Basic Math Examples

$5$ , $8$ , $8$ , $8$ , $8$ , $10$ , $10$ , $10$ , $10$ , $13$

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

Add $5$ and $8$ .

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

Step 2.2

Add $13$ and $8$ .

$\overline{x} = \frac{21 + 8 + 8 + 10 + 10 + 10 + 10 + 13}{10}$

Step 2.3

Add $21$ and $8$ .

$\overline{x} = \frac{29 + 8 + 10 + 10 + 10 + 10 + 13}{10}$

Step 2.4

Add $29$ and $8$ .

$\overline{x} = \frac{37 + 10 + 10 + 10 + 10 + 13}{10}$

Step 2.5

Add $37$ and $10$ .

$\overline{x} = \frac{47 + 10 + 10 + 10 + 13}{10}$

Step 2.6

Add $47$ and $10$ .

$\overline{x} = \frac{57 + 10 + 10 + 13}{10}$

Step 2.7

Add $57$ and $10$ .

$\overline{x} = \frac{67 + 10 + 13}{10}$

Step 2.8

Add $67$ and $10$ .

$\overline{x} = \frac{77 + 13}{10}$

Step 2.9

Add $77$ and $13$ .

$\overline{x} = \frac{90}{10}$

Step 3

Divide $90$ by $10$ .

$\overline{x} = 9$

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.

$s = \frac{{(5 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $9$ from $5$ .

$s = \frac{{(- 4)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.2

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

$s = \frac{16 + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.3

Subtract $9$ from $8$ .

$s = \frac{16 + {(- 1)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.4

Raise $- 1$ to the power of $2$ .

$s = \frac{16 + 1 + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.5

Subtract $9$ from $8$ .

$s = \frac{16 + 1 + {(- 1)}^{2} + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.6

Raise $- 1$ to the power of $2$ .

$s = \frac{16 + 1 + 1 + {(8 - 9)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.7

Subtract $9$ from $8$ .

$s = \frac{16 + 1 + 1 + {(- 1)}^{2} + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.8

Raise $- 1$ to the power of $2$ .

$s = \frac{16 + 1 + 1 + 1 + {(8 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.9

Subtract $9$ from $8$ .

$s = \frac{16 + 1 + 1 + 1 + {(- 1)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.10

Raise $- 1$ to the power of $2$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.11

Subtract $9$ from $10$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + 1^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.12

One to any power is one.

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.13

Subtract $9$ from $10$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1^{2} + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.14

One to any power is one.

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + {(10 - 9)}^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.15

Subtract $9$ from $10$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1^{2} + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.16

One to any power is one.

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + {(10 - 9)}^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.17

Subtract $9$ from $10$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1^{2} + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.18

One to any power is one.

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + {(13 - 9)}^{2}}{10 - 1}$

Step 6.1.19

Subtract $9$ from $13$ .

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 4^{2}}{10 - 1}$

Step 6.1.20

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

$s = \frac{16 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Step 6.1.21

Add $16$ and $1$ .

$s = \frac{17 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Step 6.1.22

Add $17$ and $1$ .

$s = \frac{18 + 1 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Step 6.1.23

Add $18$ and $1$ .

$s = \frac{19 + 1 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Step 6.1.24

Add $19$ and $1$ .

$s = \frac{20 + 1 + 1 + 1 + 1 + 16}{10 - 1}$

Step 6.1.25

Add $20$ and $1$ .

$s = \frac{21 + 1 + 1 + 1 + 16}{10 - 1}$

Step 6.1.26

Add $21$ and $1$ .

$s = \frac{22 + 1 + 1 + 16}{10 - 1}$

Step 6.1.27

Add $22$ and $1$ .

$s = \frac{23 + 1 + 16}{10 - 1}$

Step 6.1.28

Add $23$ and $1$ .

$s = \frac{24 + 16}{10 - 1}$

Step 6.1.29

Add $24$ and $16$ .

$s = \frac{40}{10 - 1}$

Step 6.2

Subtract $1$ from $10$ .

$s = \frac{40}{9}$

Step 7

Approximate the result.

$s^{2} \approx 4.4444$