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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