Find the Variance 14/21 , 8/12

$\frac{14}{21}$ , $\frac{8}{12}$

Step 1

Cancel the common factor of $14$ and $21$ .

Factor $7$ out of $14$ .

$\overline{x} = \frac{7 (2)}{21}, \frac{8}{12}$

Cancel the common factors.

Factor $7$ out of $21$ .

$\overline{x} = \frac{7 \cdot 2}{7 \cdot 3}, \frac{8}{12}$

Cancel the common factor.

$\overline{x} = \frac{7 \cdot 2}{7 \cdot 3}, \frac{8}{12}$

Rewrite the expression.

$\overline{x} = \frac{2}{3}, \frac{8}{12}$

Step 2

Cancel the common factor of $8$ and $12$ .

Factor $4$ out of $8$ .

$\overline{x} = \frac{2}{3}, \frac{4 (2)}{12}$

Cancel the common factors.

Factor $4$ out of $12$ .

$\overline{x} = \frac{2}{3}, \frac{4 \cdot 2}{4 \cdot 3}$

Cancel the common factor.

$\overline{x} = \frac{2}{3}, \frac{4 \cdot 2}{4 \cdot 3}$

Rewrite the expression.

$\overline{x} = \frac{2}{3}, \frac{2}{3}$

Step 3

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

$\overline{x} = \frac{\frac{2}{3} + \frac{2}{3}}{2}$

Step 4

Simplify the numerator.

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{2 + 2}{3}}{2}$

Add $2$ and $2$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\overline{x} = \frac{2 (2)}{3} \cdot \frac{1}{2}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 2}{3} \cdot \frac{1}{2}$

Rewrite the expression.

$\overline{x} = \frac{2}{3}$

Step 7

Divide.

$\overline{x} = 0.\overline{6}$

Step 8

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.

$\overline{x} = 0.7$

Step 9

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 10

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

$s = \frac{{(\frac{2}{3} - 0.7)}^{2} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

Step 11

Simplify the result.

Simplify the numerator.

To write $- 0.7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$s = \frac{{(\frac{2}{3} - 0.7 \cdot \frac{3}{3})}^{2} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

Combine $- 0.7$ and $\frac{3}{3}$ .

$s = \frac{{(\frac{2}{3} + \frac{- 0.7 \cdot 3}{3})}^{2} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{{(\frac{2 - 0.7 \cdot 3}{3})}^{2} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

Simplify the numerator.

Multiply $- 0.7$ by $3$ .

$s = \frac{{(\frac{2 - 2.1}{3})}^{2} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

Subtract $2.1$ from $2$ .

$s = \frac{{(\frac{- 0.1}{3})}^{2} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

Divide $- 0.1$ by $3$ .

$s = \frac{{(- 0.0\overline{3})}^{2} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

Raise $- 0.0\overline{3}$ to the power of $2$ .

$s = \frac{0.00\overline{1} + {(\frac{2}{3} - 0.7)}^{2}}{2 - 1}$

To write $- 0.7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$s = \frac{0.00\overline{1} + {(\frac{2}{3} - 0.7 \cdot \frac{3}{3})}^{2}}{2 - 1}$

Combine $- 0.7$ and $\frac{3}{3}$ .

$s = \frac{0.00\overline{1} + {(\frac{2}{3} + \frac{- 0.7 \cdot 3}{3})}^{2}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{0.00\overline{1} + {(\frac{2 - 0.7 \cdot 3}{3})}^{2}}{2 - 1}$

Simplify the numerator.

Multiply $- 0.7$ by $3$ .

$s = \frac{0.00\overline{1} + {(\frac{2 - 2.1}{3})}^{2}}{2 - 1}$

Subtract $2.1$ from $2$ .

$s = \frac{0.00\overline{1} + {(\frac{- 0.1}{3})}^{2}}{2 - 1}$

Divide $- 0.1$ by $3$ .

$s = \frac{0.00\overline{1} + {(- 0.0\overline{3})}^{2}}{2 - 1}$

Raise $- 0.0\overline{3}$ to the power of $2$ .

$s = \frac{0.00\overline{1} + 0.00\overline{1}}{2 - 1}$

Add $0.00\overline{1}$ and $0.00\overline{1}$ .

$s = \frac{0.00\overline{2}}{2 - 1}$

Simplify the expression.

Subtract $1$ from $2$ .

$s = \frac{0.00\overline{2}}{1}$

Divide $0.00\overline{2}$ by $1$ .

$s = 0.00\overline{2}$

Step 12

Approximate the result.

$s^{2} \approx 0.0022$

Find the Variance 14/21 , 8/12

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