Finite Math Examples

$- 2$ , $- 4$ , $- 8$ , $- 16$

Step 1

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

$\overline{x} = \frac{- 2 - 4 - 8 - 16}{4}$

Step 2

Cancel the common factor of $- 2 - 4 - 8 - 16$ and $4$ .

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Step 2.1

Factor $2$ out of $- 2$ .

$\overline{x} = \frac{2 (- 1) - 4 - 8 - 16}{4}$

Step 2.2

Factor $2$ out of $- 4$ .

$\overline{x} = \frac{2 \cdot - 1 + 2 \cdot - 2 - 8 - 16}{4}$

Step 2.3

Factor $2$ out of $2 \cdot - 1 + 2 \cdot - 2$ .

$\overline{x} = \frac{2 \cdot (- 1 - 2) - 8 - 16}{4}$

Step 2.4

Factor $2$ out of $- 8$ .

$\overline{x} = \frac{2 \cdot (- 1 - 2) + 2 (- 4) - 16}{4}$

Step 2.5

Factor $2$ out of $2 \cdot (- 1 - 2) + 2 (- 4)$ .

$\overline{x} = \frac{2 \cdot (- 1 - 2 - 4) - 16}{4}$

Step 2.6

Factor $2$ out of $- 16$ .

$\overline{x} = \frac{2 \cdot (- 1 - 2 - 4) + 2 (- 8)}{4}$

Step 2.7

Factor $2$ out of $2 \cdot (- 1 - 2 - 4) + 2 (- 8)$ .

$\overline{x} = \frac{2 \cdot (- 1 - 2 - 4 - 8)}{4}$

Step 2.8

Cancel the common factors.

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Step 2.8.1

Factor $2$ out of $4$ .

$\overline{x} = \frac{2 \cdot (- 1 - 2 - 4 - 8)}{2 (2)}$

Step 2.8.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot (- 1 - 2 - 4 - 8)}{2 \cdot 2}$

Step 2.8.3

Rewrite the expression.

$\overline{x} = \frac{- 1 - 2 - 4 - 8}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Subtract $2$ from $- 1$ .

$\overline{x} = \frac{- 3 - 4 - 8}{2}$

Step 3.2

Subtract $4$ from $- 3$ .

$\overline{x} = \frac{- 7 - 8}{2}$

Step 3.3

Subtract $8$ from $- 7$ .

$\overline{x} = \frac{- 15}{2}$

Step 4

Move the negative in front of the fraction.

$\overline{x} = - \frac{15}{2}$

Step 5

Divide.

$\overline{x} = - 7.5$

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.

$s = \frac{{(- 2 + 7.5)}^{2} + {(- 4 + 7.5)}^{2} + {(- 8 + 7.5)}^{2} + {(- 16 + 7.5)}^{2}}{4 - 1}$

Step 8

Simplify the result.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Add $- 2$ and $7.5$ .

$s = \frac{{5.5}^{2} + {(- 4 + 7.5)}^{2} + {(- 8 + 7.5)}^{2} + {(- 16 + 7.5)}^{2}}{4 - 1}$

Step 8.1.2

Raise $5.5$ to the power of $2$ .

$s = \frac{30.25 + {(- 4 + 7.5)}^{2} + {(- 8 + 7.5)}^{2} + {(- 16 + 7.5)}^{2}}{4 - 1}$

Step 8.1.3

Add $- 4$ and $7.5$ .

$s = \frac{30.25 + {3.5}^{2} + {(- 8 + 7.5)}^{2} + {(- 16 + 7.5)}^{2}}{4 - 1}$

Step 8.1.4

Raise $3.5$ to the power of $2$ .

$s = \frac{30.25 + 12.25 + {(- 8 + 7.5)}^{2} + {(- 16 + 7.5)}^{2}}{4 - 1}$

Step 8.1.5

Add $- 8$ and $7.5$ .

$s = \frac{30.25 + 12.25 + {(- 0.5)}^{2} + {(- 16 + 7.5)}^{2}}{4 - 1}$

Step 8.1.6

Raise $- 0.5$ to the power of $2$ .

$s = \frac{30.25 + 12.25 + 0.25 + {(- 16 + 7.5)}^{2}}{4 - 1}$

Step 8.1.7

Add $- 16$ and $7.5$ .

$s = \frac{30.25 + 12.25 + 0.25 + {(- 8.5)}^{2}}{4 - 1}$

Step 8.1.8

Raise $- 8.5$ to the power of $2$ .

$s = \frac{30.25 + 12.25 + 0.25 + 72.25}{4 - 1}$

Step 8.1.9

Add $30.25$ and $12.25$ .

$s = \frac{42.5 + 0.25 + 72.25}{4 - 1}$

Step 8.1.10

Add $42.5$ and $0.25$ .

$s = \frac{42.75 + 72.25}{4 - 1}$

Step 8.1.11

Add $42.75$ and $72.25$ .

$s = \frac{115}{4 - 1}$

Step 8.2

Reduce the expression by cancelling the common factors.

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Step 8.2.1

Subtract $1$ from $4$ .

$s = \frac{115}{3}$

Step 8.2.2

Cancel the common factor of $115$ and $3$ .

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Step 8.2.2.1

Rewrite $115$ as $1 (115)$ .

$s = \frac{1 (115)}{3}$

Step 8.2.2.2

Cancel the common factors.

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Step 8.2.2.2.1

Rewrite $3$ as $1 (3)$ .

$s = \frac{1 \cdot 115}{1 \cdot 3}$

Step 8.2.2.2.2

Cancel the common factor.

$s = \frac{1 \cdot 115}{1 \cdot 3}$

Step 8.2.2.2.3

Rewrite the expression.

$s = \frac{115}{3}$

Step 9

Approximate the result.

$s^{2} \approx 38.3333$