Find the Variance 12 , 13 , 8 , 13 , 9 , 11 , 12 , 9 , 11

$12$ , $13$ , $8$ , $13$ , $9$ , $11$ , $12$ , $9$ , $11$

Step 1

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

$\overline{x} = \frac{12 + 13 + 8 + 13 + 9 + 11 + 12 + 9 + 11}{9}$

Step 2

Simplify the numerator.

Add $12$ and $13$ .

$\overline{x} = \frac{25 + 8 + 13 + 9 + 11 + 12 + 9 + 11}{9}$

Add $25$ and $8$ .

$\overline{x} = \frac{33 + 13 + 9 + 11 + 12 + 9 + 11}{9}$

Add $33$ and $13$ .

$\overline{x} = \frac{46 + 9 + 11 + 12 + 9 + 11}{9}$

Add $46$ and $9$ .

$\overline{x} = \frac{55 + 11 + 12 + 9 + 11}{9}$

Add $55$ and $11$ .

$\overline{x} = \frac{66 + 12 + 9 + 11}{9}$

Add $66$ and $12$ .

$\overline{x} = \frac{78 + 9 + 11}{9}$

Add $78$ and $9$ .

$\overline{x} = \frac{87 + 11}{9}$

Add $87$ and $11$ .

$\overline{x} = \frac{98}{9}$

Step 3

Divide.

$\overline{x} = 10.\overline{8}$

Step 4

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.

$\overline{x} = 10.9$

Step 5

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

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

Step 6

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

$s = \frac{{(12 - 10.9)}^{2} + {(13 - 10.9)}^{2} + {(8 - 10.9)}^{2} + {(13 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract $10.9$ from $12$ .

$s = \frac{{1.1}^{2} + {(13 - 10.9)}^{2} + {(8 - 10.9)}^{2} + {(13 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $1.1$ to the power of $2$ .

$s = \frac{1.21 + {(13 - 10.9)}^{2} + {(8 - 10.9)}^{2} + {(13 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $13$ .

$s = \frac{1.21 + {2.1}^{2} + {(8 - 10.9)}^{2} + {(13 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $2.1$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + {(8 - 10.9)}^{2} + {(13 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $8$ .

$s = \frac{1.21 + 4.41 + {(- 2.9)}^{2} + {(13 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $- 2.9$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + 8.41 + {(13 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $13$ .

$s = \frac{1.21 + 4.41 + 8.41 + {2.1}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $2.1$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $9$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + {(- 1.9)}^{2} + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $- 1.9$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + {(11 - 10.9)}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $11$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + {0.1}^{2} + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $0.1$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + 0.01 + {(12 - 10.9)}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $12$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + 0.01 + {1.1}^{2} + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $1.1$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + 0.01 + 1.21 + {(9 - 10.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $9$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + 0.01 + 1.21 + {(- 1.9)}^{2} + {(11 - 10.9)}^{2}}{9 - 1}$

Raise $- 1.9$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + 0.01 + 1.21 + 3.61 + {(11 - 10.9)}^{2}}{9 - 1}$

Subtract $10.9$ from $11$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + 0.01 + 1.21 + 3.61 + {0.1}^{2}}{9 - 1}$

Raise $0.1$ to the power of $2$ .

$s = \frac{1.21 + 4.41 + 8.41 + 4.41 + 3.61 + 0.01 + 1.21 + 3.61 + 0.01}{9 - 1}$

Add $1.21$ and $4.41$ .

$s = \frac{5.62 + 8.41 + 4.41 + 3.61 + 0.01 + 1.21 + 3.61 + 0.01}{9 - 1}$

Add $5.62$ and $8.41$ .

$s = \frac{14.03 + 4.41 + 3.61 + 0.01 + 1.21 + 3.61 + 0.01}{9 - 1}$

Add $14.03$ and $4.41$ .

$s = \frac{18.44 + 3.61 + 0.01 + 1.21 + 3.61 + 0.01}{9 - 1}$

Add $18.44$ and $3.61$ .

$s = \frac{22.05 + 0.01 + 1.21 + 3.61 + 0.01}{9 - 1}$

Add $22.05$ and $0.01$ .

$s = \frac{22.06 + 1.21 + 3.61 + 0.01}{9 - 1}$

Add $22.06$ and $1.21$ .

$s = \frac{23.27 + 3.61 + 0.01}{9 - 1}$

Add $23.27$ and $3.61$ .

$s = \frac{26.88 + 0.01}{9 - 1}$

Add $26.88$ and $0.01$ .

$s = \frac{26.89}{9 - 1}$

Simplify the expression.

Subtract $1$ from $9$ .

$s = \frac{26.89}{8}$

Divide $26.89$ by $8$ .

$s = 3.36125$

Step 8

Approximate the result.

$s^{2} \approx 3.3612$

Find the Variance 12 , 13 , 8 , 13 , 9 , 11 , 12 , 9 , 11

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