Finite Math Examples

$2$ , $1$ , $9$ , $15$ , $7$ , $9$ , $9$ , $18$ , $5$

Step 1

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

$\overline{x} = \frac{2 + 1 + 9 + 15 + 7 + 9 + 9 + 18 + 5}{9}$

Step 2

Simplify the numerator.

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Step 2.1

Add $2$ and $1$ .

$\overline{x} = \frac{3 + 9 + 15 + 7 + 9 + 9 + 18 + 5}{9}$

Step 2.2

Add $3$ and $9$ .

$\overline{x} = \frac{12 + 15 + 7 + 9 + 9 + 18 + 5}{9}$

Step 2.3

Add $12$ and $15$ .

$\overline{x} = \frac{27 + 7 + 9 + 9 + 18 + 5}{9}$

Step 2.4

Add $27$ and $7$ .

$\overline{x} = \frac{34 + 9 + 9 + 18 + 5}{9}$

Step 2.5

Add $34$ and $9$ .

$\overline{x} = \frac{43 + 9 + 18 + 5}{9}$

Step 2.6

Add $43$ and $9$ .

$\overline{x} = \frac{52 + 18 + 5}{9}$

Step 2.7

Add $52$ and $18$ .

$\overline{x} = \frac{70 + 5}{9}$

Step 2.8

Add $70$ and $5$ .

$\overline{x} = \frac{75}{9}$

Step 3

Cancel the common factor of $75$ and $9$ .

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Step 3.1

Factor $3$ out of $75$ .

$\overline{x} = \frac{3 (25)}{9}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $9$ .

$\overline{x} = \frac{3 \cdot 25}{3 \cdot 3}$

Step 3.2.2

Cancel the common factor.

$\overline{x} = \frac{3 \cdot 25}{3 \cdot 3}$

Step 3.2.3

Rewrite the expression.

$\overline{x} = \frac{25}{3}$

Step 4

Divide.

$\overline{x} = 8.\overline{3}$

Step 5

The mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$\overline{x} = 8.3$

Step 6

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 7

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

$s = \frac{{(2 - 8.3)}^{2} + {(1 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(15 - 8.3)}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8

Simplify the result.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Subtract $8.3$ from $2$ .

$s = \frac{{(- 6.3)}^{2} + {(1 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(15 - 8.3)}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.2

Raise $- 6.3$ to the power of $2$ .

$s = \frac{39.69 + {(1 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(15 - 8.3)}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.3

Subtract $8.3$ from $1$ .

$s = \frac{39.69 + {(- 7.3)}^{2} + {(9 - 8.3)}^{2} + {(15 - 8.3)}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.4

Raise $- 7.3$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + {(9 - 8.3)}^{2} + {(15 - 8.3)}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.5

Subtract $8.3$ from $9$ .

$s = \frac{39.69 + 53.29 + {0.7}^{2} + {(15 - 8.3)}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.6

Raise $0.7$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + 0.49 + {(15 - 8.3)}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.7

Subtract $8.3$ from $15$ .

$s = \frac{39.69 + 53.29 + 0.49 + {6.7}^{2} + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.8

Raise $6.7$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + {(7 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.9

Subtract $8.3$ from $7$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + {(- 1.3)}^{2} + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.10

Raise $- 1.3$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + {(9 - 8.3)}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.11

Subtract $8.3$ from $9$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + {0.7}^{2} + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.12

Raise $0.7$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + 0.49 + {(9 - 8.3)}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.13

Subtract $8.3$ from $9$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + 0.49 + {0.7}^{2} + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.14

Raise $0.7$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + 0.49 + 0.49 + {(18 - 8.3)}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.15

Subtract $8.3$ from $18$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + 0.49 + 0.49 + {9.7}^{2} + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.16

Raise $9.7$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + 0.49 + 0.49 + 94.09 + {(5 - 8.3)}^{2}}{9 - 1}$

Step 8.1.17

Subtract $8.3$ from $5$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + 0.49 + 0.49 + 94.09 + {(- 3.3)}^{2}}{9 - 1}$

Step 8.1.18

Raise $- 3.3$ to the power of $2$ .

$s = \frac{39.69 + 53.29 + 0.49 + 44.89 + 1.69 + 0.49 + 0.49 + 94.09 + 10.89}{9 - 1}$

Step 8.1.19

Add $39.69$ and $53.29$ .

$s = \frac{92.98 + 0.49 + 44.89 + 1.69 + 0.49 + 0.49 + 94.09 + 10.89}{9 - 1}$

Step 8.1.20

Add $92.98$ and $0.49$ .

$s = \frac{93.47 + 44.89 + 1.69 + 0.49 + 0.49 + 94.09 + 10.89}{9 - 1}$

Step 8.1.21

Add $93.47$ and $44.89$ .

$s = \frac{138.36 + 1.69 + 0.49 + 0.49 + 94.09 + 10.89}{9 - 1}$

Step 8.1.22

Add $138.36$ and $1.69$ .

$s = \frac{140.05 + 0.49 + 0.49 + 94.09 + 10.89}{9 - 1}$

Step 8.1.23

Add $140.05$ and $0.49$ .

$s = \frac{140.54 + 0.49 + 94.09 + 10.89}{9 - 1}$

Step 8.1.24

Add $140.54$ and $0.49$ .

$s = \frac{141.03 + 94.09 + 10.89}{9 - 1}$

Step 8.1.25

Add $141.03$ and $94.09$ .

$s = \frac{235.12 + 10.89}{9 - 1}$

Step 8.1.26

Add $235.12$ and $10.89$ .

$s = \frac{246.01}{9 - 1}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Subtract $1$ from $9$ .

$s = \frac{246.01}{8}$

Step 8.2.2

Divide $246.01$ by $8$ .

$s = 30.75125$

Step 9

Approximate the result.

$s^{2} \approx 30.7512$