Find the Variance 22 , 15 , 69 , 65 , 85 , 74 , 45 , 45 , 45 , 22

$22$ ,

$15$ ,

$69$ ,

$65$ ,

$85$ ,

$74$ ,

$45$ ,

$22$

Step 1

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

$\overline{x} = \frac{22 + 15 + 69 + 65 + 85 + 74 + 45 + 45 + 45 + 22}{10}$

Step 2

Simplify the numerator.

Add

$22$ and

$15$ .

$\overline{x} = \frac{37 + 69 + 65 + 85 + 74 + 45 + 45 + 45 + 22}{10}$

Add

$37$ and

$69$ .

$\overline{x} = \frac{106 + 65 + 85 + 74 + 45 + 45 + 45 + 22}{10}$

Add

$106$ and

$65$ .

$\overline{x} = \frac{171 + 85 + 74 + 45 + 45 + 45 + 22}{10}$

Add

$171$ and

$85$ .

$\overline{x} = \frac{256 + 74 + 45 + 45 + 45 + 22}{10}$

Add

$256$ and

$74$ .

$\overline{x} = \frac{330 + 45 + 45 + 45 + 22}{10}$

Add

$330$ and

$45$ .

$\overline{x} = \frac{375 + 45 + 45 + 22}{10}$

Add

$375$ and

$45$ .

$\overline{x} = \frac{420 + 45 + 22}{10}$

Add

$420$ and

$45$ .

$\overline{x} = \frac{465 + 22}{10}$

Add

$465$ and

$22$ .

$\overline{x} = \frac{487}{10}$

Step 3

Divide.

$\overline{x} = 48.7$

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.

$s = \frac{{(22 - 48.7)}^{2} + {(15 - 48.7)}^{2} + {(69 - 48.7)}^{2} + {(65 - 48.7)}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract

$48.7$ from

$22$ .

$s = \frac{{(- 26.7)}^{2} + {(15 - 48.7)}^{2} + {(69 - 48.7)}^{2} + {(65 - 48.7)}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$- 26.7$ to the power of

$2$ .

$s = \frac{712.89 + {(15 - 48.7)}^{2} + {(69 - 48.7)}^{2} + {(65 - 48.7)}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$15$ .

$s = \frac{712.89 + {(- 33.7)}^{2} + {(69 - 48.7)}^{2} + {(65 - 48.7)}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$- 33.7$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + {(69 - 48.7)}^{2} + {(65 - 48.7)}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$69$ .

$s = \frac{712.89 + 1135.69 + {20.3}^{2} + {(65 - 48.7)}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$20.3$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + {(65 - 48.7)}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$65$ .

$s = \frac{712.89 + 1135.69 + 412.09 + {16.3}^{2} + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$16.3$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + {(85 - 48.7)}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$85$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + {36.3}^{2} + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$36.3$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + {(74 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$74$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + {25.3}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$25.3$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$45$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + {(- 3.7)}^{2} + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$- 3.7$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + {(45 - 48.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$45$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + {(- 3.7)}^{2} + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$- 3.7$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + 13.69 + {(45 - 48.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$45$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + 13.69 + {(- 3.7)}^{2} + {(22 - 48.7)}^{2}}{10 - 1}$

Raise

$- 3.7$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + 13.69 + 13.69 + {(22 - 48.7)}^{2}}{10 - 1}$

Subtract

$48.7$ from

$22$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + 13.69 + 13.69 + {(- 26.7)}^{2}}{10 - 1}$

Raise

$- 26.7$ to the power of

$2$ .

$s = \frac{712.89 + 1135.69 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + 13.69 + 13.69 + 712.89}{10 - 1}$

Add

$712.89$ and

$1135.69$ .

$s = \frac{1848.58 + 412.09 + 265.69 + 1317.69 + 640.09 + 13.69 + 13.69 + 13.69 + 712.89}{10 - 1}$

Add

$1848.58$ and

$412.09$ .

$s = \frac{2260.67 + 265.69 + 1317.69 + 640.09 + 13.69 + 13.69 + 13.69 + 712.89}{10 - 1}$

Add

$2260.67$ and

$265.69$ .

$s = \frac{2526.36 + 1317.69 + 640.09 + 13.69 + 13.69 + 13.69 + 712.89}{10 - 1}$

Add

$2526.36$ and

$1317.69$ .

$s = \frac{3844.05 + 640.09 + 13.69 + 13.69 + 13.69 + 712.89}{10 - 1}$

Add

$3844.05$ and

$640.09$ .

$s = \frac{4484.14 + 13.69 + 13.69 + 13.69 + 712.89}{10 - 1}$

Add

$4484.14$ and

$13.69$ .

$s = \frac{4497.83 + 13.69 + 13.69 + 712.89}{10 - 1}$

Add

$4497.83$ and

$13.69$ .

$s = \frac{4511.52 + 13.69 + 712.89}{10 - 1}$

Add

$4511.52$ and

$13.69$ .

$s = \frac{4525.21 + 712.89}{10 - 1}$

Add

$4525.21$ and

$712.89$ .

$s = \frac{5238.1}{10 - 1}$

Simplify the expression.

Subtract

$1$ from

$10$ .

$s = \frac{5238.1}{9}$

Divide

$5238.1$ by

$9$ .

$s = 582.0\overline{1}$

Step 7

Approximate the result.

$s^{2} \approx 582.0111$

Find the Variance 22 , 15 , 69 , 65 , 85 , 74 , 45 , 45 , 45 , 22

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