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

$22$ ,

$78$ ,

$45$ ,

$60$ ,

$33$ ,

$58$

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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