Finite Math Examples

$- 23$ , $- 17$ , $- 19$ , $- 5$ , $- 4$ , $- 11$ , $- 31$

Step 1

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

$\overline{x} = \frac{- 23 - 17 - 19 - 5 - 4 - 11 - 31}{7}$

Step 2

Simplify the numerator.

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Step 2.1

Subtract $17$ from $- 23$ .

$\overline{x} = \frac{- 40 - 19 - 5 - 4 - 11 - 31}{7}$

Step 2.2

Subtract $19$ from $- 40$ .

$\overline{x} = \frac{- 59 - 5 - 4 - 11 - 31}{7}$

Step 2.3

Subtract $5$ from $- 59$ .

$\overline{x} = \frac{- 64 - 4 - 11 - 31}{7}$

Step 2.4

Subtract $4$ from $- 64$ .

$\overline{x} = \frac{- 68 - 11 - 31}{7}$

Step 2.5

Subtract $11$ from $- 68$ .

$\overline{x} = \frac{- 79 - 31}{7}$

Step 2.6

Subtract $31$ from $- 79$ .

$\overline{x} = \frac{- 110}{7}$

Step 3

Move the negative in front of the fraction.

$\overline{x} = - \frac{110}{7}$

Step 4

Divide.

$\overline{x} = - 15.71428571$

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.

$\overline{x} = - 15.7$

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.

$s = \frac{{(- 23 + 15.7)}^{2} + {(- 17 + 15.7)}^{2} + {(- 19 + 15.7)}^{2} + {(- 5 + 15.7)}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8

Simplify the result.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Add $- 23$ and $15.7$ .

$s = \frac{{(- 7.3)}^{2} + {(- 17 + 15.7)}^{2} + {(- 19 + 15.7)}^{2} + {(- 5 + 15.7)}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.2

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

$s = \frac{53.29 + {(- 17 + 15.7)}^{2} + {(- 19 + 15.7)}^{2} + {(- 5 + 15.7)}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.3

Add $- 17$ and $15.7$ .

$s = \frac{53.29 + {(- 1.3)}^{2} + {(- 19 + 15.7)}^{2} + {(- 5 + 15.7)}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.4

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

$s = \frac{53.29 + 1.69 + {(- 19 + 15.7)}^{2} + {(- 5 + 15.7)}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.5

Add $- 19$ and $15.7$ .

$s = \frac{53.29 + 1.69 + {(- 3.3)}^{2} + {(- 5 + 15.7)}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.6

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

$s = \frac{53.29 + 1.69 + 10.89 + {(- 5 + 15.7)}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.7

Add $- 5$ and $15.7$ .

$s = \frac{53.29 + 1.69 + 10.89 + {10.7}^{2} + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.8

Raise $10.7$ to the power of $2$ .

$s = \frac{53.29 + 1.69 + 10.89 + 114.49 + {(- 4 + 15.7)}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.9

Add $- 4$ and $15.7$ .

$s = \frac{53.29 + 1.69 + 10.89 + 114.49 + {11.7}^{2} + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.10

Raise $11.7$ to the power of $2$ .

$s = \frac{53.29 + 1.69 + 10.89 + 114.49 + 136.89 + {(- 11 + 15.7)}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.11

Add $- 11$ and $15.7$ .

$s = \frac{53.29 + 1.69 + 10.89 + 114.49 + 136.89 + {4.7}^{2} + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.12

Raise $4.7$ to the power of $2$ .

$s = \frac{53.29 + 1.69 + 10.89 + 114.49 + 136.89 + 22.09 + {(- 31 + 15.7)}^{2}}{7 - 1}$

Step 8.1.13

Add $- 31$ and $15.7$ .

$s = \frac{53.29 + 1.69 + 10.89 + 114.49 + 136.89 + 22.09 + {(- 15.3)}^{2}}{7 - 1}$

Step 8.1.14

Raise $- 15.3$ to the power of $2$ .

$s = \frac{53.29 + 1.69 + 10.89 + 114.49 + 136.89 + 22.09 + 234.09}{7 - 1}$

Step 8.1.15

Add $53.29$ and $1.69$ .

$s = \frac{54.98 + 10.89 + 114.49 + 136.89 + 22.09 + 234.09}{7 - 1}$

Step 8.1.16

Add $54.98$ and $10.89$ .

$s = \frac{65.87 + 114.49 + 136.89 + 22.09 + 234.09}{7 - 1}$

Step 8.1.17

Add $65.87$ and $114.49$ .

$s = \frac{180.36 + 136.89 + 22.09 + 234.09}{7 - 1}$

Step 8.1.18

Add $180.36$ and $136.89$ .

$s = \frac{317.25 + 22.09 + 234.09}{7 - 1}$

Step 8.1.19

Add $317.25$ and $22.09$ .

$s = \frac{339.34 + 234.09}{7 - 1}$

Step 8.1.20

Add $339.34$ and $234.09$ .

$s = \frac{573.43}{7 - 1}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Subtract $1$ from $7$ .

$s = \frac{573.43}{6}$

Step 8.2.2

Divide $573.43$ by $6$ .

$s = 95.571\overline{6}$

Step 9

Approximate the result.

$s^{2} \approx 95.5717$