Find the Variance 88 , 102 , 106 , 116 , 128 , 134 , 166

$88$ ,

$102$ ,

$106$ ,

$116$ ,

$128$ ,

$134$ ,

$166$

Step 1

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

$\overline{x} = \frac{88 + 102 + 106 + 116 + 128 + 134 + 166}{7}$

Step 2

Simplify the numerator.

Add

$88$ and

$102$ .

$\overline{x} = \frac{190 + 106 + 116 + 128 + 134 + 166}{7}$

Add

$190$ and

$106$ .

$\overline{x} = \frac{296 + 116 + 128 + 134 + 166}{7}$

Add

$296$ and

$116$ .

$\overline{x} = \frac{412 + 128 + 134 + 166}{7}$

Add

$412$ and

$128$ .

$\overline{x} = \frac{540 + 134 + 166}{7}$

Add

$540$ and

$134$ .

$\overline{x} = \frac{674 + 166}{7}$

Add

$674$ and

$166$ .

$\overline{x} = \frac{840}{7}$

Step 3

Divide

$840$ by

$7$ .

$\overline{x} = 120$

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.

$s = \frac{{(88 - 120)}^{2} + {(102 - 120)}^{2} + {(106 - 120)}^{2} + {(116 - 120)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract

$120$ from

$88$ .

$s = \frac{{(- 32)}^{2} + {(102 - 120)}^{2} + {(106 - 120)}^{2} + {(116 - 120)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Raise

$- 32$ to the power of

$2$ .

$s = \frac{1024 + {(102 - 120)}^{2} + {(106 - 120)}^{2} + {(116 - 120)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Subtract

$120$ from

$102$ .

$s = \frac{1024 + {(- 18)}^{2} + {(106 - 120)}^{2} + {(116 - 120)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Raise

$- 18$ to the power of

$2$ .

$s = \frac{1024 + 324 + {(106 - 120)}^{2} + {(116 - 120)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Subtract

$120$ from

$106$ .

$s = \frac{1024 + 324 + {(- 14)}^{2} + {(116 - 120)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Raise

$- 14$ to the power of

$2$ .

$s = \frac{1024 + 324 + 196 + {(116 - 120)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Subtract

$120$ from

$116$ .

$s = \frac{1024 + 324 + 196 + {(- 4)}^{2} + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Raise

$- 4$ to the power of

$2$ .

$s = \frac{1024 + 324 + 196 + 16 + {(128 - 120)}^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Subtract

$120$ from

$128$ .

$s = \frac{1024 + 324 + 196 + 16 + 8^{2} + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Raise

$8$ to the power of

$2$ .

$s = \frac{1024 + 324 + 196 + 16 + 64 + {(134 - 120)}^{2} + {(166 - 120)}^{2}}{7 - 1}$

Subtract

$120$ from

$134$ .

$s = \frac{1024 + 324 + 196 + 16 + 64 + 14^{2} + {(166 - 120)}^{2}}{7 - 1}$

Raise

$14$ to the power of

$2$ .

$s = \frac{1024 + 324 + 196 + 16 + 64 + 196 + {(166 - 120)}^{2}}{7 - 1}$

Subtract

$120$ from

$166$ .

$s = \frac{1024 + 324 + 196 + 16 + 64 + 196 + 46^{2}}{7 - 1}$

Raise

$46$ to the power of

$2$ .

$s = \frac{1024 + 324 + 196 + 16 + 64 + 196 + 2116}{7 - 1}$

Add

$1024$ and

$324$ .

$s = \frac{1348 + 196 + 16 + 64 + 196 + 2116}{7 - 1}$

Add

$1348$ and

$196$ .

$s = \frac{1544 + 16 + 64 + 196 + 2116}{7 - 1}$

Add

$1544$ and

$16$ .

$s = \frac{1560 + 64 + 196 + 2116}{7 - 1}$

Add

$1560$ and

$64$ .

$s = \frac{1624 + 196 + 2116}{7 - 1}$

Add

$1624$ and

$196$ .

$s = \frac{1820 + 2116}{7 - 1}$

Add

$1820$ and

$2116$ .

$s = \frac{3936}{7 - 1}$

Simplify the expression.

Subtract

$1$ from

$7$ .

$s = \frac{3936}{6}$

Divide

$3936$ by

$6$ .

$s = 656$

Step 7

Approximate the result.

$s^{2} \approx 656$

Find the Variance 88 , 102 , 106 , 116 , 128 , 134 , 166

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