Find the Variance 22 , 78 , 45 , 60 , 33 , 33 , 58

$22$ , $78$ , $45$ , $60$ , $33$ , $33$ , $58$

Step 1

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

$\overline{x} = \frac{22 + 78 + 45 + 60 + 33 + 33 + 58}{7}$

Step 2

Simplify the numerator.

Add $22$ and $78$ .

$\overline{x} = \frac{100 + 45 + 60 + 33 + 33 + 58}{7}$

Add $100$ and $45$ .

$\overline{x} = \frac{145 + 60 + 33 + 33 + 58}{7}$

Add $145$ and $60$ .

$\overline{x} = \frac{205 + 33 + 33 + 58}{7}$

Add $205$ and $33$ .

$\overline{x} = \frac{238 + 33 + 58}{7}$

Add $238$ and $33$ .

$\overline{x} = \frac{271 + 58}{7}$

Add $271$ and $58$ .

$\overline{x} = \frac{329}{7}$

Step 3

Divide $329$ by $7$ .

$\overline{x} = 47$

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.

$s = \frac{{(22 - 47)}^{2} + {(78 - 47)}^{2} + {(45 - 47)}^{2} + {(60 - 47)}^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $47$ from $22$ .

$s = \frac{{(- 25)}^{2} + {(78 - 47)}^{2} + {(45 - 47)}^{2} + {(60 - 47)}^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Raise $- 25$ to the power of $2$ .

$s = \frac{625 + {(78 - 47)}^{2} + {(45 - 47)}^{2} + {(60 - 47)}^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Subtract $47$ from $78$ .

$s = \frac{625 + 31^{2} + {(45 - 47)}^{2} + {(60 - 47)}^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Raise $31$ to the power of $2$ .

$s = \frac{625 + 961 + {(45 - 47)}^{2} + {(60 - 47)}^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Subtract $47$ from $45$ .

$s = \frac{625 + 961 + {(- 2)}^{2} + {(60 - 47)}^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

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

$s = \frac{625 + 961 + 4 + {(60 - 47)}^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Subtract $47$ from $60$ .

$s = \frac{625 + 961 + 4 + 13^{2} + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Raise $13$ to the power of $2$ .

$s = \frac{625 + 961 + 4 + 169 + {(33 - 47)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Subtract $47$ from $33$ .

$s = \frac{625 + 961 + 4 + 169 + {(- 14)}^{2} + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Raise $- 14$ to the power of $2$ .

$s = \frac{625 + 961 + 4 + 169 + 196 + {(33 - 47)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Subtract $47$ from $33$ .

$s = \frac{625 + 961 + 4 + 169 + 196 + {(- 14)}^{2} + {(58 - 47)}^{2}}{7 - 1}$

Raise $- 14$ to the power of $2$ .

$s = \frac{625 + 961 + 4 + 169 + 196 + 196 + {(58 - 47)}^{2}}{7 - 1}$

Subtract $47$ from $58$ .

$s = \frac{625 + 961 + 4 + 169 + 196 + 196 + 11^{2}}{7 - 1}$

Raise $11$ to the power of $2$ .

$s = \frac{625 + 961 + 4 + 169 + 196 + 196 + 121}{7 - 1}$

Add $625$ and $961$ .

$s = \frac{1586 + 4 + 169 + 196 + 196 + 121}{7 - 1}$

Add $1586$ and $4$ .

$s = \frac{1590 + 169 + 196 + 196 + 121}{7 - 1}$

Add $1590$ and $169$ .

$s = \frac{1759 + 196 + 196 + 121}{7 - 1}$

Add $1759$ and $196$ .

$s = \frac{1955 + 196 + 121}{7 - 1}$

Add $1955$ and $196$ .

$s = \frac{2151 + 121}{7 - 1}$

Add $2151$ and $121$ .

$s = \frac{2272}{7 - 1}$

Reduce the expression by cancelling the common factors.

Subtract $1$ from $7$ .

$s = \frac{2272}{6}$

Cancel the common factor of $2272$ and $6$ .

Factor $2$ out of $2272$ .

$s = \frac{2 (1136)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$s = \frac{2 \cdot 1136}{2 \cdot 3}$

Cancel the common factor.

$s = \frac{2 \cdot 1136}{2 \cdot 3}$

Rewrite the expression.

$s = \frac{1136}{3}$

Step 7

Approximate the result.

$s^{2} \approx 378.6667$

Find the Variance 22 , 78 , 45 , 60 , 33 , 33 , 58

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