Find the Variance 13 , 15 , 17 , 18 , 12 , 21 , 10

$13$ , $15$ , $17$ , $18$ , $12$ , $21$ , $10$

Step 1

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

$\overline{x} = \frac{13 + 15 + 17 + 18 + 12 + 21 + 10}{7}$

Step 2

Simplify the numerator.

Add $13$ and $15$ .

$\overline{x} = \frac{28 + 17 + 18 + 12 + 21 + 10}{7}$

Add $28$ and $17$ .

$\overline{x} = \frac{45 + 18 + 12 + 21 + 10}{7}$

Add $45$ and $18$ .

$\overline{x} = \frac{63 + 12 + 21 + 10}{7}$

Add $63$ and $12$ .

$\overline{x} = \frac{75 + 21 + 10}{7}$

Add $75$ and $21$ .

$\overline{x} = \frac{96 + 10}{7}$

Add $96$ and $10$ .

$\overline{x} = \frac{106}{7}$

Step 3

Divide.

$\overline{x} = 15.14285714$

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.

$\overline{x} = 15.1$

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.

$s = \frac{{(13 - 15.1)}^{2} + {(15 - 15.1)}^{2} + {(17 - 15.1)}^{2} + {(18 - 15.1)}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract $15.1$ from $13$ .

$s = \frac{{(- 2.1)}^{2} + {(15 - 15.1)}^{2} + {(17 - 15.1)}^{2} + {(18 - 15.1)}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

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

$s = \frac{4.41 + {(15 - 15.1)}^{2} + {(17 - 15.1)}^{2} + {(18 - 15.1)}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Subtract $15.1$ from $15$ .

$s = \frac{4.41 + {(- 0.1)}^{2} + {(17 - 15.1)}^{2} + {(18 - 15.1)}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

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

$s = \frac{4.41 + 0.01 + {(17 - 15.1)}^{2} + {(18 - 15.1)}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Subtract $15.1$ from $17$ .

$s = \frac{4.41 + 0.01 + {1.9}^{2} + {(18 - 15.1)}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

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

$s = \frac{4.41 + 0.01 + 3.61 + {(18 - 15.1)}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Subtract $15.1$ from $18$ .

$s = \frac{4.41 + 0.01 + 3.61 + {2.9}^{2} + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

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

$s = \frac{4.41 + 0.01 + 3.61 + 8.41 + {(12 - 15.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Subtract $15.1$ from $12$ .

$s = \frac{4.41 + 0.01 + 3.61 + 8.41 + {(- 3.1)}^{2} + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Raise $- 3.1$ to the power of $2$ .

$s = \frac{4.41 + 0.01 + 3.61 + 8.41 + 9.61 + {(21 - 15.1)}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Subtract $15.1$ from $21$ .

$s = \frac{4.41 + 0.01 + 3.61 + 8.41 + 9.61 + {5.9}^{2} + {(10 - 15.1)}^{2}}{7 - 1}$

Raise $5.9$ to the power of $2$ .

$s = \frac{4.41 + 0.01 + 3.61 + 8.41 + 9.61 + 34.81 + {(10 - 15.1)}^{2}}{7 - 1}$

Subtract $15.1$ from $10$ .

$s = \frac{4.41 + 0.01 + 3.61 + 8.41 + 9.61 + 34.81 + {(- 5.1)}^{2}}{7 - 1}$

Raise $- 5.1$ to the power of $2$ .

$s = \frac{4.41 + 0.01 + 3.61 + 8.41 + 9.61 + 34.81 + 26.01}{7 - 1}$

Add $4.41$ and $0.01$ .

$s = \frac{4.42 + 3.61 + 8.41 + 9.61 + 34.81 + 26.01}{7 - 1}$

Add $4.42$ and $3.61$ .

$s = \frac{8.03 + 8.41 + 9.61 + 34.81 + 26.01}{7 - 1}$

Add $8.03$ and $8.41$ .

$s = \frac{16.44 + 9.61 + 34.81 + 26.01}{7 - 1}$

Add $16.44$ and $9.61$ .

$s = \frac{26.05 + 34.81 + 26.01}{7 - 1}$

Add $26.05$ and $34.81$ .

$s = \frac{60.86 + 26.01}{7 - 1}$

Add $60.86$ and $26.01$ .

$s = \frac{86.87}{7 - 1}$

Simplify the expression.

Subtract $1$ from $7$ .

$s = \frac{86.87}{6}$

Divide $86.87$ by $6$ .

$s = 14.478\overline{3}$

Step 8

Approximate the result.

$s^{2} \approx 14.4783$

Find the Variance 13 , 15 , 17 , 18 , 12 , 21 , 10

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