Find the Variance 10 , 2 , -6 , -14 , -22

$10$ , $2$ , $- 6$ , $- 14$ , $- 22$

Step 1

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

$\overline{x} = \frac{10 + 2 - 6 - 14 - 22}{5}$

Step 2

Simplify the numerator.

Add $10$ and $2$ .

$\overline{x} = \frac{12 - 6 - 14 - 22}{5}$

Subtract $6$ from $12$ .

$\overline{x} = \frac{6 - 14 - 22}{5}$

Subtract $14$ from $6$ .

$\overline{x} = \frac{- 8 - 22}{5}$

Subtract $22$ from $- 8$ .

$\overline{x} = \frac{- 30}{5}$

Step 3

Divide $- 30$ by $5$ .

$\overline{x} = - 6$

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.

$s = \frac{{(10 + 6)}^{2} + {(2 + 6)}^{2} + {(- 6 + 6)}^{2} + {(- 14 + 6)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Add $10$ and $6$ .

$s = \frac{16^{2} + {(2 + 6)}^{2} + {(- 6 + 6)}^{2} + {(- 14 + 6)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Raise $16$ to the power of $2$ .

$s = \frac{256 + {(2 + 6)}^{2} + {(- 6 + 6)}^{2} + {(- 14 + 6)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Add $2$ and $6$ .

$s = \frac{256 + 8^{2} + {(- 6 + 6)}^{2} + {(- 14 + 6)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Raise $8$ to the power of $2$ .

$s = \frac{256 + 64 + {(- 6 + 6)}^{2} + {(- 14 + 6)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Add $- 6$ and $6$ .

$s = \frac{256 + 64 + 0^{2} + {(- 14 + 6)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Raising $0$ to any positive power yields $0$ .

$s = \frac{256 + 64 + 0 + {(- 14 + 6)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Add $- 14$ and $6$ .

$s = \frac{256 + 64 + 0 + {(- 8)}^{2} + {(- 22 + 6)}^{2}}{5 - 1}$

Raise $- 8$ to the power of $2$ .

$s = \frac{256 + 64 + 0 + 64 + {(- 22 + 6)}^{2}}{5 - 1}$

Add $- 22$ and $6$ .

$s = \frac{256 + 64 + 0 + 64 + {(- 16)}^{2}}{5 - 1}$

Raise $- 16$ to the power of $2$ .

$s = \frac{256 + 64 + 0 + 64 + 256}{5 - 1}$

Add $256$ and $64$ .

$s = \frac{320 + 0 + 64 + 256}{5 - 1}$

Add $320$ and $0$ .

$s = \frac{320 + 64 + 256}{5 - 1}$

Add $320$ and $64$ .

$s = \frac{384 + 256}{5 - 1}$

Add $384$ and $256$ .

$s = \frac{640}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$s = \frac{640}{4}$

Divide $640$ by $4$ .

$s = 160$

Step 7

Approximate the result.

$s^{2} \approx 160$

Find the Variance 10 , 2 , -6 , -14 , -22

Related Questions