Find the Variance 14 , 12 , 16 , 20 , 10 , 18

$14$ ,

$12$ ,

$16$ ,

$20$ ,

$10$ ,

$18$

Step 1

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

$\overline{x} = \frac{14 + 12 + 16 + 20 + 10 + 18}{6}$

Step 2

Cancel the common factor of

$14 + 12 + 16 + 20 + 10 + 18$ and

$6$ .

Factor

$2$ out of

$14$ .

$\overline{x} = \frac{2 (7) + 12 + 16 + 20 + 10 + 18}{6}$

Factor

$2$ out of

$12$ .

$\overline{x} = \frac{2 \cdot 7 + 2 \cdot 6 + 16 + 20 + 10 + 18}{6}$

Factor

$2$ out of

$2 \cdot 7 + 2 \cdot 6$ .

$\overline{x} = \frac{2 \cdot (7 + 6) + 16 + 20 + 10 + 18}{6}$

Factor

$2$ out of

$16$ .

$\overline{x} = \frac{2 \cdot (7 + 6) + 2 (8) + 20 + 10 + 18}{6}$

Factor

$2$ out of

$2 \cdot (7 + 6) + 2 (8)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8) + 20 + 10 + 18}{6}$

Factor

$2$ out of

$20$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8) + 2 (10) + 10 + 18}{6}$

Factor

$2$ out of

$2 \cdot (7 + 6 + 8) + 2 (10)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10) + 10 + 18}{6}$

Factor

$2$ out of

$10$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10) + 2 (5) + 18}{6}$

Factor

$2$ out of

$2 \cdot (7 + 6 + 8 + 10) + 2 (5)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5) + 18}{6}$

Factor

$2$ out of

$18$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5) + 2 (9)}{6}$

Factor

$2$ out of

$2 \cdot (7 + 6 + 8 + 10 + 5) + 2 (9)$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5 + 9)}{6}$

Cancel the common factors.

Factor

$2$ out of

$6$ .

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5 + 9)}{2 (3)}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot (7 + 6 + 8 + 10 + 5 + 9)}{2 \cdot 3}$

Rewrite the expression.

$\overline{x} = \frac{7 + 6 + 8 + 10 + 5 + 9}{3}$

Step 3

Simplify the numerator.

Add

$7$ and

$6$ .

$\overline{x} = \frac{13 + 8 + 10 + 5 + 9}{3}$

Add

$13$ and

$8$ .

$\overline{x} = \frac{21 + 10 + 5 + 9}{3}$

Add

$21$ and

$10$ .

$\overline{x} = \frac{31 + 5 + 9}{3}$

Add

$31$ and

$5$ .

$\overline{x} = \frac{36 + 9}{3}$

Add

$36$ and

$9$ .

$\overline{x} = \frac{45}{3}$

Step 4

Divide

$45$ by

$3$ .

$\overline{x} = 15$

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.

$s = \frac{{(14 - 15)}^{2} + {(12 - 15)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract

$15$ from

$14$ .

$s = \frac{{(- 1)}^{2} + {(12 - 15)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{1 + {(12 - 15)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Subtract

$15$ from

$12$ .

$s = \frac{1 + {(- 3)}^{2} + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Raise

$- 3$ to the power of

$2$ .

$s = \frac{1 + 9 + {(16 - 15)}^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Subtract

$15$ from

$16$ .

$s = \frac{1 + 9 + 1^{2} + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

One to any power is one.

$s = \frac{1 + 9 + 1 + {(20 - 15)}^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Subtract

$15$ from

$20$ .

$s = \frac{1 + 9 + 1 + 5^{2} + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Raise

$5$ to the power of

$2$ .

$s = \frac{1 + 9 + 1 + 25 + {(10 - 15)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Subtract

$15$ from

$10$ .

$s = \frac{1 + 9 + 1 + 25 + {(- 5)}^{2} + {(18 - 15)}^{2}}{6 - 1}$

Raise

$- 5$ to the power of

$2$ .

$s = \frac{1 + 9 + 1 + 25 + 25 + {(18 - 15)}^{2}}{6 - 1}$

Subtract

$15$ from

$18$ .

$s = \frac{1 + 9 + 1 + 25 + 25 + 3^{2}}{6 - 1}$

Raise

$3$ to the power of

$2$ .

$s = \frac{1 + 9 + 1 + 25 + 25 + 9}{6 - 1}$

Add

$1$ and

$9$ .

$s = \frac{10 + 1 + 25 + 25 + 9}{6 - 1}$

Add

$10$ and

$1$ .

$s = \frac{11 + 25 + 25 + 9}{6 - 1}$

Add

$11$ and

$25$ .

$s = \frac{36 + 25 + 9}{6 - 1}$

Add

$36$ and

$25$ .

$s = \frac{61 + 9}{6 - 1}$

Add

$61$ and

$9$ .

$s = \frac{70}{6 - 1}$

Simplify the expression.

Subtract

$1$ from

$6$ .

$s = \frac{70}{5}$

Divide

$70$ by

$5$ .

$s = 14$

Step 8

Approximate the result.

$s^{2} \approx 14$

Find the Variance 14 , 12 , 16 , 20 , 10 , 18

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