Find the Variance 9 , 3 , 8 , 6

$9$ , $3$ , $8$ , $6$

Step 1

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

$\overline{x} = \frac{9 + 3 + 8 + 6}{4}$

Step 2

Simplify the numerator.

Add $9$ and $3$ .

$\overline{x} = \frac{12 + 8 + 6}{4}$

Add $12$ and $8$ .

$\overline{x} = \frac{20 + 6}{4}$

Add $20$ and $6$ .

$\overline{x} = \frac{26}{4}$

Step 3

Cancel the common factor of $26$ and $4$ .

Factor $2$ out of $26$ .

$\overline{x} = \frac{2 (13)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\overline{x} = \frac{2 \cdot 13}{2 \cdot 2}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 13}{2 \cdot 2}$

Rewrite the expression.

$\overline{x} = \frac{13}{2}$

Step 4

Divide.

$\overline{x} = 6.5$

Step 5

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 6

Set up the formula for variance for this set of numbers.

$s = \frac{{(9 - 6.5)}^{2} + {(3 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(6 - 6.5)}^{2}}{4 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract $6.5$ from $9$ .

$s = \frac{{2.5}^{2} + {(3 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(6 - 6.5)}^{2}}{4 - 1}$

Raise $2.5$ to the power of $2$ .

$s = \frac{6.25 + {(3 - 6.5)}^{2} + {(8 - 6.5)}^{2} + {(6 - 6.5)}^{2}}{4 - 1}$

Subtract $6.5$ from $3$ .

$s = \frac{6.25 + {(- 3.5)}^{2} + {(8 - 6.5)}^{2} + {(6 - 6.5)}^{2}}{4 - 1}$

Raise $- 3.5$ to the power of $2$ .

$s = \frac{6.25 + 12.25 + {(8 - 6.5)}^{2} + {(6 - 6.5)}^{2}}{4 - 1}$

Subtract $6.5$ from $8$ .

$s = \frac{6.25 + 12.25 + {1.5}^{2} + {(6 - 6.5)}^{2}}{4 - 1}$

Raise $1.5$ to the power of $2$ .

$s = \frac{6.25 + 12.25 + 2.25 + {(6 - 6.5)}^{2}}{4 - 1}$

Subtract $6.5$ from $6$ .

$s = \frac{6.25 + 12.25 + 2.25 + {(- 0.5)}^{2}}{4 - 1}$

Raise $- 0.5$ to the power of $2$ .

$s = \frac{6.25 + 12.25 + 2.25 + 0.25}{4 - 1}$

Add $6.25$ and $12.25$ .

$s = \frac{18.5 + 2.25 + 0.25}{4 - 1}$

Add $18.5$ and $2.25$ .

$s = \frac{20.75 + 0.25}{4 - 1}$

Add $20.75$ and $0.25$ .

$s = \frac{21}{4 - 1}$

Simplify the expression.

Subtract $1$ from $4$ .

$s = \frac{21}{3}$

Divide $21$ by $3$ .

$s = 7$

Step 8

Approximate the result.

$s^{2} \approx 7$

Find the Variance 9 , 3 , 8 , 6

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