Find the Variance 23 , 4 , 19 , 21 , 17 , 26 , 29 , 17 , 6

$23$ ,

$4$ ,

$19$ ,

$21$ ,

$17$ ,

$26$ ,

$29$ ,

$17$ ,

$6$

Step 1

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

$\overline{x} = \frac{23 + 4 + 19 + 21 + 17 + 26 + 29 + 17 + 6}{9}$

Step 2

Simplify the numerator.

Add

$23$ and

$4$ .

$\overline{x} = \frac{27 + 19 + 21 + 17 + 26 + 29 + 17 + 6}{9}$

Add

$27$ and

$19$ .

$\overline{x} = \frac{46 + 21 + 17 + 26 + 29 + 17 + 6}{9}$

Add

$46$ and

$21$ .

$\overline{x} = \frac{67 + 17 + 26 + 29 + 17 + 6}{9}$

Add

$67$ and

$17$ .

$\overline{x} = \frac{84 + 26 + 29 + 17 + 6}{9}$

Add

$84$ and

$26$ .

$\overline{x} = \frac{110 + 29 + 17 + 6}{9}$

Add

$110$ and

$29$ .

$\overline{x} = \frac{139 + 17 + 6}{9}$

Add

$139$ and

$17$ .

$\overline{x} = \frac{156 + 6}{9}$

Add

$156$ and

$6$ .

$\overline{x} = \frac{162}{9}$

Step 3

Divide

$162$ by

$9$ .

$\overline{x} = 18$

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.

$s = \frac{{(23 - 18)}^{2} + {(4 - 18)}^{2} + {(19 - 18)}^{2} + {(21 - 18)}^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract

$18$ from

$23$ .

$s = \frac{5^{2} + {(4 - 18)}^{2} + {(19 - 18)}^{2} + {(21 - 18)}^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Raise

$5$ to the power of

$2$ .

$s = \frac{25 + {(4 - 18)}^{2} + {(19 - 18)}^{2} + {(21 - 18)}^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$4$ .

$s = \frac{25 + {(- 14)}^{2} + {(19 - 18)}^{2} + {(21 - 18)}^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Raise

$- 14$ to the power of

$2$ .

$s = \frac{25 + 196 + {(19 - 18)}^{2} + {(21 - 18)}^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$19$ .

$s = \frac{25 + 196 + 1^{2} + {(21 - 18)}^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

One to any power is one.

$s = \frac{25 + 196 + 1 + {(21 - 18)}^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$21$ .

$s = \frac{25 + 196 + 1 + 3^{2} + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Raise

$3$ to the power of

$2$ .

$s = \frac{25 + 196 + 1 + 9 + {(17 - 18)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$17$ .

$s = \frac{25 + 196 + 1 + 9 + {(- 1)}^{2} + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + {(26 - 18)}^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$26$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 8^{2} + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Raise

$8$ to the power of

$2$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 64 + {(29 - 18)}^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$29$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 64 + 11^{2} + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Raise

$11$ to the power of

$2$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 64 + 121 + {(17 - 18)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$17$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 64 + 121 + {(- 1)}^{2} + {(6 - 18)}^{2}}{9 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 64 + 121 + 1 + {(6 - 18)}^{2}}{9 - 1}$

Subtract

$18$ from

$6$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 64 + 121 + 1 + {(- 12)}^{2}}{9 - 1}$

Raise

$- 12$ to the power of

$2$ .

$s = \frac{25 + 196 + 1 + 9 + 1 + 64 + 121 + 1 + 144}{9 - 1}$

Add

$25$ and

$196$ .

$s = \frac{221 + 1 + 9 + 1 + 64 + 121 + 1 + 144}{9 - 1}$

Add

$221$ and

$1$ .

$s = \frac{222 + 9 + 1 + 64 + 121 + 1 + 144}{9 - 1}$

Add

$222$ and

$9$ .

$s = \frac{231 + 1 + 64 + 121 + 1 + 144}{9 - 1}$

Add

$231$ and

$1$ .

$s = \frac{232 + 64 + 121 + 1 + 144}{9 - 1}$

Add

$232$ and

$64$ .

$s = \frac{296 + 121 + 1 + 144}{9 - 1}$

Add

$296$ and

$121$ .

$s = \frac{417 + 1 + 144}{9 - 1}$

Add

$417$ and

$1$ .

$s = \frac{418 + 144}{9 - 1}$

Add

$418$ and

$144$ .

$s = \frac{562}{9 - 1}$

Reduce the expression by cancelling the common factors.

Subtract

$1$ from

$9$ .

$s = \frac{562}{8}$

Cancel the common factor of

$562$ and

$8$ .

Factor

$2$ out of

$562$ .

$s = \frac{2 (281)}{8}$

Cancel the common factors.

Factor

$2$ out of

$8$ .

$s = \frac{2 \cdot 281}{2 \cdot 4}$

Cancel the common factor.

$s = \frac{2 \cdot 281}{2 \cdot 4}$

Rewrite the expression.

$s = \frac{281}{4}$

Step 7

Approximate the result.

$s^{2} \approx 70.25$

Find the Variance 23 , 4 , 19 , 21 , 17 , 26 , 29 , 17 , 6

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