Find the Variance -21 , -35 , -17 , -31 , -18 , -2 , 5

$- 21$ ,

$- 35$ ,

$- 17$ ,

$- 31$ ,

$- 18$ ,

$- 2$ ,

$5$

Step 1

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

$\overline{x} = \frac{- 21 - 35 - 17 - 31 - 18 - 2 + 5}{7}$

Step 2

Simplify the numerator.

Subtract

$35$ from

$- 21$ .

$\overline{x} = \frac{- 56 - 17 - 31 - 18 - 2 + 5}{7}$

Subtract

$17$ from

$- 56$ .

$\overline{x} = \frac{- 73 - 31 - 18 - 2 + 5}{7}$

Subtract

$31$ from

$- 73$ .

$\overline{x} = \frac{- 104 - 18 - 2 + 5}{7}$

Subtract

$18$ from

$- 104$ .

$\overline{x} = \frac{- 122 - 2 + 5}{7}$

Subtract

$2$ from

$- 122$ .

$\overline{x} = \frac{- 124 + 5}{7}$

Add

$- 124$ and

$5$ .

$\overline{x} = \frac{- 119}{7}$

Step 3

Divide

$- 119$ by

$7$ .

$\overline{x} = - 17$

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.

$s = \frac{{(- 21 + 17)}^{2} + {(- 35 + 17)}^{2} + {(- 17 + 17)}^{2} + {(- 31 + 17)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Add

$- 21$ and

$17$ .

$s = \frac{{(- 4)}^{2} + {(- 35 + 17)}^{2} + {(- 17 + 17)}^{2} + {(- 31 + 17)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Raise

$- 4$ to the power of

$2$ .

$s = \frac{16 + {(- 35 + 17)}^{2} + {(- 17 + 17)}^{2} + {(- 31 + 17)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Add

$- 35$ and

$17$ .

$s = \frac{16 + {(- 18)}^{2} + {(- 17 + 17)}^{2} + {(- 31 + 17)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Raise

$- 18$ to the power of

$2$ .

$s = \frac{16 + 324 + {(- 17 + 17)}^{2} + {(- 31 + 17)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Add

$- 17$ and

$17$ .

$s = \frac{16 + 324 + 0^{2} + {(- 31 + 17)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Raising

$0$ to any positive power yields

$0$ .

$s = \frac{16 + 324 + 0 + {(- 31 + 17)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Add

$- 31$ and

$17$ .

$s = \frac{16 + 324 + 0 + {(- 14)}^{2} + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Raise

$- 14$ to the power of

$2$ .

$s = \frac{16 + 324 + 0 + 196 + {(- 18 + 17)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Add

$- 18$ and

$17$ .

$s = \frac{16 + 324 + 0 + 196 + {(- 1)}^{2} + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{16 + 324 + 0 + 196 + 1 + {(- 2 + 17)}^{2} + {(5 + 17)}^{2}}{7 - 1}$

Add

$- 2$ and

$17$ .

$s = \frac{16 + 324 + 0 + 196 + 1 + 15^{2} + {(5 + 17)}^{2}}{7 - 1}$

Raise

$15$ to the power of

$2$ .

$s = \frac{16 + 324 + 0 + 196 + 1 + 225 + {(5 + 17)}^{2}}{7 - 1}$

Add

$5$ and

$17$ .

$s = \frac{16 + 324 + 0 + 196 + 1 + 225 + 22^{2}}{7 - 1}$

Raise

$22$ to the power of

$2$ .

$s = \frac{16 + 324 + 0 + 196 + 1 + 225 + 484}{7 - 1}$

Add

$16$ and

$324$ .

$s = \frac{340 + 0 + 196 + 1 + 225 + 484}{7 - 1}$

Add

$340$ and

$0$ .

$s = \frac{340 + 196 + 1 + 225 + 484}{7 - 1}$

Add

$340$ and

$196$ .

$s = \frac{536 + 1 + 225 + 484}{7 - 1}$

Add

$536$ and

$1$ .

$s = \frac{537 + 225 + 484}{7 - 1}$

Add

$537$ and

$225$ .

$s = \frac{762 + 484}{7 - 1}$

Add

$762$ and

$484$ .

$s = \frac{1246}{7 - 1}$

Reduce the expression by cancelling the common factors.

Subtract

$1$ from

$7$ .

$s = \frac{1246}{6}$

Cancel the common factor of

$1246$ and

$6$ .

Factor

$2$ out of

$1246$ .

$s = \frac{2 (623)}{6}$

Cancel the common factors.

Factor

$2$ out of

$6$ .

$s = \frac{2 \cdot 623}{2 \cdot 3}$

Cancel the common factor.

$s = \frac{2 \cdot 623}{2 \cdot 3}$

Rewrite the expression.

$s = \frac{623}{3}$

Step 7

Approximate the result.

$s^{2} \approx 207.6667$

Find the Variance -21 , -35 , -17 , -31 , -18 , -2 , 5

Related Questions