Finite Math Examples

$4$ , $- \frac{1}{2}$ , $\frac{1}{16}$

Step 1

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

$\overline{x} = \frac{4 - \frac{1}{2} + \frac{1}{16}}{3}$

Step 2

Simplify the numerator.

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Step 2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\overline{x} = \frac{4 \cdot \frac{2}{2} - \frac{1}{2} + \frac{1}{16}}{3}$

Step 2.2

Combine $4$ and $\frac{2}{2}$ .

$\overline{x} = \frac{\frac{4 \cdot 2}{2} - \frac{1}{2} + \frac{1}{16}}{3}$

Step 2.3

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{4 \cdot 2 - 1}{2} + \frac{1}{16}}{3}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Multiply $4$ by $2$ .

$\overline{x} = \frac{\frac{8 - 1}{2} + \frac{1}{16}}{3}$

Step 2.4.2

Subtract $1$ from $8$ .

$\overline{x} = \frac{\frac{7}{2} + \frac{1}{16}}{3}$

Step 2.5

To write $\frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\overline{x} = \frac{\frac{7}{2} \cdot \frac{8}{8} + \frac{1}{16}}{3}$

Step 2.6

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.1

Multiply $\frac{7}{2}$ by $\frac{8}{8}$ .

$\overline{x} = \frac{\frac{7 \cdot 8}{2 \cdot 8} + \frac{1}{16}}{3}$

Step 2.6.2

Multiply $2$ by $8$ .

$\overline{x} = \frac{\frac{7 \cdot 8}{16} + \frac{1}{16}}{3}$

Step 2.7

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{7 \cdot 8 + 1}{16}}{3}$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $7$ by $8$ .

$\overline{x} = \frac{\frac{56 + 1}{16}}{3}$

Step 2.8.2

Add $56$ and $1$ .

$\overline{x} = \frac{\frac{57}{16}}{3}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\overline{x} = \frac{57}{16} \cdot \frac{1}{3}$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Factor $3$ out of $57$ .

$\overline{x} = \frac{3 (19)}{16} \cdot \frac{1}{3}$

Step 4.2

Cancel the common factor.

$\overline{x} = \frac{3 \cdot 19}{16} \cdot \frac{1}{3}$

Step 4.3

Rewrite the expression.

$\overline{x} = \frac{19}{16}$

Step 5

Divide.

$\overline{x} = 1.1875$

Step 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.

$\overline{x} = 1.2$

Step 7

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 8

Set up the formula for variance for this set of numbers.

$s = \frac{{(4 - 1.2)}^{2} + {(- \frac{1}{2} - 1.2)}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9

Simplify the result.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Subtract $1.2$ from $4$ .

$s = \frac{{2.8}^{2} + {(- \frac{1}{2} - 1.2)}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.2

Raise $2.8$ to the power of $2$ .

$s = \frac{7.84 + {(- \frac{1}{2} - 1.2)}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.3

To write $- 1.2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$s = \frac{7.84 + {(- \frac{1}{2} - 1.2 \cdot \frac{2}{2})}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.4

Combine $- 1.2$ and $\frac{2}{2}$ .

$s = \frac{7.84 + {(- \frac{1}{2} + \frac{- 1.2 \cdot 2}{2})}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.5

Combine the numerators over the common denominator.

$s = \frac{7.84 + {(\frac{- 1 - 1.2 \cdot 2}{2})}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.6

Simplify the numerator.

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Step 9.1.6.1

Multiply $- 1.2$ by $2$ .

$s = \frac{7.84 + {(\frac{- 1 - 2.4}{2})}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.6.2

Subtract $2.4$ from $- 1$ .

$s = \frac{7.84 + {(\frac{- 3.4}{2})}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.7

Divide $- 3.4$ by $2$ .

$s = \frac{7.84 + {(- 1.7)}^{2} + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.8

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

$s = \frac{7.84 + 2.89 + {(\frac{1}{16} - 1.2)}^{2}}{3 - 1}$

Step 9.1.9

To write $- 1.2$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$s = \frac{7.84 + 2.89 + {(\frac{1}{16} - 1.2 \cdot \frac{16}{16})}^{2}}{3 - 1}$

Step 9.1.10

Combine $- 1.2$ and $\frac{16}{16}$ .

$s = \frac{7.84 + 2.89 + {(\frac{1}{16} + \frac{- 1.2 \cdot 16}{16})}^{2}}{3 - 1}$

Step 9.1.11

Combine the numerators over the common denominator.

$s = \frac{7.84 + 2.89 + {(\frac{1 - 1.2 \cdot 16}{16})}^{2}}{3 - 1}$

Step 9.1.12

Simplify the numerator.

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Step 9.1.12.1

Multiply $- 1.2$ by $16$ .

$s = \frac{7.84 + 2.89 + {(\frac{1 - 19.2}{16})}^{2}}{3 - 1}$

Step 9.1.12.2

Subtract $19.2$ from $1$ .

$s = \frac{7.84 + 2.89 + {(\frac{- 18.2}{16})}^{2}}{3 - 1}$

Step 9.1.13

Divide $- 18.2$ by $16$ .

$s = \frac{7.84 + 2.89 + {(- 1.1375)}^{2}}{3 - 1}$

Step 9.1.14

Raise $- 1.1375$ to the power of $2$ .

$s = \frac{7.84 + 2.89 + 1.29390625}{3 - 1}$

Step 9.1.15

Add $7.84$ and $2.89$ .

$s = \frac{10.73 + 1.29390625}{3 - 1}$

Step 9.1.16

Add $10.73$ and $1.29390625$ .

$s = \frac{12.02390625}{3 - 1}$

Step 9.2

Simplify the expression.

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Step 9.2.1

Subtract $1$ from $3$ .

$s = \frac{12.02390625}{2}$

Step 9.2.2

Divide $12.02390625$ by $2$ .

$s = 6.01195312$

Step 10

Approximate the result.

$s^{2} \approx 6.012$