Finite Math Examples

6

$6$ ,

12

$12$ ,

15

$15$ ,

22

$22$ ,

18

$18$ ,

14

$14$ ,

8

$8$ ,

17

$17$

Step 1

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

\overline{x} = \frac{6 + 12 + 15 + 22 + 18 + 14 + 8 + 17}{8}

$\overline{x} = \frac{6 + 12 + 15 + 22 + 18 + 14 + 8 + 17}{8}$

Step 2

Simplify the numerator.

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Step 2.1

Add

6

$6$ and

12

$12$ .

\overline{x} = \frac{18 + 15 + 22 + 18 + 14 + 8 + 17}{8}

$\overline{x} = \frac{18 + 15 + 22 + 18 + 14 + 8 + 17}{8}$

Step 2.2

Add

18

$18$ and

15

$15$ .

\overline{x} = \frac{33 + 22 + 18 + 14 + 8 + 17}{8}

$\overline{x} = \frac{33 + 22 + 18 + 14 + 8 + 17}{8}$

Step 2.3

Add

33

$33$ and

22

$22$ .

\overline{x} = \frac{55 + 18 + 14 + 8 + 17}{8}

$\overline{x} = \frac{55 + 18 + 14 + 8 + 17}{8}$

Step 2.4

Add

55

$55$ and

18

$18$ .

\overline{x} = \frac{73 + 14 + 8 + 17}{8}

$\overline{x} = \frac{73 + 14 + 8 + 17}{8}$

Step 2.5

Add

73

$73$ and

14

$14$ .

\overline{x} = \frac{87 + 8 + 17}{8}

$\overline{x} = \frac{87 + 8 + 17}{8}$

Step 2.6

Add

87

$87$ and

8

$8$ .

\overline{x} = \frac{95 + 17}{8}

$\overline{x} = \frac{95 + 17}{8}$

Step 2.7

Add

95

$95$ and

17

$17$ .

\overline{x} = \frac{112}{8}

$\overline{x} = \frac{112}{8}$

\overline{x} = \frac{112}{8}

$\overline{x} = \frac{112}{8}$

Step 3

Divide

112

$112$ by

8

$8$ .

\overline{x} = 14

$\overline{x} = 14$

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}

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

Step 5

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

s = \frac{{(6 - 14)}^{2} + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{{(6 - 14)}^{2} + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract

14

$14$ from

6

$6$ .

s = \frac{{(- 8)}^{2} + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{{(- 8)}^{2} + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.2

Raise

- 8

$- 8$ to the power of

2

$2$ .

s = \frac{64 + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + {(12 - 14)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.3

Subtract

14

$14$ from

12

$12$ .

s = \frac{64 + {(- 2)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + {(- 2)}^{2} + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.4

Raise

- 2

$- 2$ to the power of

2

$2$ .

s = \frac{64 + 4 + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + {(15 - 14)}^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.5

Subtract

14

$14$ from

15

$15$ .

s = \frac{64 + 4 + 1^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1^{2} + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.6

One to any power is one.

s = \frac{64 + 4 + 1 + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + {(22 - 14)}^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.7

Subtract

14

$14$ from

22

$22$ .

s = \frac{64 + 4 + 1 + 8^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 8^{2} + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.8

Raise

8

$8$ to the power of

2

$2$ .

s = \frac{64 + 4 + 1 + 64 + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + {(18 - 14)}^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.9

Subtract

14

$14$ from

18

$18$ .

s = \frac{64 + 4 + 1 + 64 + 4^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 4^{2} + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.10

Raise

4

$4$ to the power of

2

$2$ .

s = \frac{64 + 4 + 1 + 64 + 16 + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 16 + {(14 - 14)}^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.11

Subtract

14

$14$ from

14

$14$ .

s = \frac{64 + 4 + 1 + 64 + 16 + 0^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 16 + 0^{2} + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.12

Raising

0

$0$ to any positive power yields

0

$0$ .

s = \frac{64 + 4 + 1 + 64 + 16 + 0 + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + {(8 - 14)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.13

Subtract

14

$14$ from

8

$8$ .

s = \frac{64 + 4 + 1 + 64 + 16 + 0 + {(- 6)}^{2} + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + {(- 6)}^{2} + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.14

Raise

- 6

$- 6$ to the power of

2

$2$ .

s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + {(17 - 14)}^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + {(17 - 14)}^{2}}{8 - 1}$

Step 6.1.15

Subtract

14

$14$ from

17

$17$ .

s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + 3^{2}}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + 3^{2}}{8 - 1}$

Step 6.1.16

Raise

3

$3$ to the power of

2

$2$ .

s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + 9}{8 - 1}

$s = \frac{64 + 4 + 1 + 64 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.17

Add

64

$64$ and

4

$4$ .

s = \frac{68 + 1 + 64 + 16 + 0 + 36 + 9}{8 - 1}

$s = \frac{68 + 1 + 64 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.18

Add

68

$68$ and

1

$1$ .

s = \frac{69 + 64 + 16 + 0 + 36 + 9}{8 - 1}

$s = \frac{69 + 64 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.19

Add

69

$69$ and

64

$64$ .

s = \frac{133 + 16 + 0 + 36 + 9}{8 - 1}

$s = \frac{133 + 16 + 0 + 36 + 9}{8 - 1}$

Step 6.1.20

Add

133

$133$ and

16

$16$ .

s = \frac{149 + 0 + 36 + 9}{8 - 1}

$s = \frac{149 + 0 + 36 + 9}{8 - 1}$

Step 6.1.21

Add

149

$149$ and

0

$0$ .

s = \frac{149 + 36 + 9}{8 - 1}

$s = \frac{149 + 36 + 9}{8 - 1}$

Step 6.1.22

Add

149

$149$ and

36

$36$ .

s = \frac{185 + 9}{8 - 1}

$s = \frac{185 + 9}{8 - 1}$

Step 6.1.23

Add

185

$185$ and

9

$9$ .

s = \frac{194}{8 - 1}

$s = \frac{194}{8 - 1}$

s = \frac{194}{8 - 1}

$s = \frac{194}{8 - 1}$

Step 6.2

Subtract

1

$1$ from

8

$8$ .

s = \frac{194}{7}

$s = \frac{194}{7}$

s = \frac{194}{7}

$s = \frac{194}{7}$

Step 7

Approximate the result.

s^{2} \approx 27.7143

$s^{2} \approx 27.7143$