Find the Variance 106 , 109 , 129 , 136 , 144 , 151 , 198

$106$ , $109$ , $129$ , $136$ , $144$ , $151$ , $198$

Step 1

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

$\overline{x} = \frac{106 + 109 + 129 + 136 + 144 + 151 + 198}{7}$

Step 2

Simplify the numerator.

Add $106$ and $109$ .

$\overline{x} = \frac{215 + 129 + 136 + 144 + 151 + 198}{7}$

Add $215$ and $129$ .

$\overline{x} = \frac{344 + 136 + 144 + 151 + 198}{7}$

Add $344$ and $136$ .

$\overline{x} = \frac{480 + 144 + 151 + 198}{7}$

Add $480$ and $144$ .

$\overline{x} = \frac{624 + 151 + 198}{7}$

Add $624$ and $151$ .

$\overline{x} = \frac{775 + 198}{7}$

Add $775$ and $198$ .

$\overline{x} = \frac{973}{7}$

Step 3

Divide $973$ by $7$ .

$\overline{x} = 139$

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.

$s = \frac{{(106 - 139)}^{2} + {(109 - 139)}^{2} + {(129 - 139)}^{2} + {(136 - 139)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $139$ from $106$ .

$s = \frac{{(- 33)}^{2} + {(109 - 139)}^{2} + {(129 - 139)}^{2} + {(136 - 139)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Raise $- 33$ to the power of $2$ .

$s = \frac{1089 + {(109 - 139)}^{2} + {(129 - 139)}^{2} + {(136 - 139)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Subtract $139$ from $109$ .

$s = \frac{1089 + {(- 30)}^{2} + {(129 - 139)}^{2} + {(136 - 139)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Raise $- 30$ to the power of $2$ .

$s = \frac{1089 + 900 + {(129 - 139)}^{2} + {(136 - 139)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Subtract $139$ from $129$ .

$s = \frac{1089 + 900 + {(- 10)}^{2} + {(136 - 139)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Raise $- 10$ to the power of $2$ .

$s = \frac{1089 + 900 + 100 + {(136 - 139)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Subtract $139$ from $136$ .

$s = \frac{1089 + 900 + 100 + {(- 3)}^{2} + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Raise $- 3$ to the power of $2$ .

$s = \frac{1089 + 900 + 100 + 9 + {(144 - 139)}^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Subtract $139$ from $144$ .

$s = \frac{1089 + 900 + 100 + 9 + 5^{2} + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Raise $5$ to the power of $2$ .

$s = \frac{1089 + 900 + 100 + 9 + 25 + {(151 - 139)}^{2} + {(198 - 139)}^{2}}{7 - 1}$

Subtract $139$ from $151$ .

$s = \frac{1089 + 900 + 100 + 9 + 25 + 12^{2} + {(198 - 139)}^{2}}{7 - 1}$

Raise $12$ to the power of $2$ .

$s = \frac{1089 + 900 + 100 + 9 + 25 + 144 + {(198 - 139)}^{2}}{7 - 1}$

Subtract $139$ from $198$ .

$s = \frac{1089 + 900 + 100 + 9 + 25 + 144 + 59^{2}}{7 - 1}$

Raise $59$ to the power of $2$ .

$s = \frac{1089 + 900 + 100 + 9 + 25 + 144 + 3481}{7 - 1}$

Add $1089$ and $900$ .

$s = \frac{1989 + 100 + 9 + 25 + 144 + 3481}{7 - 1}$

Add $1989$ and $100$ .

$s = \frac{2089 + 9 + 25 + 144 + 3481}{7 - 1}$

Add $2089$ and $9$ .

$s = \frac{2098 + 25 + 144 + 3481}{7 - 1}$

Add $2098$ and $25$ .

$s = \frac{2123 + 144 + 3481}{7 - 1}$

Add $2123$ and $144$ .

$s = \frac{2267 + 3481}{7 - 1}$

Add $2267$ and $3481$ .

$s = \frac{5748}{7 - 1}$

Simplify the expression.

Subtract $1$ from $7$ .

$s = \frac{5748}{6}$

Divide $5748$ by $6$ .

$s = 958$

Step 7

Approximate the result.

$s^{2} \approx 958$

Find the Variance 106 , 109 , 129 , 136 , 144 , 151 , 198

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