Find the Variance -2 , -2 , 5 , 4 , 2 , -2 , 8 , 3

$- 2$ ,

$5$ ,

$4$ ,

$2$ ,

$- 2$ ,

$8$ ,

$3$

Step 1

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

$\overline{x} = \frac{- 2 - 2 + 5 + 4 + 2 - 2 + 8 + 3}{8}$

Step 2

Simplify the numerator.

Subtract

$2$ from

$- 2$ .

$\overline{x} = \frac{- 4 + 5 + 4 + 2 - 2 + 8 + 3}{8}$

Add

$- 4$ and

$5$ .

$\overline{x} = \frac{1 + 4 + 2 - 2 + 8 + 3}{8}$

Add

$1$ and

$4$ .

$\overline{x} = \frac{5 + 2 - 2 + 8 + 3}{8}$

Add

$5$ and

$2$ .

$\overline{x} = \frac{7 - 2 + 8 + 3}{8}$

Subtract

$2$ from

$7$ .

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

Add

$5$ and

$8$ .

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

Add

$13$ and

$3$ .

$\overline{x} = \frac{16}{8}$

Step 3

Divide

$16$ by

$8$ .

$\overline{x} = 2$

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.

$s = \frac{{(- 2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(5 - 2)}^{2} + {(4 - 2)}^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract

$2$ from

$- 2$ .

$s = \frac{{(- 4)}^{2} + {(- 2 - 2)}^{2} + {(5 - 2)}^{2} + {(4 - 2)}^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Raise

$- 4$ to the power of

$2$ .

$s = \frac{16 + {(- 2 - 2)}^{2} + {(5 - 2)}^{2} + {(4 - 2)}^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Subtract

$2$ from

$- 2$ .

$s = \frac{16 + {(- 4)}^{2} + {(5 - 2)}^{2} + {(4 - 2)}^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Raise

$- 4$ to the power of

$2$ .

$s = \frac{16 + 16 + {(5 - 2)}^{2} + {(4 - 2)}^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Subtract

$2$ from

$5$ .

$s = \frac{16 + 16 + 3^{2} + {(4 - 2)}^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Raise

$3$ to the power of

$2$ .

$s = \frac{16 + 16 + 9 + {(4 - 2)}^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Subtract

$2$ from

$4$ .

$s = \frac{16 + 16 + 9 + 2^{2} + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Raise

$2$ to the power of

$2$ .

$s = \frac{16 + 16 + 9 + 4 + {(2 - 2)}^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Subtract

$2$ from

$2$ .

$s = \frac{16 + 16 + 9 + 4 + 0^{2} + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Raising

$0$ to any positive power yields

$0$ .

$s = \frac{16 + 16 + 9 + 4 + 0 + {(- 2 - 2)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Subtract

$2$ from

$- 2$ .

$s = \frac{16 + 16 + 9 + 4 + 0 + {(- 4)}^{2} + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Raise

$- 4$ to the power of

$2$ .

$s = \frac{16 + 16 + 9 + 4 + 0 + 16 + {(8 - 2)}^{2} + {(3 - 2)}^{2}}{8 - 1}$

Subtract

$2$ from

$8$ .

$s = \frac{16 + 16 + 9 + 4 + 0 + 16 + 6^{2} + {(3 - 2)}^{2}}{8 - 1}$

Raise

$6$ to the power of

$2$ .

$s = \frac{16 + 16 + 9 + 4 + 0 + 16 + 36 + {(3 - 2)}^{2}}{8 - 1}$

Subtract

$2$ from

$3$ .

$s = \frac{16 + 16 + 9 + 4 + 0 + 16 + 36 + 1^{2}}{8 - 1}$

One to any power is one.

$s = \frac{16 + 16 + 9 + 4 + 0 + 16 + 36 + 1}{8 - 1}$

Add

$16$ and

$16$ .

$s = \frac{32 + 9 + 4 + 0 + 16 + 36 + 1}{8 - 1}$

Add

$32$ and

$9$ .

$s = \frac{41 + 4 + 0 + 16 + 36 + 1}{8 - 1}$

Add

$41$ and

$4$ .

$s = \frac{45 + 0 + 16 + 36 + 1}{8 - 1}$

Add

$45$ and

$0$ .

$s = \frac{45 + 16 + 36 + 1}{8 - 1}$

Add

$45$ and

$16$ .

$s = \frac{61 + 36 + 1}{8 - 1}$

Add

$61$ and

$36$ .

$s = \frac{97 + 1}{8 - 1}$

Add

$97$ and

$1$ .

$s = \frac{98}{8 - 1}$

Simplify the expression.

Subtract

$1$ from

$8$ .

$s = \frac{98}{7}$

Divide

$98$ by

$7$ .

$s = 14$

Step 7

Approximate the result.

$s^{2} \approx 14$

Find the Variance -2 , -2 , 5 , 4 , 2 , -2 , 8 , 3

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