

Approximate the result.

$s^{2} \approx 27.7143$