Find the Variance 101/6750 , 31/108

$\frac{101}{6750}$ , $\frac{31}{108}$

Step 1

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

$\overline{x} = \frac{\frac{101}{6750} + \frac{31}{108}}{2}$

Step 2

Simplify the numerator.

To write $\frac{101}{6750}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\overline{x} = \frac{\frac{101}{6750} \cdot \frac{2}{2} + \frac{31}{108}}{2}$

To write $\frac{31}{108}$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$\overline{x} = \frac{\frac{101}{6750} \cdot \frac{2}{2} + \frac{31}{108} \cdot \frac{125}{125}}{2}$

Write each expression with a common denominator of $13500$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{101}{6750}$ by $\frac{2}{2}$ .

$\overline{x} = \frac{\frac{101 \cdot 2}{6750 \cdot 2} + \frac{31}{108} \cdot \frac{125}{125}}{2}$

Multiply $6750$ by $2$ .

$\overline{x} = \frac{\frac{101 \cdot 2}{13500} + \frac{31}{108} \cdot \frac{125}{125}}{2}$

Multiply $\frac{31}{108}$ by $\frac{125}{125}$ .

$\overline{x} = \frac{\frac{101 \cdot 2}{13500} + \frac{31 \cdot 125}{108 \cdot 125}}{2}$

Multiply $108$ by $125$ .

$\overline{x} = \frac{\frac{101 \cdot 2}{13500} + \frac{31 \cdot 125}{13500}}{2}$

Combine the numerators over the common denominator.

$\overline{x} = \frac{\frac{101 \cdot 2 + 31 \cdot 125}{13500}}{2}$

Simplify the numerator.

Multiply $101$ by $2$ .

$\overline{x} = \frac{\frac{202 + 31 \cdot 125}{13500}}{2}$

Multiply $31$ by $125$ .

$\overline{x} = \frac{\frac{202 + 3875}{13500}}{2}$

Add $202$ and $3875$ .

$\overline{x} = \frac{\frac{4077}{13500}}{2}$

Cancel the common factor of $4077$ and $13500$ .

Factor $27$ out of $4077$ .

$\overline{x} = \frac{\frac{27 (151)}{13500}}{2}$

Cancel the common factors.

Factor $27$ out of $13500$ .

$\overline{x} = \frac{\frac{27 \cdot 151}{27 \cdot 500}}{2}$

Cancel the common factor.

$\overline{x} = \frac{\frac{27 \cdot 151}{27 \cdot 500}}{2}$

Rewrite the expression.

$\overline{x} = \frac{\frac{151}{500}}{2}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\overline{x} = \frac{151}{500} \cdot \frac{1}{2}$

Step 4

Multiply $\frac{151}{500} \cdot \frac{1}{2}$ .

Multiply $\frac{151}{500}$ by $\frac{1}{2}$ .

$\overline{x} = \frac{151}{500 \cdot 2}$

Multiply $500$ by $2$ .

$\overline{x} = \frac{151}{1000}$

Step 5

Divide.

$\overline{x} = 0.151$

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.

$\overline{x} = 0.2$

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.

$s = \frac{{(\frac{101}{6750} - 0.2)}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Step 9

Simplify the result.

Simplify the numerator.

To write $- 0.2$ as a fraction with a common denominator, multiply by $\frac{6750}{6750}$ .

$s = \frac{{(\frac{101}{6750} - 0.2 \cdot \frac{6750}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Combine $- 0.2$ and $\frac{6750}{6750}$ .

$s = \frac{{(\frac{101}{6750} + \frac{- 0.2 \cdot 6750}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{{(\frac{101 - 0.2 \cdot 6750}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Simplify the numerator.

Multiply $- 0.2$ by $6750$ .

$s = \frac{{(\frac{101 - 1350}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Subtract $1350$ from $101$ .

$s = \frac{{(\frac{- 1249}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Cancel the common factor of $- 1249$ and $6750$ .

Rewrite $- 1249$ as $1 (- 1249)$ .

$s = \frac{{(\frac{1 (- 1249)}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Cancel the common factors.

Rewrite $6750$ as $1 (6750)$ .

$s = \frac{{(\frac{1 \cdot - 1249}{1 \cdot 6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Cancel the common factor.

$s = \frac{{(\frac{1 \cdot - 1249}{1 \cdot 6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Rewrite the expression.

$s = \frac{{(\frac{- 1249}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Move the negative in front of the fraction.

$s = \frac{{(- \frac{1249}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1249}{6750}$ .

$s = \frac{{(- 1)}^{2} {(\frac{1249}{6750})}^{2} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Apply the product rule to $\frac{1249}{6750}$ .

$s = \frac{{(- 1)}^{2} (\frac{1249^{2}}{6750^{2}}) + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

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

$s = \frac{1 (\frac{1249^{2}}{6750^{2}}) + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Multiply $\frac{1249^{2}}{6750^{2}}$ by $1$ .

$s = \frac{\frac{1249^{2}}{6750^{2}} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Raise $1249$ to the power of $2$ .

$s = \frac{\frac{1560001}{6750^{2}} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

Raise $6750$ to the power of $2$ .

$s = \frac{\frac{1560001}{45562500} + {(\frac{31}{108} - 0.2)}^{2}}{2 - 1}$

To write $- 0.2$ as a fraction with a common denominator, multiply by $\frac{108}{108}$ .

$s = \frac{\frac{1560001}{45562500} + {(\frac{31}{108} - 0.2 \cdot \frac{108}{108})}^{2}}{2 - 1}$

Combine $- 0.2$ and $\frac{108}{108}$ .

$s = \frac{\frac{1560001}{45562500} + {(\frac{31}{108} + \frac{- 0.2 \cdot 108}{108})}^{2}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{\frac{1560001}{45562500} + {(\frac{31 - 0.2 \cdot 108}{108})}^{2}}{2 - 1}$

Simplify the numerator.

Multiply $- 0.2$ by $108$ .

$s = \frac{\frac{1560001}{45562500} + {(\frac{31 - 21.6}{108})}^{2}}{2 - 1}$

Subtract $21.6$ from $31$ .

$s = \frac{\frac{1560001}{45562500} + {(\frac{9.4}{108})}^{2}}{2 - 1}$

Divide $9.4$ by $108$ .

$s = \frac{\frac{1560001}{45562500} + {0.08\overline{703}}^{2}}{2 - 1}$

Raise $0.08\overline{703}$ to the power of $2$ .

$s = \frac{\frac{1560001}{45562500} + 0.00757544}{2 - 1}$

To write $0.00757544$ as a fraction with a common denominator, multiply by $\frac{45562500}{45562500}$ .

$s = \frac{\frac{1560001}{45562500} + 0.00757544 \cdot \frac{45562500}{45562500}}{2 - 1}$

Combine $0.00757544$ and $\frac{45562500}{45562500}$ .

$s = \frac{\frac{1560001}{45562500} + \frac{0.00757544 \cdot 45562500}{45562500}}{2 - 1}$

Combine the numerators over the common denominator.

$s = \frac{\frac{1560001 + 0.00757544 \cdot 45562500}{45562500}}{2 - 1}$

Rewrite $\frac{1560001 + 0.00757544 \cdot 45562500}{45562500}$ in a factored form.

Multiply $0.00757544$ by $45562500$ .

$s = \frac{\frac{1560001 + 345156.25}{45562500}}{2 - 1}$

Add $1560001$ and $345156.25$ .

$s = \frac{\frac{1905157.25}{45562500}}{2 - 1}$

Divide $1905157.25$ by $45562500$ .

$s = \frac{0.04181415}{2 - 1}$

Simplify the expression.

Subtract $1$ from $2$ .

$s = \frac{0.04181415}{1}$

Divide $0.04181415$ by $1$ .

$s = 0.04181415$

Step 10

Approximate the result.

$s^{2} \approx 0.0418$

Find the Variance 101/6750 , 31/108

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