Finite Math Examples

$4$ , $5$ , $6$ , $7$ , $8$ , $9$ , $9$ , $10$ , $12$ , $12$ , $14$ , $16$ , $17$ , $17$ , $19$

Step 1

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

$\overline{x} = \frac{4 + 5 + 6 + 7 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2

Simplify the numerator.

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Step 2.1

Add $4$ and $5$ .

$\overline{x} = \frac{9 + 6 + 7 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.2

Add $9$ and $6$ .

$\overline{x} = \frac{15 + 7 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.3

Add $15$ and $7$ .

$\overline{x} = \frac{22 + 8 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.4

Add $22$ and $8$ .

$\overline{x} = \frac{30 + 9 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.5

Add $30$ and $9$ .

$\overline{x} = \frac{39 + 9 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.6

Add $39$ and $9$ .

$\overline{x} = \frac{48 + 10 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.7

Add $48$ and $10$ .

$\overline{x} = \frac{58 + 12 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.8

Add $58$ and $12$ .

$\overline{x} = \frac{70 + 12 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.9

Add $70$ and $12$ .

$\overline{x} = \frac{82 + 14 + 16 + 17 + 17 + 19}{15}$

Step 2.10

Add $82$ and $14$ .

$\overline{x} = \frac{96 + 16 + 17 + 17 + 19}{15}$

Step 2.11

Add $96$ and $16$ .

$\overline{x} = \frac{112 + 17 + 17 + 19}{15}$

Step 2.12

Add $112$ and $17$ .

$\overline{x} = \frac{129 + 17 + 19}{15}$

Step 2.13

Add $129$ and $17$ .

$\overline{x} = \frac{146 + 19}{15}$

Step 2.14

Add $146$ and $19$ .

$\overline{x} = \frac{165}{15}$

Step 3

Divide $165$ by $15$ .

$\overline{x} = 11$

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.

$s = \frac{{(4 - 11)}^{2} + {(5 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $11$ from $4$ .

$s = \frac{{(- 7)}^{2} + {(5 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.2

Raise $- 7$ to the power of $2$ .

$s = \frac{49 + {(5 - 11)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.3

Subtract $11$ from $5$ .

$s = \frac{49 + {(- 6)}^{2} + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.4

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

$s = \frac{49 + 36 + {(6 - 11)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.5

Subtract $11$ from $6$ .

$s = \frac{49 + 36 + {(- 5)}^{2} + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.6

Raise $- 5$ to the power of $2$ .

$s = \frac{49 + 36 + 25 + {(7 - 11)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.7

Subtract $11$ from $7$ .

$s = \frac{49 + 36 + 25 + {(- 4)}^{2} + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.8

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

$s = \frac{49 + 36 + 25 + 16 + {(8 - 11)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.9

Subtract $11$ from $8$ .

$s = \frac{49 + 36 + 25 + 16 + {(- 3)}^{2} + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.10

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

$s = \frac{49 + 36 + 25 + 16 + 9 + {(9 - 11)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.11

Subtract $11$ from $9$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + {(- 2)}^{2} + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.12

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

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + {(9 - 11)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.13

Subtract $11$ from $9$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + {(- 2)}^{2} + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.14

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

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + {(10 - 11)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.15

Subtract $11$ from $10$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + {(- 1)}^{2} + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.16

Raise $- 1$ to the power of $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + {(12 - 11)}^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.17

Subtract $11$ from $12$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1^{2} + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.18

One to any power is one.

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + {(12 - 11)}^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.19

Subtract $11$ from $12$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1^{2} + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.20

One to any power is one.

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + {(14 - 11)}^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.21

Subtract $11$ from $14$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 3^{2} + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.22

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

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + {(16 - 11)}^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.23

Subtract $11$ from $16$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 5^{2} + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.24

Raise $5$ to the power of $2$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + {(17 - 11)}^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.25

Subtract $11$ from $17$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 6^{2} + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.26

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

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + {(17 - 11)}^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.27

Subtract $11$ from $17$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 6^{2} + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.28

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

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + {(19 - 11)}^{2}}{15 - 1}$

Step 6.1.29

Subtract $11$ from $19$ .

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 8^{2}}{15 - 1}$

Step 6.1.30

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

$s = \frac{49 + 36 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.31

Add $49$ and $36$ .

$s = \frac{85 + 25 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.32

Add $85$ and $25$ .

$s = \frac{110 + 16 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.33

Add $110$ and $16$ .

$s = \frac{126 + 9 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.34

Add $126$ and $9$ .

$s = \frac{135 + 4 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.35

Add $135$ and $4$ .

$s = \frac{139 + 4 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.36

Add $139$ and $4$ .

$s = \frac{143 + 1 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.37

Add $143$ and $1$ .

$s = \frac{144 + 1 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.38

Add $144$ and $1$ .

$s = \frac{145 + 1 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.39

Add $145$ and $1$ .

$s = \frac{146 + 9 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.40

Add $146$ and $9$ .

$s = \frac{155 + 25 + 36 + 36 + 64}{15 - 1}$

Step 6.1.41

Add $155$ and $25$ .

$s = \frac{180 + 36 + 36 + 64}{15 - 1}$

Step 6.1.42

Add $180$ and $36$ .

$s = \frac{216 + 36 + 64}{15 - 1}$

Step 6.1.43

Add $216$ and $36$ .

$s = \frac{252 + 64}{15 - 1}$

Step 6.1.44

Add $252$ and $64$ .

$s = \frac{316}{15 - 1}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Subtract $1$ from $15$ .

$s = \frac{316}{14}$

Step 6.2.2

Cancel the common factor of $316$ and $14$ .

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Step 6.2.2.1

Factor $2$ out of $316$ .

$s = \frac{2 (158)}{14}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $2$ out of $14$ .

$s = \frac{2 \cdot 158}{2 \cdot 7}$

Step 6.2.2.2.2

Cancel the common factor.

$s = \frac{2 \cdot 158}{2 \cdot 7}$

Step 6.2.2.2.3

Rewrite the expression.

$s = \frac{158}{7}$

Step 7

Approximate the result.

$s^{2} \approx 22.5714$