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

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