Find the Variance 14 , 11 , 5 , 21 , 12 , 4 , 22 , 2 , 12 , 13

$14$ , $11$ , $5$ , $21$ , $12$ , $4$ , $22$ , $2$ , $12$ , $13$

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

$\overline{x} = \frac{14 + 11 + 5 + 21 + 12 + 4 + 22 + 2 + 12 + 13}{10}$

Simplify the numerator.

Add $14$ and $11$ .

$\overline{x} = \frac{25 + 5 + 21 + 12 + 4 + 22 + 2 + 12 + 13}{10}$

Add $25$ and $5$ .

$\overline{x} = \frac{30 + 21 + 12 + 4 + 22 + 2 + 12 + 13}{10}$

Add $30$ and $21$ .

$\overline{x} = \frac{51 + 12 + 4 + 22 + 2 + 12 + 13}{10}$

Add $51$ and $12$ .

$\overline{x} = \frac{63 + 4 + 22 + 2 + 12 + 13}{10}$

Add $63$ and $4$ .

$\overline{x} = \frac{67 + 22 + 2 + 12 + 13}{10}$

Add $67$ and $22$ .

$\overline{x} = \frac{89 + 2 + 12 + 13}{10}$

Add $89$ and $2$ .

$\overline{x} = \frac{91 + 12 + 13}{10}$

Add $91$ and $12$ .

$\overline{x} = \frac{103 + 13}{10}$

Add $103$ and $13$ .

$\overline{x} = \frac{116}{10}$

Cancel the common factor of $116$ and $10$ .

Factor $2$ out of $116$ .

$\overline{x} = \frac{2 (58)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\overline{x} = \frac{2 \cdot 58}{2 \cdot 5}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 58}{2 \cdot 5}$

Rewrite the expression.

$\overline{x} = \frac{58}{5}$

Divide.

$\overline{x} = 11.6$

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.

$s = \frac{{(14 - 11.6)}^{2} + {(11 - 11.6)}^{2} + {(5 - 11.6)}^{2} + {(21 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Simplify the result.

Simplify the numerator.

Subtract $11.6$ from $14$ .

$s = \frac{{2.4}^{2} + {(11 - 11.6)}^{2} + {(5 - 11.6)}^{2} + {(21 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $2.4$ to the power of $2$ .

$s = \frac{5.76 + {(11 - 11.6)}^{2} + {(5 - 11.6)}^{2} + {(21 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $11$ .

$s = \frac{5.76 + {(- 0.6)}^{2} + {(5 - 11.6)}^{2} + {(21 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $- 0.6$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + {(5 - 11.6)}^{2} + {(21 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $5$ .

$s = \frac{5.76 + 0.36 + {(- 6.6)}^{2} + {(21 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $- 6.6$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + {(21 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $21$ .

$s = \frac{5.76 + 0.36 + 43.56 + {9.4}^{2} + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $9.4$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + {(12 - 11.6)}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $12$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + {0.4}^{2} + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $0.4$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + {(4 - 11.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $4$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + {(- 7.6)}^{2} + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $- 7.6$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + {(22 - 11.6)}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $22$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + {10.4}^{2} + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $10.4$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + {(2 - 11.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + {(- 9.6)}^{2} + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $- 9.6$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + 92.16 + {(12 - 11.6)}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $12$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + 92.16 + {0.4}^{2} + {(13 - 11.6)}^{2}}{10 - 1}$

Raise $0.4$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + 92.16 + 0.16 + {(13 - 11.6)}^{2}}{10 - 1}$

Subtract $11.6$ from $13$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + 92.16 + 0.16 + {1.4}^{2}}{10 - 1}$

Raise $1.4$ to the power of $2$ .

$s = \frac{5.76 + 0.36 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + 92.16 + 0.16 + 1.96}{10 - 1}$

Add $5.76$ and $0.36$ .

$s = \frac{6.12 + 43.56 + 88.36 + 0.16 + 57.76 + 108.16 + 92.16 + 0.16 + 1.96}{10 - 1}$

Add $6.12$ and $43.56$ .

$s = \frac{49.68 + 88.36 + 0.16 + 57.76 + 108.16 + 92.16 + 0.16 + 1.96}{10 - 1}$

Add $49.68$ and $88.36$ .

$s = \frac{138.04 + 0.16 + 57.76 + 108.16 + 92.16 + 0.16 + 1.96}{10 - 1}$

Add $138.04$ and $0.16$ .

$s = \frac{138.2 + 57.76 + 108.16 + 92.16 + 0.16 + 1.96}{10 - 1}$

Add $138.2$ and $57.76$ .

$s = \frac{195.96 + 108.16 + 92.16 + 0.16 + 1.96}{10 - 1}$

Add $195.96$ and $108.16$ .

$s = \frac{304.12 + 92.16 + 0.16 + 1.96}{10 - 1}$

Add $304.12$ and $92.16$ .

$s = \frac{396.28 + 0.16 + 1.96}{10 - 1}$

Add $396.28$ and $0.16$ .

$s = \frac{396.44 + 1.96}{10 - 1}$

Add $396.44$ and $1.96$ .

$s = \frac{398.4}{10 - 1}$

Simplify the expression.

Subtract $1$ from $10$ .

$s = \frac{398.4}{9}$

Divide $398.4$ by $9$ .

$s = 44.2\overline{6}$

Approximate the result.

$s^{2} \approx 44.2667$

Find the Variance 14 , 11 , 5 , 21 , 12 , 4 , 22 , 2 , 12 , 13

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