Find the Variance 5 , 8 , 8 , 8 , 8 , 10 , 10 , 10 , 10 , 13

$5$ ,

$8$ ,

$10$ ,

$13$

Step 1

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

$\overline{x} = \frac{5 + 8 + 8 + 8 + 8 + 10 + 10 + 10 + 10 + 13}{10}$

Step 2

Simplify the numerator.

Add

$5$ and

$8$ .

$\overline{x} = \frac{13 + 8 + 8 + 8 + 10 + 10 + 10 + 10 + 13}{10}$

Add

$13$ and

$8$ .

$\overline{x} = \frac{21 + 8 + 8 + 10 + 10 + 10 + 10 + 13}{10}$

Add

$21$ and

$8$ .

$\overline{x} = \frac{29 + 8 + 10 + 10 + 10 + 10 + 13}{10}$

Add

$29$ and

$8$ .

$\overline{x} = \frac{37 + 10 + 10 + 10 + 10 + 13}{10}$

Add

$37$ and

$10$ .

$\overline{x} = \frac{47 + 10 + 10 + 10 + 13}{10}$

Add

$47$ and

$10$ .

$\overline{x} = \frac{57 + 10 + 10 + 13}{10}$

Add

$57$ and

$10$ .

$\overline{x} = \frac{67 + 10 + 13}{10}$

Add

$67$ and

$10$ .

$\overline{x} = \frac{77 + 13}{10}$

Add

$77$ and

$13$ .

$\overline{x} = \frac{90}{10}$

Step 3

Divide

$90$ by

$10$ .

$\overline{x} = 9$

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.

$s = \frac{{(5 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract

$9$ from

$5$ .

$s = \frac{{(- 4)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Raise

$- 4$ to the power of

$2$ .

$s = \frac{16 + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Subtract

$9$ from

$8$ .

$s = \frac{16 + {(- 1)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{16 + 1 + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Subtract

$9$ from

$8$ .

$s = \frac{16 + 1 + {(- 1)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{16 + 1 + 1 + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Subtract

$9$ from

$8$ .

$s = \frac{16 + 1 + 1 + {(- 1)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{16 + 1 + 1 + 1 + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Subtract

$9$ from

$8$ .

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

Raise

$- 1$ to the power of

$2$ .

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

Subtract

$9$ from

$10$ .

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

One to any power is one.

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

Subtract

$9$ from

$10$ .

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

One to any power is one.

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

Subtract

$9$ from

$10$ .

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

One to any power is one.

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

Subtract

$9$ from

$10$ .

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

One to any power is one.

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

Subtract

$9$ from

$13$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 4^{2}}{10 - 1}$

Raise

$4$ to the power of

$2$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Add

$16$ and

$1$ .

$s = \frac{17 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Add

$17$ and

$1$ .

$s = \frac{18 + 1 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Add

$18$ and

$1$ .

$s = \frac{19 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Add

$19$ and

$1$ .

$s = \frac{20 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Add

$20$ and

$1$ .

$s = \frac{21 + 1 + 1 + 1 + 16}{10 - 1}$

Add

$21$ and

$1$ .

$s = \frac{22 + 1 + 1 + 16}{10 - 1}$

Add

$22$ and

$1$ .

$s = \frac{23 + 1 + 16}{10 - 1}$

Add

$23$ and

$1$ .

$s = \frac{24 + 16}{10 - 1}$

Add

$24$ and

$16$ .

$s = \frac{40}{10 - 1}$

Subtract

$1$ from

$10$ .

$s = \frac{40}{9}$

Step 7

Approximate the result.

$s^{2} \approx 4.4444$

Find the Variance 5 , 8 , 8 , 8 , 8 , 10 , 10 , 10 , 10 , 13

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