Finite Math Examples

2

$2$ ,

$4$ ,

$4.5$ ,

$6$ ,

$7.3$ ,

$7.9$ ,

$9$ ,

$10$ ,

$30$

Step 1

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

$\overline{x} = \frac{2 + 4 + 4.5 + 6 + 7.3 + 7.9 + 9 + 10 + 30}{9}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Add

$2$ and

$4$ .

$\overline{x} = \frac{6 + 4.5 + 6 + 7.3 + 7.9 + 9 + 10 + 30}{9}$

Step 2.2

Add

$6$ and

$4.5$ .

$\overline{x} = \frac{10.5 + 6 + 7.3 + 7.9 + 9 + 10 + 30}{9}$

Step 2.3

Add

$10.5$ and

$6$ .

$\overline{x} = \frac{16.5 + 7.3 + 7.9 + 9 + 10 + 30}{9}$

Step 2.4

Add

$16.5$ and

$7.3$ .

$\overline{x} = \frac{23.8 + 7.9 + 9 + 10 + 30}{9}$

Step 2.5

Add

$23.8$ and

$7.9$ .

$\overline{x} = \frac{31.7 + 9 + 10 + 30}{9}$

Step 2.6

Add

$31.7$ and

$9$ .

$\overline{x} = \frac{40.7 + 10 + 30}{9}$

Step 2.7

Add

$40.7$ and

$10$ .

$\overline{x} = \frac{50.7 + 30}{9}$

Step 2.8

Add

$50.7$ and

$30$ .

$\overline{x} = \frac{80.7}{9}$

Step 3

Divide

$80.7$ by

$9$ .

$\overline{x} = 8.9\overline{6}$

Step 4

Divide.

$\overline{x} = 8.9\overline{6}$

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.

$\overline{x} = 9$

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.

$s = \frac{{(2 - 9)}^{2} + {(4 - 9)}^{2} + {(4.5 - 9)}^{2} + {(6 - 9)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8

Simplify the result.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

Subtract

$9$ from

$2$ .

$s = \frac{{(- 7)}^{2} + {(4 - 9)}^{2} + {(4.5 - 9)}^{2} + {(6 - 9)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.2

Raise

$- 7$ to the power of

$2$ .

$s = \frac{49 + {(4 - 9)}^{2} + {(4.5 - 9)}^{2} + {(6 - 9)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.3

Subtract

$9$ from

$4$ .

$s = \frac{49 + {(- 5)}^{2} + {(4.5 - 9)}^{2} + {(6 - 9)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.4

Raise

$- 5$ to the power of

$2$ .

$s = \frac{49 + 25 + {(4.5 - 9)}^{2} + {(6 - 9)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.5

Subtract

$9$ from

$4.5$ .

$s = \frac{49 + 25 + {(- 4.5)}^{2} + {(6 - 9)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.6

Raise

$- 4.5$ to the power of

$2$ .

$s = \frac{49 + 25 + 20.25 + {(6 - 9)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.7

Subtract

$9$ from

$6$ .

$s = \frac{49 + 25 + 20.25 + {(- 3)}^{2} + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.8

Raise

$- 3$ to the power of

$2$ .

$s = \frac{49 + 25 + 20.25 + 9 + {(7.3 - 9)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.9

Subtract

$9$ from

$7.3$ .

$s = \frac{49 + 25 + 20.25 + 9 + {(- 1.7)}^{2} + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.10

Raise

$- 1.7$ to the power of

$2$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + {(7.9 - 9)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.11

Subtract

$9$ from

$7.9$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + {(- 1.1)}^{2} + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.12

Raise

$- 1.1$ to the power of

$2$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + 1.21 + {(9 - 9)}^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.13

Subtract

$9$ from

$9$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + 1.21 + 0^{2} + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.14

Raising

$0$ to any positive power yields

$0$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + 1.21 + 0 + {(10 - 9)}^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.15

Subtract

$9$ from

$10$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + 1.21 + 0 + 1^{2} + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.16

One to any power is one.

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + 1.21 + 0 + 1 + {(30 - 9)}^{2}}{9 - 1}$

Step 8.1.17

Subtract

$9$ from

$30$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + 1.21 + 0 + 1 + 21^{2}}{9 - 1}$

Step 8.1.18

Raise

$21$ to the power of

$2$ .

$s = \frac{49 + 25 + 20.25 + 9 + 2.89 + 1.21 + 0 + 1 + 441}{9 - 1}$

Step 8.1.19

Add

$49$ and

$25$ .

$s = \frac{74 + 20.25 + 9 + 2.89 + 1.21 + 0 + 1 + 441}{9 - 1}$

Step 8.1.20

Add

$74$ and

$20.25$ .

$s = \frac{94.25 + 9 + 2.89 + 1.21 + 0 + 1 + 441}{9 - 1}$

Step 8.1.21

Add

$94.25$ and

$9$ .

$s = \frac{103.25 + 2.89 + 1.21 + 0 + 1 + 441}{9 - 1}$

Step 8.1.22

Add

$103.25$ and

$2.89$ .

$s = \frac{106.14 + 1.21 + 0 + 1 + 441}{9 - 1}$

Step 8.1.23

Add

$106.14$ and

$1.21$ .

$s = \frac{107.35 + 0 + 1 + 441}{9 - 1}$

Step 8.1.24

Add

$107.35$ and

$0$ .

$s = \frac{107.35 + 1 + 441}{9 - 1}$

Step 8.1.25

Add

$107.35$ and

$1$ .

$s = \frac{108.35 + 441}{9 - 1}$

Step 8.1.26

Add

$108.35$ and

$441$ .

$s = \frac{549.35}{9 - 1}$

Step 8.2

Simplify the expression.

Tap for more steps...

Step 8.2.1

Subtract

$1$ from

$9$ .

$s = \frac{549.35}{8}$

Step 8.2.2

Divide

$549.35$ by

$8$ .

$s = 68.66875$

Step 9

Approximate the result.

$s^{2} \approx 68.6688$