Finite Math Examples

8

$8$ ,

47

$47$ ,

10

$10$ ,

50

$50$ ,

38

$38$ ,

23

$23$ ,

28

$28$ ,

27

$27$ ,

34

$34$ ,

28

$28$

Step 1

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

¯ x = \frac{8 + 47 + 10 + 50 + 38 + 23 + 28 + 27 + 34 + 28}{10}

$\overline{x} = \frac{8 + 47 + 10 + 50 + 38 + 23 + 28 + 27 + 34 + 28}{10}$

Step 2

Simplify the numerator.

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Step 2.1

Add

8

$8$ and

47

$47$ .

¯ x = \frac{55 + 10 + 50 + 38 + 23 + 28 + 27 + 34 + 28}{10}

$\overline{x} = \frac{55 + 10 + 50 + 38 + 23 + 28 + 27 + 34 + 28}{10}$

Step 2.2

Add

55

$55$ and

10

$10$ .

¯ x = \frac{65 + 50 + 38 + 23 + 28 + 27 + 34 + 28}{10}

$\overline{x} = \frac{65 + 50 + 38 + 23 + 28 + 27 + 34 + 28}{10}$

Step 2.3

Add

65

$65$ and

50

$50$ .

¯ x = \frac{115 + 38 + 23 + 28 + 27 + 34 + 28}{10}

$\overline{x} = \frac{115 + 38 + 23 + 28 + 27 + 34 + 28}{10}$

Step 2.4

Add

115

$115$ and

38

$38$ .

¯ x = \frac{153 + 23 + 28 + 27 + 34 + 28}{10}

$\overline{x} = \frac{153 + 23 + 28 + 27 + 34 + 28}{10}$

Step 2.5

Add

153

$153$ and

23

$23$ .

¯ x = \frac{176 + 28 + 27 + 34 + 28}{10}

$\overline{x} = \frac{176 + 28 + 27 + 34 + 28}{10}$

Step 2.6

Add

176

$176$ and

28

$28$ .

¯ x = \frac{204 + 27 + 34 + 28}{10}

$\overline{x} = \frac{204 + 27 + 34 + 28}{10}$

Step 2.7

Add

204

$204$ and

27

$27$ .

¯ x = \frac{231 + 34 + 28}{10}

$\overline{x} = \frac{231 + 34 + 28}{10}$

Step 2.8

Add

231

$231$ and

34

$34$ .

¯ x = \frac{265 + 28}{10}

$\overline{x} = \frac{265 + 28}{10}$

Step 2.9

Add

265

$265$ and

28

$28$ .

¯ x = \frac{293}{10}

$\overline{x} = \frac{293}{10}$

¯ x = \frac{293}{10}

$\overline{x} = \frac{293}{10}$

Step 3

Divide.

¯ x = 29.3

$\overline{x} = 29.3$

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} = n \sum i = 1 \frac{{(x_{i} - x_{a v g})}_{i}^{2}}{n - 1}

$s^{2} = \sum_{i = 1}^{n} \frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}$

Step 5

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

s = \frac{{(8 - 29.3)}^{2} + {(47 - 29.3)}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{{(8 - 29.3)}^{2} + {(47 - 29.3)}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract

29.3

$29.3$ from

8

$8$ .

s = \frac{{(- 21.3)}^{2} + {(47 - 29.3)}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{{(- 21.3)}^{2} + {(47 - 29.3)}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.2

Raise

- 21.3

$- 21.3$ to the power of

2

$2$ .

s = \frac{453.69 + {(47 - 29.3)}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + {(47 - 29.3)}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.3

Subtract

29.3

$29.3$ from

47

$47$ .

s = \frac{453.69 + {17.7}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + {17.7}^{2} + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.4

Raise

17.7

$17.7$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + {(10 - 29.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.5

Subtract

29.3

$29.3$ from

10

$10$ .

s = \frac{453.69 + 313.29 + {(- 19.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + {(- 19.3)}^{2} + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.6

Raise

- 19.3

$- 19.3$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + {(50 - 29.3)}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.7

Subtract

29.3

$29.3$ from

50

$50$ .

s = \frac{453.69 + 313.29 + 372.49 + {20.7}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + {20.7}^{2} + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.8

Raise

20.7

$20.7$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + {(38 - 29.3)}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.9

Subtract

29.3

$29.3$ from

38

$38$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + {8.7}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + {8.7}^{2} + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.10

Raise

8.7

$8.7$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + {(23 - 29.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.11

Subtract

29.3

$29.3$ from

23

$23$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + {(- 6.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + {(- 6.3)}^{2} + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.12

Raise

- 6.3

$- 6.3$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + {(28 - 29.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.13

Subtract

29.3

$29.3$ from

28

$28$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + {(- 1.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + {(- 1.3)}^{2} + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.14

Raise

- 1.3

$- 1.3$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + {(27 - 29.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.15

Subtract

29.3

$29.3$ from

27

$27$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + {(- 2.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + {(- 2.3)}^{2} + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.16

Raise

- 2.3

$- 2.3$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + {(34 - 29.3)}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.17

Subtract

29.3

$29.3$ from

34

$34$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + {4.7}^{2} + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + {4.7}^{2} + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.18

Raise

4.7

$4.7$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + {(28 - 29.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + {(28 - 29.3)}^{2}}{10 - 1}$

Step 6.1.19

Subtract

29.3

$29.3$ from

28

$28$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + {(- 1.3)}^{2}}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + {(- 1.3)}^{2}}{10 - 1}$

Step 6.1.20

Raise

- 1.3

$- 1.3$ to the power of

2

$2$ .

s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}

$s = \frac{453.69 + 313.29 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}$

Step 6.1.21

Add

453.69

$453.69$ and

313.29

$313.29$ .

s = \frac{766.98 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}

$s = \frac{766.98 + 372.49 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}$

Step 6.1.22

Add

766.98

$766.98$ and

372.49

$372.49$ .

s = \frac{1139.47 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}

$s = \frac{1139.47 + 428.49 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}$

Step 6.1.23

Add

$1139.47$ and

$428.49$ .

$s = \frac{1567.96 + 75.69 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}$

Step 6.1.24

Add

$1567.96$ and

$75.69$ .

$s = \frac{1643.65 + 39.69 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}$

Step 6.1.25

Add

$1643.65$ and

$39.69$ .

$s = \frac{1683.34 + 1.69 + 5.29 + 22.09 + 1.69}{10 - 1}$

Step 6.1.26

Add

$1683.34$ and

$1.69$ .

$s = \frac{1685.03 + 5.29 + 22.09 + 1.69}{10 - 1}$

Step 6.1.27

Add

$1685.03$ and

$5.29$ .

$s = \frac{1690.32 + 22.09 + 1.69}{10 - 1}$

Step 6.1.28

Add

$1690.32$ and

$22.09$ .

$s = \frac{1712.41 + 1.69}{10 - 1}$

Step 6.1.29

Add

$1712.41$ and

$1.69$ .

$s = \frac{1714.1}{10 - 1}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract

$1$ from

$10$ .

$s = \frac{1714.1}{9}$

Step 6.2.2

Divide

$1714.1$ by

$9$ .

$s = 190.4\overline{5}$

Step 7

Approximate the result.

$s^{2} \approx 190.4556$