Finite Math Examples

$5$ , $10$ , $19$ , $22$ , $16$ , $13$ , $8$ , $17$

Step 1

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

$\overline{x} = \frac{5 + 10 + 19 + 22 + 16 + 13 + 8 + 17}{8}$

Step 2

Simplify the numerator.

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Step 2.1

Add $5$ and $10$ .

$\overline{x} = \frac{15 + 19 + 22 + 16 + 13 + 8 + 17}{8}$

Step 2.2

Add $15$ and $19$ .

$\overline{x} = \frac{34 + 22 + 16 + 13 + 8 + 17}{8}$

Step 2.3

Add $34$ and $22$ .

$\overline{x} = \frac{56 + 16 + 13 + 8 + 17}{8}$

Step 2.4

Add $56$ and $16$ .

$\overline{x} = \frac{72 + 13 + 8 + 17}{8}$

Step 2.5

Add $72$ and $13$ .

$\overline{x} = \frac{85 + 8 + 17}{8}$

Step 2.6

Add $85$ and $8$ .

$\overline{x} = \frac{93 + 17}{8}$

Step 2.7

Add $93$ and $17$ .

$\overline{x} = \frac{110}{8}$

Step 3

Cancel the common factor of $110$ and $8$ .

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Step 3.1

Factor $2$ out of $110$ .

$\overline{x} = \frac{2 (55)}{8}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $8$ .

$\overline{x} = \frac{2 \cdot 55}{2 \cdot 4}$

Step 3.2.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 55}{2 \cdot 4}$

Step 3.2.3

Rewrite the expression.

$\overline{x} = \frac{55}{4}$

Step 4

Divide.

$\overline{x} = 13.75$

Step 5

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.

$\overline{x} = 13.8$

Step 6

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 7

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

$s = \frac{{(5 - 13.8)}^{2} + {(10 - 13.8)}^{2} + {(19 - 13.8)}^{2} + {(22 - 13.8)}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8

Simplify the result.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Subtract $13.8$ from $5$ .

$s = \frac{{(- 8.8)}^{2} + {(10 - 13.8)}^{2} + {(19 - 13.8)}^{2} + {(22 - 13.8)}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.2

Raise $- 8.8$ to the power of $2$ .

$s = \frac{77.44 + {(10 - 13.8)}^{2} + {(19 - 13.8)}^{2} + {(22 - 13.8)}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.3

Subtract $13.8$ from $10$ .

$s = \frac{77.44 + {(- 3.8)}^{2} + {(19 - 13.8)}^{2} + {(22 - 13.8)}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.4

Raise $- 3.8$ to the power of $2$ .

$s = \frac{77.44 + 14.44 + {(19 - 13.8)}^{2} + {(22 - 13.8)}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.5

Subtract $13.8$ from $19$ .

$s = \frac{77.44 + 14.44 + {5.2}^{2} + {(22 - 13.8)}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.6

Raise $5.2$ to the power of $2$ .

$s = \frac{77.44 + 14.44 + 27.04 + {(22 - 13.8)}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.7

Subtract $13.8$ from $22$ .

$s = \frac{77.44 + 14.44 + 27.04 + {8.2}^{2} + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.8

Raise $8.2$ to the power of $2$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + {(16 - 13.8)}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.9

Subtract $13.8$ from $16$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + {2.2}^{2} + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.10

Raise $2.2$ to the power of $2$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + 4.84 + {(13 - 13.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.11

Subtract $13.8$ from $13$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + 4.84 + {(- 0.8)}^{2} + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.12

Raise $- 0.8$ to the power of $2$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + 4.84 + 0.64 + {(8 - 13.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.13

Subtract $13.8$ from $8$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + 4.84 + 0.64 + {(- 5.8)}^{2} + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.14

Raise $- 5.8$ to the power of $2$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + 4.84 + 0.64 + 33.64 + {(17 - 13.8)}^{2}}{8 - 1}$

Step 8.1.15

Subtract $13.8$ from $17$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + 4.84 + 0.64 + 33.64 + {3.2}^{2}}{8 - 1}$

Step 8.1.16

Raise $3.2$ to the power of $2$ .

$s = \frac{77.44 + 14.44 + 27.04 + 67.24 + 4.84 + 0.64 + 33.64 + 10.24}{8 - 1}$

Step 8.1.17

Add $77.44$ and $14.44$ .

$s = \frac{91.88 + 27.04 + 67.24 + 4.84 + 0.64 + 33.64 + 10.24}{8 - 1}$

Step 8.1.18

Add $91.88$ and $27.04$ .

$s = \frac{118.92 + 67.24 + 4.84 + 0.64 + 33.64 + 10.24}{8 - 1}$

Step 8.1.19

Add $118.92$ and $67.24$ .

$s = \frac{186.16 + 4.84 + 0.64 + 33.64 + 10.24}{8 - 1}$

Step 8.1.20

Add $186.16$ and $4.84$ .

$s = \frac{191 + 0.64 + 33.64 + 10.24}{8 - 1}$

Step 8.1.21

Add $191$ and $0.64$ .

$s = \frac{191.64 + 33.64 + 10.24}{8 - 1}$

Step 8.1.22

Add $191.64$ and $33.64$ .

$s = \frac{225.28 + 10.24}{8 - 1}$

Step 8.1.23

Add $225.28$ and $10.24$ .

$s = \frac{235.52}{8 - 1}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Subtract $1$ from $8$ .

$s = \frac{235.52}{7}$

Step 8.2.2

Divide $235.52$ by $7$ .

$s = 33.64571428$

Step 9

Approximate the result.

$s^{2} \approx 33.6457$