Finite Math Examples

$9$ , $5$ , $6$ , $11$ , $6$ , $10$ , $7$

Step 1

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

$\overline{x} = \frac{9 + 5 + 6 + 11 + 6 + 10 + 7}{7}$

Step 2

Simplify the numerator.

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Step 2.1

Add $9$ and $5$ .

$\overline{x} = \frac{14 + 6 + 11 + 6 + 10 + 7}{7}$

Step 2.2

Add $14$ and $6$ .

$\overline{x} = \frac{20 + 11 + 6 + 10 + 7}{7}$

Step 2.3

Add $20$ and $11$ .

$\overline{x} = \frac{31 + 6 + 10 + 7}{7}$

Step 2.4

Add $31$ and $6$ .

$\overline{x} = \frac{37 + 10 + 7}{7}$

Step 2.5

Add $37$ and $10$ .

$\overline{x} = \frac{47 + 7}{7}$

Step 2.6

Add $47$ and $7$ .

$\overline{x} = \frac{54}{7}$

Step 3

Divide.

$\overline{x} = 7.\overline{714285}$

Step 4

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.

$\overline{x} = 7.7$

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.

$s = \frac{{(9 - 7.7)}^{2} + {(5 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(11 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $7.7$ from $9$ .

$s = \frac{{1.3}^{2} + {(5 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(11 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.2

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

$s = \frac{1.69 + {(5 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(11 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.3

Subtract $7.7$ from $5$ .

$s = \frac{1.69 + {(- 2.7)}^{2} + {(6 - 7.7)}^{2} + {(11 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.4

Raise $- 2.7$ to the power of $2$ .

$s = \frac{1.69 + 7.29 + {(6 - 7.7)}^{2} + {(11 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.5

Subtract $7.7$ from $6$ .

$s = \frac{1.69 + 7.29 + {(- 1.7)}^{2} + {(11 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.6

Raise $- 1.7$ to the power of $2$ .

$s = \frac{1.69 + 7.29 + 2.89 + {(11 - 7.7)}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.7

Subtract $7.7$ from $11$ .

$s = \frac{1.69 + 7.29 + 2.89 + {3.3}^{2} + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.8

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

$s = \frac{1.69 + 7.29 + 2.89 + 10.89 + {(6 - 7.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.9

Subtract $7.7$ from $6$ .

$s = \frac{1.69 + 7.29 + 2.89 + 10.89 + {(- 1.7)}^{2} + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.10

Raise $- 1.7$ to the power of $2$ .

$s = \frac{1.69 + 7.29 + 2.89 + 10.89 + 2.89 + {(10 - 7.7)}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.11

Subtract $7.7$ from $10$ .

$s = \frac{1.69 + 7.29 + 2.89 + 10.89 + 2.89 + {2.3}^{2} + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.12

Raise $2.3$ to the power of $2$ .

$s = \frac{1.69 + 7.29 + 2.89 + 10.89 + 2.89 + 5.29 + {(7 - 7.7)}^{2}}{7 - 1}$

Step 7.1.13

Subtract $7.7$ from $7$ .

$s = \frac{1.69 + 7.29 + 2.89 + 10.89 + 2.89 + 5.29 + {(- 0.7)}^{2}}{7 - 1}$

Step 7.1.14

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

$s = \frac{1.69 + 7.29 + 2.89 + 10.89 + 2.89 + 5.29 + 0.49}{7 - 1}$

Step 7.1.15

Add $1.69$ and $7.29$ .

$s = \frac{8.98 + 2.89 + 10.89 + 2.89 + 5.29 + 0.49}{7 - 1}$

Step 7.1.16

Add $8.98$ and $2.89$ .

$s = \frac{11.87 + 10.89 + 2.89 + 5.29 + 0.49}{7 - 1}$

Step 7.1.17

Add $11.87$ and $10.89$ .

$s = \frac{22.76 + 2.89 + 5.29 + 0.49}{7 - 1}$

Step 7.1.18

Add $22.76$ and $2.89$ .

$s = \frac{25.65 + 5.29 + 0.49}{7 - 1}$

Step 7.1.19

Add $25.65$ and $5.29$ .

$s = \frac{30.94 + 0.49}{7 - 1}$

Step 7.1.20

Add $30.94$ and $0.49$ .

$s = \frac{31.43}{7 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $7$ .

$s = \frac{31.43}{6}$

Step 7.2.2

Divide $31.43$ by $6$ .

$s = 5.238\overline{3}$

Step 8

Approximate the result.

$s^{2} \approx 5.2383$