Find the Variance 12 , 16 , 32 , 14 , 19 , 27

$12$ ,

$16$ ,

$32$ ,

$14$ ,

$19$ ,

$27$

Step 1

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

$\overline{x} = \frac{12 + 16 + 32 + 14 + 19 + 27}{6}$

Step 2

Simplify the numerator.

Add

$12$ and

$16$ .

$\overline{x} = \frac{28 + 32 + 14 + 19 + 27}{6}$

Add

$28$ and

$32$ .

$\overline{x} = \frac{60 + 14 + 19 + 27}{6}$

Add

$60$ and

$14$ .

$\overline{x} = \frac{74 + 19 + 27}{6}$

Add

$74$ and

$19$ .

$\overline{x} = \frac{93 + 27}{6}$

Add

$93$ and

$27$ .

$\overline{x} = \frac{120}{6}$

Step 3

Divide

$120$ by

$6$ .

$\overline{x} = 20$

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.

$s = \frac{{(12 - 20)}^{2} + {(16 - 20)}^{2} + {(32 - 20)}^{2} + {(14 - 20)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract

$20$ from

$12$ .

$s = \frac{{(- 8)}^{2} + {(16 - 20)}^{2} + {(32 - 20)}^{2} + {(14 - 20)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Raise

$- 8$ to the power of

$2$ .

$s = \frac{64 + {(16 - 20)}^{2} + {(32 - 20)}^{2} + {(14 - 20)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Subtract

$20$ from

$16$ .

$s = \frac{64 + {(- 4)}^{2} + {(32 - 20)}^{2} + {(14 - 20)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Raise

$- 4$ to the power of

$2$ .

$s = \frac{64 + 16 + {(32 - 20)}^{2} + {(14 - 20)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Subtract

$20$ from

$32$ .

$s = \frac{64 + 16 + 12^{2} + {(14 - 20)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Raise

$12$ to the power of

$2$ .

$s = \frac{64 + 16 + 144 + {(14 - 20)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Subtract

$20$ from

$14$ .

$s = \frac{64 + 16 + 144 + {(- 6)}^{2} + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Raise

$- 6$ to the power of

$2$ .

$s = \frac{64 + 16 + 144 + 36 + {(19 - 20)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Subtract

$20$ from

$19$ .

$s = \frac{64 + 16 + 144 + 36 + {(- 1)}^{2} + {(27 - 20)}^{2}}{6 - 1}$

Raise

$- 1$ to the power of

$2$ .

$s = \frac{64 + 16 + 144 + 36 + 1 + {(27 - 20)}^{2}}{6 - 1}$

Subtract

$20$ from

$27$ .

$s = \frac{64 + 16 + 144 + 36 + 1 + 7^{2}}{6 - 1}$

Raise

$7$ to the power of

$2$ .

$s = \frac{64 + 16 + 144 + 36 + 1 + 49}{6 - 1}$

Add

$64$ and

$16$ .

$s = \frac{80 + 144 + 36 + 1 + 49}{6 - 1}$

Add

$80$ and

$144$ .

$s = \frac{224 + 36 + 1 + 49}{6 - 1}$

Add

$224$ and

$36$ .

$s = \frac{260 + 1 + 49}{6 - 1}$

Add

$260$ and

$1$ .

$s = \frac{261 + 49}{6 - 1}$

Add

$261$ and

$49$ .

$s = \frac{310}{6 - 1}$

Simplify the expression.

Subtract

$1$ from

$6$ .

$s = \frac{310}{5}$

Divide

$310$ by

$5$ .

$s = 62$

Step 7

Approximate the result.

$s^{2} \approx 62$

Find the Variance 12 , 16 , 32 , 14 , 19 , 27

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