Find the Variance -12 , -12 , 5 , 5 , -12 , 5 , -14

$- 12$ , $- 12$ , $5$ , $5$ , $- 12$ , $5$ , $- 14$

Step 1

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

$\overline{x} = \frac{- 12 - 12 + 5 + 5 - 12 + 5 - 14}{7}$

Step 2

Simplify the numerator.

Subtract $12$ from $- 12$ .

$\overline{x} = \frac{- 24 + 5 + 5 - 12 + 5 - 14}{7}$

Add $- 24$ and $5$ .

$\overline{x} = \frac{- 19 + 5 - 12 + 5 - 14}{7}$

Add $- 19$ and $5$ .

$\overline{x} = \frac{- 14 - 12 + 5 - 14}{7}$

Subtract $12$ from $- 14$ .

$\overline{x} = \frac{- 26 + 5 - 14}{7}$

Add $- 26$ and $5$ .

$\overline{x} = \frac{- 21 - 14}{7}$

Subtract $14$ from $- 21$ .

$\overline{x} = \frac{- 35}{7}$

Step 3

Divide $- 35$ by $7$ .

$\overline{x} = - 5$

Step 4

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 5

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

$s = \frac{{(- 12 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(5 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Add $- 12$ and $5$ .

$s = \frac{{(- 7)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(5 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Raise $- 7$ to the power of $2$ .

$s = \frac{49 + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(5 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Add $- 12$ and $5$ .

$s = \frac{49 + {(- 7)}^{2} + {(5 + 5)}^{2} + {(5 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Raise $- 7$ to the power of $2$ .

$s = \frac{49 + 49 + {(5 + 5)}^{2} + {(5 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Add $5$ and $5$ .

$s = \frac{49 + 49 + 10^{2} + {(5 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Raise $10$ to the power of $2$ .

$s = \frac{49 + 49 + 100 + {(5 + 5)}^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Add $5$ and $5$ .

$s = \frac{49 + 49 + 100 + 10^{2} + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Raise $10$ to the power of $2$ .

$s = \frac{49 + 49 + 100 + 100 + {(- 12 + 5)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Add $- 12$ and $5$ .

$s = \frac{49 + 49 + 100 + 100 + {(- 7)}^{2} + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Raise $- 7$ to the power of $2$ .

$s = \frac{49 + 49 + 100 + 100 + 49 + {(5 + 5)}^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Add $5$ and $5$ .

$s = \frac{49 + 49 + 100 + 100 + 49 + 10^{2} + {(- 14 + 5)}^{2}}{7 - 1}$

Raise $10$ to the power of $2$ .

$s = \frac{49 + 49 + 100 + 100 + 49 + 100 + {(- 14 + 5)}^{2}}{7 - 1}$

Add $- 14$ and $5$ .

$s = \frac{49 + 49 + 100 + 100 + 49 + 100 + {(- 9)}^{2}}{7 - 1}$

Raise $- 9$ to the power of $2$ .

$s = \frac{49 + 49 + 100 + 100 + 49 + 100 + 81}{7 - 1}$

Add $49$ and $49$ .

$s = \frac{98 + 100 + 100 + 49 + 100 + 81}{7 - 1}$

Add $98$ and $100$ .

$s = \frac{198 + 100 + 49 + 100 + 81}{7 - 1}$

Add $198$ and $100$ .

$s = \frac{298 + 49 + 100 + 81}{7 - 1}$

Add $298$ and $49$ .

$s = \frac{347 + 100 + 81}{7 - 1}$

Add $347$ and $100$ .

$s = \frac{447 + 81}{7 - 1}$

Add $447$ and $81$ .

$s = \frac{528}{7 - 1}$

Simplify the expression.

Subtract $1$ from $7$ .

$s = \frac{528}{6}$

Divide $528$ by $6$ .

$s = 88$

Step 7

Approximate the result.

$s^{2} \approx 88$

Find the Variance -12 , -12 , 5 , 5 , -12 , 5 , -14

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