Find the Variance 11/15 , 1/12

$\frac{11}{15}$ , $\frac{1}{12}$

Step 1

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

$\overline{x} = \frac{\frac{11}{15} + \frac{1}{12}}{2}$

Step 2

Simplify the numerator.

To write $\frac{11}{15}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\overline{x} = \frac{\frac{11}{15} \cdot \frac{4}{4} + \frac{1}{12}}{2}$

To write $\frac{1}{12}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\overline{x} = \frac{\frac{11}{15} \cdot \frac{4}{4} + \frac{1}{12} \cdot \frac{5}{5}}{2}$

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{11}{15}$ by $\frac{4}{4}$ .

$\overline{x} = \frac{\frac{11 \cdot 4}{15 \cdot 4} + \frac{1}{12} \cdot \frac{5}{5}}{2}$

Multiply $15$ by $4$ .

$\overline{x} = \frac{\frac{11 \cdot 4}{60} + \frac{1}{12} \cdot \frac{5}{5}}{2}$

Multiply $\frac{1}{12}$ by $\frac{5}{5}$ .

$\overline{x} = \frac{\frac{11 \cdot 4}{60} + \frac{5}{12 \cdot 5}}{2}$

Multiply $12$ by $5$ .

$\overline{x} = \frac{\frac{11 \cdot 4}{60} + \frac{5}{60}}{2}$

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{11 \cdot 4 + 5}{60}}{2}$

Simplify the numerator.

Multiply $11$ by $4$ .

$\overline{x} = \frac{\frac{44 + 5}{60}}{2}$

Add $44$ and $5$ .

$\overline{x} = \frac{\frac{49}{60}}{2}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\overline{x} = \frac{49}{60} \cdot \frac{1}{2}$

Step 4

Multiply $\frac{49}{60} \cdot \frac{1}{2}$ .

Multiply $\frac{49}{60}$ by $\frac{1}{2}$ .

$\overline{x} = \frac{49}{60 \cdot 2}$

Multiply $60$ by $2$ .

$\overline{x} = \frac{49}{120}$

Step 5

Divide.

$\overline{x} = 0.408\overline{3}$

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.

$\overline{x} = 0.4$

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.

$s = \frac{{(\frac{11}{15} - 0.4)}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Step 9

Simplify the result.

Simplify the numerator.

To write $- 0.4$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$s = \frac{{(\frac{11}{15} - 0.4 \cdot \frac{15}{15})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Combine $- 0.4$ and $\frac{15}{15}$ .

$s = \frac{{(\frac{11}{15} + \frac{- 0.4 \cdot 15}{15})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{{(\frac{11 - 0.4 \cdot 15}{15})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Simplify the numerator.

Multiply $- 0.4$ by $15$ .

$s = \frac{{(\frac{11 - 6}{15})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Subtract $6$ from $11$ .

$s = \frac{{(\frac{5}{15})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Cancel the common factor of $5$ and $15$ .

Factor $5$ out of $5$ .

$s = \frac{{(\frac{5 (1)}{15})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Cancel the common factors.

Factor $5$ out of $15$ .

$s = \frac{{(\frac{5 \cdot 1}{5 \cdot 3})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Cancel the common factor.

$s = \frac{{(\frac{5 \cdot 1}{5 \cdot 3})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Rewrite the expression.

$s = \frac{{(\frac{1}{3})}^{2} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Apply the product rule to $\frac{1}{3}$ .

$s = \frac{\frac{1^{2}}{3^{2}} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

One to any power is one.

$s = \frac{\frac{1}{3^{2}} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

Raise $3$ to the power of $2$ .

$s = \frac{\frac{1}{9} + {(\frac{1}{12} - 0.4)}^{2}}{2 - 1}$

To write $- 0.4$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$s = \frac{\frac{1}{9} + {(\frac{1}{12} - 0.4 \cdot \frac{12}{12})}^{2}}{2 - 1}$

Combine $- 0.4$ and $\frac{12}{12}$ .

$s = \frac{\frac{1}{9} + {(\frac{1}{12} + \frac{- 0.4 \cdot 12}{12})}^{2}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{\frac{1}{9} + {(\frac{1 - 0.4 \cdot 12}{12})}^{2}}{2 - 1}$

Simplify the numerator.

Multiply $- 0.4$ by $12$ .

$s = \frac{\frac{1}{9} + {(\frac{1 - 4.8}{12})}^{2}}{2 - 1}$

Subtract $4.8$ from $1$ .

$s = \frac{\frac{1}{9} + {(\frac{- 3.8}{12})}^{2}}{2 - 1}$

Divide $- 3.8$ by $12$ .

$s = \frac{\frac{1}{9} + {(- 0.31\overline{6})}^{2}}{2 - 1}$

Raise $- 0.31\overline{6}$ to the power of $2$ .

$s = \frac{\frac{1}{9} + 0.1002\overline{7}}{2 - 1}$

To write $0.1002\overline{7}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$s = \frac{\frac{1}{9} + 0.1002\overline{7} \cdot \frac{9}{9}}{2 - 1}$

Combine $0.1002\overline{7}$ and $\frac{9}{9}$ .

$s = \frac{\frac{1}{9} + \frac{0.1002\overline{7} \cdot 9}{9}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{\frac{1 + 0.1002\overline{7} \cdot 9}{9}}{2 - 1}$

Rewrite $\frac{1 + 0.1002\overline{7} \cdot 9}{9}$ in a factored form.

Multiply $0.1002\overline{7}$ by $9$ .

$s = \frac{\frac{1 + 0.9025}{9}}{2 - 1}$

Add $1$ and $0.9025$ .

$s = \frac{\frac{1.9025}{9}}{2 - 1}$

Divide $1.9025$ by $9$ .

$s = \frac{0.21138888}{2 - 1}$

Simplify the expression.

Subtract $1$ from $2$ .

$s = \frac{0.21138888}{1}$

Divide $0.21138888$ by $1$ .

$s = 0.21138888$

Step 10

Approximate the result.

$s^{2} \approx 0.2114$

Find the Variance 11/15 , 1/12

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