Finite Math Examples

$\frac{1}{2}$ , $\frac{3}{4}$ , $- 1$ , $0$ , $7$

Step 1

Find the mean.

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Step 1.1

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

$\overline{x} = \frac{\frac{1}{2} + \frac{3}{4} - 1 + 0 + 7}{5}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\overline{x} = \frac{\frac{1}{2} \cdot \frac{2}{2} + \frac{3}{4} - 1 + 0 + 7}{5}$

Step 1.2.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\overline{x} = \frac{\frac{2}{2 \cdot 2} + \frac{3}{4} - 1 + 0 + 7}{5}$

Step 1.2.2.2

Multiply $2$ by $2$ .

$\overline{x} = \frac{\frac{2}{4} + \frac{3}{4} - 1 + 0 + 7}{5}$

Step 1.2.3

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{2 + 3}{4} - 1 + 0 + 7}{5}$

Step 1.2.4

Add $2$ and $3$ .

$\overline{x} = \frac{\frac{5}{4} - 1 + 0 + 7}{5}$

Step 1.2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\overline{x} = \frac{\frac{5}{4} - 1 \cdot \frac{4}{4} + 0 + 7}{5}$

Step 1.2.6

Combine $- 1$ and $\frac{4}{4}$ .

$\overline{x} = \frac{\frac{5}{4} + \frac{- 1 \cdot 4}{4} + 0 + 7}{5}$

Step 1.2.7

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{5 - 1 \cdot 4}{4} + 0 + 7}{5}$

Step 1.2.8

Simplify the numerator.

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Step 1.2.8.1

Multiply $- 1$ by $4$ .

$\overline{x} = \frac{\frac{5 - 4}{4} + 0 + 7}{5}$

Step 1.2.8.2

Subtract $4$ from $5$ .

$\overline{x} = \frac{\frac{1}{4} + 0 + 7}{5}$

Step 1.2.9

Add $\frac{1}{4}$ and $0$ .

$\overline{x} = \frac{\frac{1}{4} + 7}{5}$

Step 1.2.10

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\overline{x} = \frac{\frac{1}{4} + 7 \cdot \frac{4}{4}}{5}$

Step 1.2.11

Combine $7$ and $\frac{4}{4}$ .

$\overline{x} = \frac{\frac{1}{4} + \frac{7 \cdot 4}{4}}{5}$

Step 1.2.12

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{1 + 7 \cdot 4}{4}}{5}$

Step 1.2.13

Simplify the numerator.

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Step 1.2.13.1

Multiply $7$ by $4$ .

$\overline{x} = \frac{\frac{1 + 28}{4}}{5}$

Step 1.2.13.2

Add $1$ and $28$ .

$\overline{x} = \frac{\frac{29}{4}}{5}$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$\overline{x} = \frac{29}{4} \cdot \frac{1}{5}$

Step 1.4

Multiply $\frac{29}{4} \cdot \frac{1}{5}$ .

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Step 1.4.1

Multiply $\frac{29}{4}$ by $\frac{1}{5}$ .

$\overline{x} = \frac{29}{4 \cdot 5}$

Step 1.4.2

Multiply $4$ by $5$ .

$\overline{x} = \frac{29}{20}$

Step 1.5

Divide.

$\overline{x} = 1.45$

Step 1.6

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.

$\overline{x} = 1.5$

Step 2

Simplify each value in the list.

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Step 2.1

Convert $\frac{1}{2}$ to a decimal value.

$0.5$

Step 2.2

Convert $\frac{3}{4}$ to a decimal value.

$0.75$

Step 2.3

Convert $- 1$ to a decimal value.

$- 1$

Step 2.4

Convert $0$ to a decimal value.

$0$

Step 2.5

Convert $7$ to a decimal value.

$7$

Step 2.6

The simplified values are $0.5, 0.75, - 1, 0, 7$ .

$0.5, 0.75, - 1, 0, 7$

Step 3

Set up the formula for sample standard deviation. The standard deviation of a set of values is a measure of the spread of its values.

$s = \sum_{i = 1}^{n} \sqrt{\frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}}$

Step 4

Set up the formula for standard deviation for this set of numbers.

$s = \sqrt{\frac{{(0.5 - 1.5)}^{2} + {(0.75 - 1.5)}^{2} + {(- 1 - 1.5)}^{2} + {(0 - 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5

Simplify the result.

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Step 5.1

Subtract $1.5$ from $0.5$ .

$s = \sqrt{\frac{{(- 1)}^{2} + {(0.75 - 1.5)}^{2} + {(- 1 - 1.5)}^{2} + {(0 - 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.2

Raise $- 1$ to the power of $2$ .

$s = \sqrt{\frac{1 + {(0.75 - 1.5)}^{2} + {(- 1 - 1.5)}^{2} + {(0 - 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.3

Subtract $1.5$ from $0.75$ .

$s = \sqrt{\frac{1 + {(- 0.75)}^{2} + {(- 1 - 1.5)}^{2} + {(0 - 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.4

Raise $- 0.75$ to the power of $2$ .

$s = \sqrt{\frac{1 + 0.5625 + {(- 1 - 1.5)}^{2} + {(0 - 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.5

Subtract $1.5$ from $- 1$ .

$s = \sqrt{\frac{1 + 0.5625 + {(- 2.5)}^{2} + {(0 - 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.6

Raise $- 2.5$ to the power of $2$ .

$s = \sqrt{\frac{1 + 0.5625 + 6.25 + {(0 - 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.7

Subtract $1.5$ from $0$ .

$s = \sqrt{\frac{1 + 0.5625 + 6.25 + {(- 1.5)}^{2} + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.8

Raise $- 1.5$ to the power of $2$ .

$s = \sqrt{\frac{1 + 0.5625 + 6.25 + 2.25 + {(7 - 1.5)}^{2}}{5 - 1}}$

Step 5.9

Subtract $1.5$ from $7$ .

$s = \sqrt{\frac{1 + 0.5625 + 6.25 + 2.25 + {5.5}^{2}}{5 - 1}}$

Step 5.10

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

$s = \sqrt{\frac{1 + 0.5625 + 6.25 + 2.25 + 30.25}{5 - 1}}$

Step 5.11

Add $1$ and $0.5625$ .

$s = \sqrt{\frac{1.5625 + 6.25 + 2.25 + 30.25}{5 - 1}}$

Step 5.12

Add $1.5625$ and $6.25$ .

$s = \sqrt{\frac{7.8125 + 2.25 + 30.25}{5 - 1}}$

Step 5.13

Add $7.8125$ and $2.25$ .

$s = \sqrt{\frac{10.0625 + 30.25}{5 - 1}}$

Step 5.14

Add $10.0625$ and $30.25$ .

$s = \sqrt{\frac{40.3125}{5 - 1}}$

Step 5.15

Subtract $1$ from $5$ .

$s = \sqrt{\frac{40.3125}{4}}$

Step 5.16

Divide $40.3125$ by $4$ .

$s = \sqrt{10.078125}$

Step 6

The standard deviation 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.

$3.2$