Finite Math Examples

$1$ , $1$ , $2$ , $2$ , $2$ , $3$ , $3$ , $4$ , $4$ , $5$

Step 1

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

$\overline{x} = \frac{1 + 1 + 2 + 2 + 2 + 3 + 3 + 4 + 4 + 5}{10}$

Step 2

Simplify the numerator.

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Step 2.1

Add $1$ and $1$ .

$\overline{x} = \frac{2 + 2 + 2 + 2 + 3 + 3 + 4 + 4 + 5}{10}$

Step 2.2

Add $2$ and $2$ .

$\overline{x} = \frac{4 + 2 + 2 + 3 + 3 + 4 + 4 + 5}{10}$

Step 2.3

Add $4$ and $2$ .

$\overline{x} = \frac{6 + 2 + 3 + 3 + 4 + 4 + 5}{10}$

Step 2.4

Add $6$ and $2$ .

$\overline{x} = \frac{8 + 3 + 3 + 4 + 4 + 5}{10}$

Step 2.5

Add $8$ and $3$ .

$\overline{x} = \frac{11 + 3 + 4 + 4 + 5}{10}$

Step 2.6

Add $11$ and $3$ .

$\overline{x} = \frac{14 + 4 + 4 + 5}{10}$

Step 2.7

Add $14$ and $4$ .

$\overline{x} = \frac{18 + 4 + 5}{10}$

Step 2.8

Add $18$ and $4$ .

$\overline{x} = \frac{22 + 5}{10}$

Step 2.9

Add $22$ and $5$ .

$\overline{x} = \frac{27}{10}$

Step 3

Divide.

$\overline{x} = 2.7$

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.

$s = \frac{{(1 - 2.7)}^{2} + {(1 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $2.7$ from $1$ .

$s = \frac{{(- 1.7)}^{2} + {(1 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.2

Raise $- 1.7$ to the power of $2$ .

$s = \frac{2.89 + {(1 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.3

Subtract $2.7$ from $1$ .

$s = \frac{2.89 + {(- 1.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.4

Raise $- 1.7$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.5

Subtract $2.7$ from $2$ .

$s = \frac{2.89 + 2.89 + {(- 0.7)}^{2} + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.6

Raise $- 0.7$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + {(2 - 2.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.7

Subtract $2.7$ from $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + {(- 0.7)}^{2} + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.8

Raise $- 0.7$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + {(2 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.9

Subtract $2.7$ from $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + {(- 0.7)}^{2} + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.10

Raise $- 0.7$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + {(3 - 2.7)}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.11

Subtract $2.7$ from $3$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + {0.3}^{2} + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.12

Raise $0.3$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + {(3 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.13

Subtract $2.7$ from $3$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + {0.3}^{2} + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.14

Raise $0.3$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + {(4 - 2.7)}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.15

Subtract $2.7$ from $4$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + {1.3}^{2} + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.16

Raise $1.3$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + 1.69 + {(4 - 2.7)}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.17

Subtract $2.7$ from $4$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + 1.69 + {1.3}^{2} + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.18

Raise $1.3$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + 1.69 + 1.69 + {(5 - 2.7)}^{2}}{10 - 1}$

Step 6.1.19

Subtract $2.7$ from $5$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + 1.69 + 1.69 + {2.3}^{2}}{10 - 1}$

Step 6.1.20

Raise $2.3$ to the power of $2$ .

$s = \frac{2.89 + 2.89 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + 1.69 + 1.69 + 5.29}{10 - 1}$

Step 6.1.21

Add $2.89$ and $2.89$ .

$s = \frac{5.78 + 0.49 + 0.49 + 0.49 + 0.09 + 0.09 + 1.69 + 1.69 + 5.29}{10 - 1}$

Step 6.1.22

Add $5.78$ and $0.49$ .

$s = \frac{6.27 + 0.49 + 0.49 + 0.09 + 0.09 + 1.69 + 1.69 + 5.29}{10 - 1}$

Step 6.1.23

Add $6.27$ and $0.49$ .

$s = \frac{6.76 + 0.49 + 0.09 + 0.09 + 1.69 + 1.69 + 5.29}{10 - 1}$

Step 6.1.24

Add $6.76$ and $0.49$ .

$s = \frac{7.25 + 0.09 + 0.09 + 1.69 + 1.69 + 5.29}{10 - 1}$

Step 6.1.25

Add $7.25$ and $0.09$ .

$s = \frac{7.34 + 0.09 + 1.69 + 1.69 + 5.29}{10 - 1}$

Step 6.1.26

Add $7.34$ and $0.09$ .

$s = \frac{7.43 + 1.69 + 1.69 + 5.29}{10 - 1}$

Step 6.1.27

Add $7.43$ and $1.69$ .

$s = \frac{9.12 + 1.69 + 5.29}{10 - 1}$

Step 6.1.28

Add $9.12$ and $1.69$ .

$s = \frac{10.81 + 5.29}{10 - 1}$

Step 6.1.29

Add $10.81$ and $5.29$ .

$s = \frac{16.1}{10 - 1}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract $1$ from $10$ .

$s = \frac{16.1}{9}$

Step 6.2.2

Divide $16.1$ by $9$ .

$s = 1.7\overline{8}$

Step 7

Approximate the result.

$s^{2} \approx 1.7889$