Finite Math Examples

$6$ , $13$ , $10$ , $15$ , $20$ , $16$ , $11$ , $19$ , $17$

Step 1

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

$\overline{x} = \frac{6 + 13 + 10 + 15 + 20 + 16 + 11 + 19 + 17}{9}$

Step 2

Simplify the numerator.

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Step 2.1

Add $6$ and $13$ .

$\overline{x} = \frac{19 + 10 + 15 + 20 + 16 + 11 + 19 + 17}{9}$

Step 2.2

Add $19$ and $10$ .

$\overline{x} = \frac{29 + 15 + 20 + 16 + 11 + 19 + 17}{9}$

Step 2.3

Add $29$ and $15$ .

$\overline{x} = \frac{44 + 20 + 16 + 11 + 19 + 17}{9}$

Step 2.4

Add $44$ and $20$ .

$\overline{x} = \frac{64 + 16 + 11 + 19 + 17}{9}$

Step 2.5

Add $64$ and $16$ .

$\overline{x} = \frac{80 + 11 + 19 + 17}{9}$

Step 2.6

Add $80$ and $11$ .

$\overline{x} = \frac{91 + 19 + 17}{9}$

Step 2.7

Add $91$ and $19$ .

$\overline{x} = \frac{110 + 17}{9}$

Step 2.8

Add $110$ and $17$ .

$\overline{x} = \frac{127}{9}$

Step 3

Divide.

$\overline{x} = 14.\overline{1}$

Step 4

The mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$\overline{x} = 14.1$

Step 5

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 6

Set up the formula for variance for this set of numbers.

$s = \frac{{(6 - 14.1)}^{2} + {(13 - 14.1)}^{2} + {(10 - 14.1)}^{2} + {(15 - 14.1)}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $14.1$ from $6$ .

$s = \frac{{(- 8.1)}^{2} + {(13 - 14.1)}^{2} + {(10 - 14.1)}^{2} + {(15 - 14.1)}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.2

Raise $- 8.1$ to the power of $2$ .

$s = \frac{65.61 + {(13 - 14.1)}^{2} + {(10 - 14.1)}^{2} + {(15 - 14.1)}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.3

Subtract $14.1$ from $13$ .

$s = \frac{65.61 + {(- 1.1)}^{2} + {(10 - 14.1)}^{2} + {(15 - 14.1)}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.4

Raise $- 1.1$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + {(10 - 14.1)}^{2} + {(15 - 14.1)}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.5

Subtract $14.1$ from $10$ .

$s = \frac{65.61 + 1.21 + {(- 4.1)}^{2} + {(15 - 14.1)}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.6

Raise $- 4.1$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + 16.81 + {(15 - 14.1)}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.7

Subtract $14.1$ from $15$ .

$s = \frac{65.61 + 1.21 + 16.81 + {0.9}^{2} + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.8

Raise $0.9$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + {(20 - 14.1)}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.9

Subtract $14.1$ from $20$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + {5.9}^{2} + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.10

Raise $5.9$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + {(16 - 14.1)}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.11

Subtract $14.1$ from $16$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + {1.9}^{2} + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.12

Raise $1.9$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + 3.61 + {(11 - 14.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.13

Subtract $14.1$ from $11$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + 3.61 + {(- 3.1)}^{2} + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.14

Raise $- 3.1$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + 3.61 + 9.61 + {(19 - 14.1)}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.15

Subtract $14.1$ from $19$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + 3.61 + 9.61 + {4.9}^{2} + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.16

Raise $4.9$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + 3.61 + 9.61 + 24.01 + {(17 - 14.1)}^{2}}{9 - 1}$

Step 7.1.17

Subtract $14.1$ from $17$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + 3.61 + 9.61 + 24.01 + {2.9}^{2}}{9 - 1}$

Step 7.1.18

Raise $2.9$ to the power of $2$ .

$s = \frac{65.61 + 1.21 + 16.81 + 0.81 + 34.81 + 3.61 + 9.61 + 24.01 + 8.41}{9 - 1}$

Step 7.1.19

Add $65.61$ and $1.21$ .

$s = \frac{66.82 + 16.81 + 0.81 + 34.81 + 3.61 + 9.61 + 24.01 + 8.41}{9 - 1}$

Step 7.1.20

Add $66.82$ and $16.81$ .

$s = \frac{83.63 + 0.81 + 34.81 + 3.61 + 9.61 + 24.01 + 8.41}{9 - 1}$

Step 7.1.21

Add $83.63$ and $0.81$ .

$s = \frac{84.44 + 34.81 + 3.61 + 9.61 + 24.01 + 8.41}{9 - 1}$

Step 7.1.22

Add $84.44$ and $34.81$ .

$s = \frac{119.25 + 3.61 + 9.61 + 24.01 + 8.41}{9 - 1}$

Step 7.1.23

Add $119.25$ and $3.61$ .

$s = \frac{122.86 + 9.61 + 24.01 + 8.41}{9 - 1}$

Step 7.1.24

Add $122.86$ and $9.61$ .

$s = \frac{132.47 + 24.01 + 8.41}{9 - 1}$

Step 7.1.25

Add $132.47$ and $24.01$ .

$s = \frac{156.48 + 8.41}{9 - 1}$

Step 7.1.26

Add $156.48$ and $8.41$ .

$s = \frac{164.89}{9 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $9$ .

$s = \frac{164.89}{8}$

Step 7.2.2

Divide $164.89$ by $8$ .

$s = 20.61125$

Step 8

Approximate the result.

$s^{2} \approx 20.6112$