Algebra Examples

$27$ , $32$ , $41$ , $9$ , $14$ , $20$ , $31$ , $33$ , $20$ , $13$

Step 1

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

$\overline{x} = \frac{27 + 32 + 41 + 9 + 14 + 20 + 31 + 33 + 20 + 13}{10}$

Step 2

Simplify the numerator.

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Add $27$ and $32$ .

$\overline{x} = \frac{59 + 41 + 9 + 14 + 20 + 31 + 33 + 20 + 13}{10}$

Add $59$ and $41$ .

$\overline{x} = \frac{100 + 9 + 14 + 20 + 31 + 33 + 20 + 13}{10}$

Add $100$ and $9$ .

$\overline{x} = \frac{109 + 14 + 20 + 31 + 33 + 20 + 13}{10}$

Add $109$ and $14$ .

$\overline{x} = \frac{123 + 20 + 31 + 33 + 20 + 13}{10}$

Add $123$ and $20$ .

$\overline{x} = \frac{143 + 31 + 33 + 20 + 13}{10}$

Add $143$ and $31$ .

$\overline{x} = \frac{174 + 33 + 20 + 13}{10}$

Add $174$ and $33$ .

$\overline{x} = \frac{207 + 20 + 13}{10}$

Add $207$ and $20$ .

$\overline{x} = \frac{227 + 13}{10}$

Add $227$ and $13$ .

$\overline{x} = \frac{240}{10}$

Step 3

Divide $240$ by $10$ .

$\overline{x} = 24$

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.

$s = \frac{{(27 - 24)}^{2} + {(32 - 24)}^{2} + {(41 - 24)}^{2} + {(9 - 24)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Step 6

Simplify the result.

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Simplify the numerator.

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Subtract $24$ from $27$ .

$s = \frac{3^{2} + {(32 - 24)}^{2} + {(41 - 24)}^{2} + {(9 - 24)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

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

$s = \frac{9 + {(32 - 24)}^{2} + {(41 - 24)}^{2} + {(9 - 24)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $32$ .

$s = \frac{9 + 8^{2} + {(41 - 24)}^{2} + {(9 - 24)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Raise $8$ to the power of $2$ .

$s = \frac{9 + 64 + {(41 - 24)}^{2} + {(9 - 24)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $41$ .

$s = \frac{9 + 64 + 17^{2} + {(9 - 24)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Raise $17$ to the power of $2$ .

$s = \frac{9 + 64 + 289 + {(9 - 24)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $9$ .

$s = \frac{9 + 64 + 289 + {(- 15)}^{2} + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Raise $- 15$ to the power of $2$ .

$s = \frac{9 + 64 + 289 + 225 + {(14 - 24)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $14$ .

$s = \frac{9 + 64 + 289 + 225 + {(- 10)}^{2} + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Raise $- 10$ to the power of $2$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + {(20 - 24)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $20$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + {(- 4)}^{2} + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

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

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + {(31 - 24)}^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $31$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 7^{2} + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Raise $7$ to the power of $2$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 49 + {(33 - 24)}^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $33$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 49 + 9^{2} + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Raise $9$ to the power of $2$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 49 + 81 + {(20 - 24)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $20$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 49 + 81 + {(- 4)}^{2} + {(13 - 24)}^{2}}{10 - 1}$

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

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 49 + 81 + 16 + {(13 - 24)}^{2}}{10 - 1}$

Subtract $24$ from $13$ .

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 49 + 81 + 16 + {(- 11)}^{2}}{10 - 1}$

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

$s = \frac{9 + 64 + 289 + 225 + 100 + 16 + 49 + 81 + 16 + 121}{10 - 1}$

Add $9$ and $64$ .

$s = \frac{73 + 289 + 225 + 100 + 16 + 49 + 81 + 16 + 121}{10 - 1}$

Add $73$ and $289$ .

$s = \frac{362 + 225 + 100 + 16 + 49 + 81 + 16 + 121}{10 - 1}$

Add $362$ and $225$ .

$s = \frac{587 + 100 + 16 + 49 + 81 + 16 + 121}{10 - 1}$

Add $587$ and $100$ .

$s = \frac{687 + 16 + 49 + 81 + 16 + 121}{10 - 1}$

Add $687$ and $16$ .

$s = \frac{703 + 49 + 81 + 16 + 121}{10 - 1}$

Add $703$ and $49$ .

$s = \frac{752 + 81 + 16 + 121}{10 - 1}$

Add $752$ and $81$ .

$s = \frac{833 + 16 + 121}{10 - 1}$

Add $833$ and $16$ .

$s = \frac{849 + 121}{10 - 1}$

Add $849$ and $121$ .

$s = \frac{970}{10 - 1}$

Subtract $1$ from $10$ .

$s = \frac{970}{9}$

Step 7

Approximate the result.

$s^{2} \approx 107.7778$