Finite Math Examples

7

$7$ ,

$13$ ,

$16$ ,

$25$ ,

$10$ ,

$15$ ,

$14$ ,

$18$

Step 1

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

$\overline{x} = \frac{7 + 13 + 16 + 25 + 10 + 15 + 14 + 18}{8}$

Step 2

Simplify the numerator.

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Step 2.1

Add

$7$ and

$13$ .

$\overline{x} = \frac{20 + 16 + 25 + 10 + 15 + 14 + 18}{8}$

Step 2.2

Add

$20$ and

$16$ .

$\overline{x} = \frac{36 + 25 + 10 + 15 + 14 + 18}{8}$

Step 2.3

Add

$36$ and

$25$ .

$\overline{x} = \frac{61 + 10 + 15 + 14 + 18}{8}$

Step 2.4

Add

$61$ and

$10$ .

$\overline{x} = \frac{71 + 15 + 14 + 18}{8}$

Step 2.5

Add

$71$ and

$15$ .

$\overline{x} = \frac{86 + 14 + 18}{8}$

Step 2.6

Add

$86$ and

$14$ .

$\overline{x} = \frac{100 + 18}{8}$

Step 2.7

Add

$100$ and

$18$ .

$\overline{x} = \frac{118}{8}$

Step 3

Cancel the common factor of

$118$ and

$8$ .

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Step 3.1

Factor

$2$ out of

$118$ .

$\overline{x} = \frac{2 (59)}{8}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor

$2$ out of

$8$ .

$\overline{x} = \frac{2 \cdot 59}{2 \cdot 4}$

Step 3.2.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 59}{2 \cdot 4}$

Step 3.2.3

Rewrite the expression.

$\overline{x} = \frac{59}{4}$

Step 4

Divide.

$\overline{x} = 14.75$

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.

$\overline{x} = 14.8$

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.

$s = \frac{{(7 - 14.8)}^{2} + {(13 - 14.8)}^{2} + {(16 - 14.8)}^{2} + {(25 - 14.8)}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8

Simplify the result.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Subtract

$14.8$ from

$7$ .

$s = \frac{{(- 7.8)}^{2} + {(13 - 14.8)}^{2} + {(16 - 14.8)}^{2} + {(25 - 14.8)}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.2

Raise

$- 7.8$ to the power of

$2$ .

$s = \frac{60.84 + {(13 - 14.8)}^{2} + {(16 - 14.8)}^{2} + {(25 - 14.8)}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.3

Subtract

$14.8$ from

$13$ .

$s = \frac{60.84 + {(- 1.8)}^{2} + {(16 - 14.8)}^{2} + {(25 - 14.8)}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.4

Raise

$- 1.8$ to the power of

$2$ .

$s = \frac{60.84 + 3.24 + {(16 - 14.8)}^{2} + {(25 - 14.8)}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.5

Subtract

$14.8$ from

$16$ .

$s = \frac{60.84 + 3.24 + {1.2}^{2} + {(25 - 14.8)}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.6

Raise

$1.2$ to the power of

$2$ .

$s = \frac{60.84 + 3.24 + 1.44 + {(25 - 14.8)}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.7

Subtract

$14.8$ from

$25$ .

$s = \frac{60.84 + 3.24 + 1.44 + {10.2}^{2} + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.8

Raise

$10.2$ to the power of

$2$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + {(10 - 14.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.9

Subtract

$14.8$ from

$10$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + {(- 4.8)}^{2} + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.10

Raise

$- 4.8$ to the power of

$2$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + 23.04 + {(15 - 14.8)}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.11

Subtract

$14.8$ from

$15$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + 23.04 + {0.2}^{2} + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.12

Raise

$0.2$ to the power of

$2$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + 23.04 + 0.04 + {(14 - 14.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.13

Subtract

$14.8$ from

$14$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + 23.04 + 0.04 + {(- 0.8)}^{2} + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.14

Raise

$- 0.8$ to the power of

$2$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + 23.04 + 0.04 + 0.64 + {(18 - 14.8)}^{2}}{8 - 1}$

Step 8.1.15

Subtract

$14.8$ from

$18$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + 23.04 + 0.04 + 0.64 + {3.2}^{2}}{8 - 1}$

Step 8.1.16

Raise

$3.2$ to the power of

$2$ .

$s = \frac{60.84 + 3.24 + 1.44 + 104.04 + 23.04 + 0.04 + 0.64 + 10.24}{8 - 1}$

Step 8.1.17

Add

$60.84$ and

$3.24$ .

$s = \frac{64.08 + 1.44 + 104.04 + 23.04 + 0.04 + 0.64 + 10.24}{8 - 1}$

Step 8.1.18

Add

$64.08$ and

$1.44$ .

$s = \frac{65.52 + 104.04 + 23.04 + 0.04 + 0.64 + 10.24}{8 - 1}$

Step 8.1.19

Add

$65.52$ and

$104.04$ .

$s = \frac{169.56 + 23.04 + 0.04 + 0.64 + 10.24}{8 - 1}$

Step 8.1.20

Add

$169.56$ and

$23.04$ .

$s = \frac{192.6 + 0.04 + 0.64 + 10.24}{8 - 1}$

Step 8.1.21

Add

$192.6$ and

$0.04$ .

$s = \frac{192.64 + 0.64 + 10.24}{8 - 1}$

Step 8.1.22

Add

$192.64$ and

$0.64$ .

$s = \frac{193.28 + 10.24}{8 - 1}$

Step 8.1.23

Add

$193.28$ and

$10.24$ .

$s = \frac{203.52}{8 - 1}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Subtract

$1$ from

$8$ .

$s = \frac{203.52}{7}$

Step 8.2.2

Divide

$203.52$ by

$7$ .

$s = 29.07428571$

Step 9

Approximate the result.

$s^{2} \approx 29.0743$