Find the Variance 6 , 7 , 10 , 11 , 11 , 13 , 16 , 18 , 25

$6$ , $7$ , $10$ , $11$ , $11$ , $13$ , $16$ , $18$ , $25$

Step 1

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

$\overline{x} = \frac{6 + 7 + 10 + 11 + 11 + 13 + 16 + 18 + 25}{9}$

Step 2

Simplify the numerator.

Add $6$ and $7$ .

$\overline{x} = \frac{13 + 10 + 11 + 11 + 13 + 16 + 18 + 25}{9}$

Add $13$ and $10$ .

$\overline{x} = \frac{23 + 11 + 11 + 13 + 16 + 18 + 25}{9}$

Add $23$ and $11$ .

$\overline{x} = \frac{34 + 11 + 13 + 16 + 18 + 25}{9}$

Add $34$ and $11$ .

$\overline{x} = \frac{45 + 13 + 16 + 18 + 25}{9}$

Add $45$ and $13$ .

$\overline{x} = \frac{58 + 16 + 18 + 25}{9}$

Add $58$ and $16$ .

$\overline{x} = \frac{74 + 18 + 25}{9}$

Add $74$ and $18$ .

$\overline{x} = \frac{92 + 25}{9}$

Add $92$ and $25$ .

$\overline{x} = \frac{117}{9}$

Step 3

Divide $117$ by $9$ .

$\overline{x} = 13$

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.

$s = \frac{{(6 - 13)}^{2} + {(7 - 13)}^{2} + {(10 - 13)}^{2} + {(11 - 13)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $13$ from $6$ .

$s = \frac{{(- 7)}^{2} + {(7 - 13)}^{2} + {(10 - 13)}^{2} + {(11 - 13)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Raise $- 7$ to the power of $2$ .

$s = \frac{49 + {(7 - 13)}^{2} + {(10 - 13)}^{2} + {(11 - 13)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $7$ .

$s = \frac{49 + {(- 6)}^{2} + {(10 - 13)}^{2} + {(11 - 13)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Raise $- 6$ to the power of $2$ .

$s = \frac{49 + 36 + {(10 - 13)}^{2} + {(11 - 13)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $10$ .

$s = \frac{49 + 36 + {(- 3)}^{2} + {(11 - 13)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

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

$s = \frac{49 + 36 + 9 + {(11 - 13)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $11$ .

$s = \frac{49 + 36 + 9 + {(- 2)}^{2} + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

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

$s = \frac{49 + 36 + 9 + 4 + {(11 - 13)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $11$ .

$s = \frac{49 + 36 + 9 + 4 + {(- 2)}^{2} + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

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

$s = \frac{49 + 36 + 9 + 4 + 4 + {(13 - 13)}^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $13$ .

$s = \frac{49 + 36 + 9 + 4 + 4 + 0^{2} + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Raising $0$ to any positive power yields $0$ .

$s = \frac{49 + 36 + 9 + 4 + 4 + 0 + {(16 - 13)}^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $16$ .

$s = \frac{49 + 36 + 9 + 4 + 4 + 0 + 3^{2} + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

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

$s = \frac{49 + 36 + 9 + 4 + 4 + 0 + 9 + {(18 - 13)}^{2} + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $18$ .

$s = \frac{49 + 36 + 9 + 4 + 4 + 0 + 9 + 5^{2} + {(25 - 13)}^{2}}{9 - 1}$

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

$s = \frac{49 + 36 + 9 + 4 + 4 + 0 + 9 + 25 + {(25 - 13)}^{2}}{9 - 1}$

Subtract $13$ from $25$ .

$s = \frac{49 + 36 + 9 + 4 + 4 + 0 + 9 + 25 + 12^{2}}{9 - 1}$

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

$s = \frac{49 + 36 + 9 + 4 + 4 + 0 + 9 + 25 + 144}{9 - 1}$

Add $49$ and $36$ .

$s = \frac{85 + 9 + 4 + 4 + 0 + 9 + 25 + 144}{9 - 1}$

Add $85$ and $9$ .

$s = \frac{94 + 4 + 4 + 0 + 9 + 25 + 144}{9 - 1}$

Add $94$ and $4$ .

$s = \frac{98 + 4 + 0 + 9 + 25 + 144}{9 - 1}$

Add $98$ and $4$ .

$s = \frac{102 + 0 + 9 + 25 + 144}{9 - 1}$

Add $102$ and $0$ .

$s = \frac{102 + 9 + 25 + 144}{9 - 1}$

Add $102$ and $9$ .

$s = \frac{111 + 25 + 144}{9 - 1}$

Add $111$ and $25$ .

$s = \frac{136 + 144}{9 - 1}$

Add $136$ and $144$ .

$s = \frac{280}{9 - 1}$

Simplify the expression.

Subtract $1$ from $9$ .

$s = \frac{280}{8}$

Divide $280$ by $8$ .

$s = 35$

Step 7

Approximate the result.

$s^{2} \approx 35$

Find the Variance 6 , 7 , 10 , 11 , 11 , 13 , 16 , 18 , 25

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