Basic Math Examples

$66$ , $77$ , $93$ , $84$ , $69$ , $75$ , $91$ , $86$ , $81$ , $78$

Step 1

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

$\overline{x} = \frac{66 + 77 + 93 + 84 + 69 + 75 + 91 + 86 + 81 + 78}{10}$

Step 2

Simplify the numerator.

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Step 2.1

Add $66$ and $77$ .

$\overline{x} = \frac{143 + 93 + 84 + 69 + 75 + 91 + 86 + 81 + 78}{10}$

Step 2.2

Add $143$ and $93$ .

$\overline{x} = \frac{236 + 84 + 69 + 75 + 91 + 86 + 81 + 78}{10}$

Step 2.3

Add $236$ and $84$ .

$\overline{x} = \frac{320 + 69 + 75 + 91 + 86 + 81 + 78}{10}$

Step 2.4

Add $320$ and $69$ .

$\overline{x} = \frac{389 + 75 + 91 + 86 + 81 + 78}{10}$

Step 2.5

Add $389$ and $75$ .

$\overline{x} = \frac{464 + 91 + 86 + 81 + 78}{10}$

Step 2.6

Add $464$ and $91$ .

$\overline{x} = \frac{555 + 86 + 81 + 78}{10}$

Step 2.7

Add $555$ and $86$ .

$\overline{x} = \frac{641 + 81 + 78}{10}$

Step 2.8

Add $641$ and $81$ .

$\overline{x} = \frac{722 + 78}{10}$

Step 2.9

Add $722$ and $78$ .

$\overline{x} = \frac{800}{10}$

Step 3

Divide $800$ by $10$ .

$\overline{x} = 80$

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.

$s = \frac{{(66 - 80)}^{2} + {(77 - 80)}^{2} + {(93 - 80)}^{2} + {(84 - 80)}^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $80$ from $66$ .

$s = \frac{{(- 14)}^{2} + {(77 - 80)}^{2} + {(93 - 80)}^{2} + {(84 - 80)}^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.2

Raise $- 14$ to the power of $2$ .

$s = \frac{196 + {(77 - 80)}^{2} + {(93 - 80)}^{2} + {(84 - 80)}^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.3

Subtract $80$ from $77$ .

$s = \frac{196 + {(- 3)}^{2} + {(93 - 80)}^{2} + {(84 - 80)}^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.4

Raise $- 3$ to the power of $2$ .

$s = \frac{196 + 9 + {(93 - 80)}^{2} + {(84 - 80)}^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.5

Subtract $80$ from $93$ .

$s = \frac{196 + 9 + 13^{2} + {(84 - 80)}^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.6

Raise $13$ to the power of $2$ .

$s = \frac{196 + 9 + 169 + {(84 - 80)}^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.7

Subtract $80$ from $84$ .

$s = \frac{196 + 9 + 169 + 4^{2} + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.8

Raise $4$ to the power of $2$ .

$s = \frac{196 + 9 + 169 + 16 + {(69 - 80)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.9

Subtract $80$ from $69$ .

$s = \frac{196 + 9 + 169 + 16 + {(- 11)}^{2} + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.10

Raise $- 11$ to the power of $2$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + {(75 - 80)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.11

Subtract $80$ from $75$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + {(- 5)}^{2} + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.12

Raise $- 5$ to the power of $2$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + {(91 - 80)}^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.13

Subtract $80$ from $91$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 11^{2} + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.14

Raise $11$ to the power of $2$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 121 + {(86 - 80)}^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.15

Subtract $80$ from $86$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 121 + 6^{2} + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.16

Raise $6$ to the power of $2$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 121 + 36 + {(81 - 80)}^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.17

Subtract $80$ from $81$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 121 + 36 + 1^{2} + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.18

One to any power is one.

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 121 + 36 + 1 + {(78 - 80)}^{2}}{10 - 1}$

Step 6.1.19

Subtract $80$ from $78$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 121 + 36 + 1 + {(- 2)}^{2}}{10 - 1}$

Step 6.1.20

Raise $- 2$ to the power of $2$ .

$s = \frac{196 + 9 + 169 + 16 + 121 + 25 + 121 + 36 + 1 + 4}{10 - 1}$

Step 6.1.21

Add $196$ and $9$ .

$s = \frac{205 + 169 + 16 + 121 + 25 + 121 + 36 + 1 + 4}{10 - 1}$

Step 6.1.22

Add $205$ and $169$ .

$s = \frac{374 + 16 + 121 + 25 + 121 + 36 + 1 + 4}{10 - 1}$

Step 6.1.23

Add $374$ and $16$ .

$s = \frac{390 + 121 + 25 + 121 + 36 + 1 + 4}{10 - 1}$

Step 6.1.24

Add $390$ and $121$ .

$s = \frac{511 + 25 + 121 + 36 + 1 + 4}{10 - 1}$

Step 6.1.25

Add $511$ and $25$ .

$s = \frac{536 + 121 + 36 + 1 + 4}{10 - 1}$

Step 6.1.26

Add $536$ and $121$ .

$s = \frac{657 + 36 + 1 + 4}{10 - 1}$

Step 6.1.27

Add $657$ and $36$ .

$s = \frac{693 + 1 + 4}{10 - 1}$

Step 6.1.28

Add $693$ and $1$ .

$s = \frac{694 + 4}{10 - 1}$

Step 6.1.29

Add $694$ and $4$ .

$s = \frac{698}{10 - 1}$

Step 6.2

Subtract $1$ from $10$ .

$s = \frac{698}{9}$

Step 7

Approximate the result.

$s^{2} \approx 77.5556$