Finite Math Examples

$0$ , $10$ , $- 10$ , $20$ , $- 20$

Step 1

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

$\overline{x} = \frac{0 + 10 - 10 + 20 - 20}{5}$

Step 2

Cancel the common factor of $0 + 10 - 10 + 20 - 20$ and $5$ .

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Step 2.1

Factor $5$ out of $0$ .

$\overline{x} = \frac{5 \cdot 0 + 10 - 10 + 20 - 20}{5}$

Step 2.2

Factor $5$ out of $10$ .

$\overline{x} = \frac{5 \cdot 0 + 5 \cdot 2 - 10 + 20 - 20}{5}$

Step 2.3

Factor $5$ out of $5 \cdot 0 + 5 \cdot 2$ .

$\overline{x} = \frac{5 \cdot (0 + 2) - 10 + 20 - 20}{5}$

Step 2.4

Factor $5$ out of $- 10$ .

$\overline{x} = \frac{5 \cdot (0 + 2) + 5 \cdot - 2 + 20 - 20}{5}$

Step 2.5

Factor $5$ out of $5 \cdot (0 + 2) + 5 (- 2)$ .

$\overline{x} = \frac{5 \cdot (0 + 2 - 2) + 20 - 20}{5}$

Step 2.6

Factor $5$ out of $20$ .

$\overline{x} = \frac{5 \cdot (0 + 2 - 2) + 5 \cdot 4 - 20}{5}$

Step 2.7

Factor $5$ out of $5 \cdot (0 + 2 - 2) + 5 (4)$ .

$\overline{x} = \frac{5 \cdot (0 + 2 - 2 + 4) - 20}{5}$

Step 2.8

Factor $5$ out of $- 20$ .

$\overline{x} = \frac{5 \cdot (0 + 2 - 2 + 4) + 5 \cdot - 4}{5}$

Step 2.9

Factor $5$ out of $5 \cdot (0 + 2 - 2 + 4) + 5 (- 4)$ .

$\overline{x} = \frac{5 \cdot (0 + 2 - 2 + 4 - 4)}{5}$

Step 2.10

Cancel the common factors.

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Step 2.10.1

Factor $5$ out of $5$ .

$\overline{x} = \frac{5 \cdot (0 + 2 - 2 + 4 - 4)}{5 (1)}$

Step 2.10.2

Cancel the common factor.

$\overline{x} = \frac{5 \cdot (0 + 2 - 2 + 4 - 4)}{5 \cdot 1}$

Step 2.10.3

Rewrite the expression.

$\overline{x} = \frac{0 + 2 - 2 + 4 - 4}{1}$

Step 2.10.4

Divide $0 + 2 - 2 + 4 - 4$ by $1$ .

$\overline{x} = 0 + 2 - 2 + 4 - 4$

Step 3

Simplify by adding and subtracting.

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Step 3.1

Add $0$ and $2$ .

$\overline{x} = 2 - 2 + 4 - 4$

Step 3.2

Subtract $2$ from $2$ .

$\overline{x} = 0 + 4 - 4$

Step 3.3

Add $0$ and $4$ .

$\overline{x} = 4 - 4$

Step 3.4

Subtract $4$ from $4$ .

$\overline{x} = 0$

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.

$s = \frac{{(0 - 0)}^{2} + {(10 - 0)}^{2} + {(- 10 - 0)}^{2} + {(20 - 0)}^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $0$ from $0$ .

$s = \frac{0^{2} + {(10 - 0)}^{2} + {(- 10 - 0)}^{2} + {(20 - 0)}^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.2

Raising $0$ to any positive power yields $0$ .

$s = \frac{0 + {(10 - 0)}^{2} + {(- 10 - 0)}^{2} + {(20 - 0)}^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.3

Subtract $0$ from $10$ .

$s = \frac{0 + 10^{2} + {(- 10 - 0)}^{2} + {(20 - 0)}^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.4

Raise $10$ to the power of $2$ .

$s = \frac{0 + 100 + {(- 10 - 0)}^{2} + {(20 - 0)}^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.5

Subtract $0$ from $- 10$ .

$s = \frac{0 + 100 + {(- 10)}^{2} + {(20 - 0)}^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.6

Raise $- 10$ to the power of $2$ .

$s = \frac{0 + 100 + 100 + {(20 - 0)}^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.7

Subtract $0$ from $20$ .

$s = \frac{0 + 100 + 100 + 20^{2} + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.8

Raise $20$ to the power of $2$ .

$s = \frac{0 + 100 + 100 + 400 + {(- 20 - 0)}^{2}}{5 - 1}$

Step 6.1.9

Subtract $0$ from $- 20$ .

$s = \frac{0 + 100 + 100 + 400 + {(- 20)}^{2}}{5 - 1}$

Step 6.1.10

Raise $- 20$ to the power of $2$ .

$s = \frac{0 + 100 + 100 + 400 + 400}{5 - 1}$

Step 6.1.11

Add $0$ and $100$ .

$s = \frac{100 + 100 + 400 + 400}{5 - 1}$

Step 6.1.12

Add $100$ and $100$ .

$s = \frac{200 + 400 + 400}{5 - 1}$

Step 6.1.13

Add $200$ and $400$ .

$s = \frac{600 + 400}{5 - 1}$

Step 6.1.14

Add $600$ and $400$ .

$s = \frac{1000}{5 - 1}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract $1$ from $5$ .

$s = \frac{1000}{4}$

Step 6.2.2

Divide $1000$ by $4$ .

$s = 250$

Step 7

Approximate the result.

$s^{2} \approx 250$