Find the Variance 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12

$1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $9$ , $10$ , $11$ , $12$

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

$\overline{x} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Simplify the numerator.

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Add $1$ and $2$ .

$\overline{x} = \frac{3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Add $3$ and $3$ .

$\overline{x} = \frac{6 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Add $6$ and $4$ .

$\overline{x} = \frac{10 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Add $10$ and $5$ .

$\overline{x} = \frac{15 + 6 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Add $15$ and $6$ .

$\overline{x} = \frac{21 + 7 + 8 + 9 + 10 + 11 + 12}{12}$

Add $21$ and $7$ .

$\overline{x} = \frac{28 + 8 + 9 + 10 + 11 + 12}{12}$

Add $28$ and $8$ .

$\overline{x} = \frac{36 + 9 + 10 + 11 + 12}{12}$

Add $36$ and $9$ .

$\overline{x} = \frac{45 + 10 + 11 + 12}{12}$

Add $45$ and $10$ .

$\overline{x} = \frac{55 + 11 + 12}{12}$

Add $55$ and $11$ .

$\overline{x} = \frac{66 + 12}{12}$

Add $66$ and $12$ .

$\overline{x} = \frac{78}{12}$

Cancel the common factor of $78$ and $12$ .

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Factor $6$ out of $78$ .

$\overline{x} = \frac{6 (13)}{12}$

Cancel the common factors.

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Factor $6$ out of $12$ .

$\overline{x} = \frac{6 \cdot 13}{6 \cdot 2}$

Cancel the common factor.

$\overline{x} = \frac{6 \cdot 13}{6 \cdot 2}$

Rewrite the expression.

$\overline{x} = \frac{13}{2}$

Divide.

$\overline{x} = 6.5$

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

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

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

$s = \frac{{(1 - 6.5)}^{2} + {(2 - 6.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Simplify the result.

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Simplify the numerator.

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Subtract $6.5$ from $1$ .

$s = \frac{{(- 5.5)}^{2} + {(2 - 6.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

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

$s = \frac{30.25 + {(2 - 6.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $2$ .

$s = \frac{30.25 + {(- 4.5)}^{2} + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Raise $- 4.5$ to the power of $2$ .

$s = \frac{30.25 + 20.25 + {(3 - 6.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $3$ .

$s = \frac{30.25 + 20.25 + {(- 3.5)}^{2} + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Raise $- 3.5$ to the power of $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + {(4 - 6.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $4$ .

$s = \frac{30.25 + 20.25 + 12.25 + {(- 2.5)}^{2} + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

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

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + {(5 - 6.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $5$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + {(- 1.5)}^{2} + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

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

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + {(6 - 6.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $6$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + {(- 0.5)}^{2} + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Raise $- 0.5$ to the power of $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + {(7 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $7$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + {0.5}^{2} + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Raise $0.5$ to the power of $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + {(8 - 6.5)}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $8$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + {1.5}^{2} + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

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

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + {(9 - 6.5)}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $9$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + {2.5}^{2} + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

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

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + {(10 - 6.5)}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $10$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + {3.5}^{2} + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Raise $3.5$ to the power of $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + {(11 - 6.5)}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $11$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + {4.5}^{2} + {(12 - 6.5)}^{2}}{12 - 1}$

Raise $4.5$ to the power of $2$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + {(12 - 6.5)}^{2}}{12 - 1}$

Subtract $6.5$ from $12$ .

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + {5.5}^{2}}{12 - 1}$

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

$s = \frac{30.25 + 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $30.25$ and $20.25$ .

$s = \frac{50.5 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $50.5$ and $12.25$ .

$s = \frac{62.75 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $62.75$ and $6.25$ .

$s = \frac{69 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $69$ and $2.25$ .

$s = \frac{71.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $71.25$ and $0.25$ .

$s = \frac{71.5 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $71.5$ and $0.25$ .

$s = \frac{71.75 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $71.75$ and $2.25$ .

$s = \frac{74 + 6.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $74$ and $6.25$ .

$s = \frac{80.25 + 12.25 + 20.25 + 30.25}{12 - 1}$

Add $80.25$ and $12.25$ .

$s = \frac{92.5 + 20.25 + 30.25}{12 - 1}$

Add $92.5$ and $20.25$ .

$s = \frac{112.75 + 30.25}{12 - 1}$

Add $112.75$ and $30.25$ .

$s = \frac{143}{12 - 1}$

Simplify the expression.

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Subtract $1$ from $12$ .

$s = \frac{143}{11}$

Divide $143$ by $11$ .

$s = 13$

Approximate the result.

$s^{2} \approx 13$

Find the Variance 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12

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