Finite Math Examples

3 (30)

$3 (30)$ ,

5 (45)

$5 (45)$ ,

7 (70)

$7 (70)$ ,

9 (55)

$9 (55)$ ,

11 (10)

$11 (10)$

Step 1

Multiply

3

$3$ by

30

$30$ .

¯ x = 90, 5 (45), 7 (70), 9 (55), 11 (10)

$\overline{x} = 90, 5 (45), 7 (70), 9 (55), 11 (10)$

Step 2

Multiply

5

$5$ by

45

$45$ .

¯ x = 90, 225, 7 (70), 9 (55), 11 (10)

$\overline{x} = 90, 225, 7 (70), 9 (55), 11 (10)$

Step 3

Multiply

7

$7$ by

70

$70$ .

¯ x = 90, 225, 490, 9 (55), 11 (10)

$\overline{x} = 90, 225, 490, 9 (55), 11 (10)$

Step 4

Multiply

9

$9$ by

55

$55$ .

¯ x = 90, 225, 490, 495, 11 (10)

$\overline{x} = 90, 225, 490, 495, 11 (10)$

Step 5

Multiply

11

$11$ by

10

$10$ .

¯ x = 90, 225, 490, 495, 110

$\overline{x} = 90, 225, 490, 495, 110$

Step 6

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

¯ x = \frac{90 + 225 + 490 + 495 + 110}{5}

$\overline{x} = \frac{90 + 225 + 490 + 495 + 110}{5}$

Step 7

Cancel the common factor of

90 + 225 + 490 + 495 + 110

$90 + 225 + 490 + 495 + 110$ and

5

$5$ .

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Step 7.1

Factor

5

$5$ out of

90

$90$ .

¯ x = \frac{5 \cdot 18 + 225 + 490 + 495 + 110}{5}

$\overline{x} = \frac{5 \cdot 18 + 225 + 490 + 495 + 110}{5}$

Step 7.2

Factor

5

$5$ out of

225

$225$ .

¯ x = \frac{5 \cdot 18 + 5 \cdot 45 + 490 + 495 + 110}{5}

$\overline{x} = \frac{5 \cdot 18 + 5 \cdot 45 + 490 + 495 + 110}{5}$

Step 7.3

Factor

5

$5$ out of

5 \cdot 18 + 5 \cdot 45

$5 \cdot 18 + 5 \cdot 45$ .

¯ x = \frac{5 \cdot (18 + 45) + 490 + 495 + 110}{5}

$\overline{x} = \frac{5 \cdot (18 + 45) + 490 + 495 + 110}{5}$

Step 7.4

Factor

5

$5$ out of

490

$490$ .

¯ x = \frac{5 \cdot (18 + 45) + 5 \cdot 98 + 495 + 110}{5}

$\overline{x} = \frac{5 \cdot (18 + 45) + 5 \cdot 98 + 495 + 110}{5}$

Step 7.5

Factor

5

$5$ out of

5 \cdot (18 + 45) + 5 (98)

$5 \cdot (18 + 45) + 5 (98)$ .

¯ x = \frac{5 \cdot (18 + 45 + 98) + 495 + 110}{5}

$\overline{x} = \frac{5 \cdot (18 + 45 + 98) + 495 + 110}{5}$

Step 7.6

Factor

5

$5$ out of

495

$495$ .

¯ x = \frac{5 \cdot (18 + 45 + 98) + 5 \cdot 99 + 110}{5}

$\overline{x} = \frac{5 \cdot (18 + 45 + 98) + 5 \cdot 99 + 110}{5}$

Step 7.7

Factor

5

$5$ out of

5 \cdot (18 + 45 + 98) + 5 (99)

$5 \cdot (18 + 45 + 98) + 5 (99)$ .

¯ x = \frac{5 \cdot (18 + 45 + 98 + 99) + 110}{5}

$\overline{x} = \frac{5 \cdot (18 + 45 + 98 + 99) + 110}{5}$

Step 7.8

Factor

5

$5$ out of

110

$110$ .

¯ x = \frac{5 \cdot (18 + 45 + 98 + 99) + 5 \cdot 22}{5}

$\overline{x} = \frac{5 \cdot (18 + 45 + 98 + 99) + 5 \cdot 22}{5}$

Step 7.9

Factor

5

$5$ out of

5 \cdot (18 + 45 + 98 + 99) + 5 (22)

$5 \cdot (18 + 45 + 98 + 99) + 5 (22)$ .

¯ x = \frac{5 \cdot (18 + 45 + 98 + 99 + 22)}{5}

$\overline{x} = \frac{5 \cdot (18 + 45 + 98 + 99 + 22)}{5}$

Step 7.10

Cancel the common factors.

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Step 7.10.1

Factor

5

$5$ out of

5

$5$ .

¯ x = \frac{5 \cdot (18 + 45 + 98 + 99 + 22)}{5 (1)}

$\overline{x} = \frac{5 \cdot (18 + 45 + 98 + 99 + 22)}{5 (1)}$

Step 7.10.2

Cancel the common factor.

$\overline{x} = \frac{5 \cdot (18 + 45 + 98 + 99 + 22)}{5 \cdot 1}$

Step 7.10.3

Rewrite the expression.

$\overline{x} = \frac{18 + 45 + 98 + 99 + 22}{1}$

Step 7.10.4

Divide

$18 + 45 + 98 + 99 + 22$ by

$1$ .

$\overline{x} = 18 + 45 + 98 + 99 + 22$

Step 8

Simplify by adding numbers.

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Step 8.1

Add

$18$ and

$45$ .

$\overline{x} = 63 + 98 + 99 + 22$

Step 8.2

Add

$63$ and

$98$ .

$\overline{x} = 161 + 99 + 22$

Step 8.3

Add

$161$ and

$99$ .

$\overline{x} = 260 + 22$

Step 8.4

Add

$260$ and

$22$ .

$\overline{x} = 282$

Step 9

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 10

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

$s = \frac{{(90 - 282)}^{2} + {(225 - 282)}^{2} + {(490 - 282)}^{2} + {(495 - 282)}^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11

Simplify the result.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Subtract

$282$ from

$90$ .

$s = \frac{{(- 192)}^{2} + {(225 - 282)}^{2} + {(490 - 282)}^{2} + {(495 - 282)}^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.2

Raise

$- 192$ to the power of

$2$ .

$s = \frac{36864 + {(225 - 282)}^{2} + {(490 - 282)}^{2} + {(495 - 282)}^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.3

Subtract

$282$ from

$225$ .

$s = \frac{36864 + {(- 57)}^{2} + {(490 - 282)}^{2} + {(495 - 282)}^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.4

Raise

$- 57$ to the power of

$2$ .

$s = \frac{36864 + 3249 + {(490 - 282)}^{2} + {(495 - 282)}^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.5

Subtract

$282$ from

$490$ .

$s = \frac{36864 + 3249 + 208^{2} + {(495 - 282)}^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.6

Raise

$208$ to the power of

$2$ .

$s = \frac{36864 + 3249 + 43264 + {(495 - 282)}^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.7

Subtract

$282$ from

$495$ .

$s = \frac{36864 + 3249 + 43264 + 213^{2} + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.8

Raise

$213$ to the power of

$2$ .

$s = \frac{36864 + 3249 + 43264 + 45369 + {(110 - 282)}^{2}}{5 - 1}$

Step 11.1.9

Subtract

$282$ from

$110$ .

$s = \frac{36864 + 3249 + 43264 + 45369 + {(- 172)}^{2}}{5 - 1}$

Step 11.1.10

Raise

$- 172$ to the power of

$2$ .

$s = \frac{36864 + 3249 + 43264 + 45369 + 29584}{5 - 1}$

Step 11.1.11

Add

$36864$ and

$3249$ .

$s = \frac{40113 + 43264 + 45369 + 29584}{5 - 1}$

Step 11.1.12

Add

$40113$ and

$43264$ .

$s = \frac{83377 + 45369 + 29584}{5 - 1}$

Step 11.1.13

Add

$83377$ and

$45369$ .

$s = \frac{128746 + 29584}{5 - 1}$

Step 11.1.14

Add

$128746$ and

$29584$ .

$s = \frac{158330}{5 - 1}$

Step 11.2

Reduce the expression by cancelling the common factors.

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Step 11.2.1

Subtract

$1$ from

$5$ .

$s = \frac{158330}{4}$

Step 11.2.2

Cancel the common factor of

$158330$ and

$4$ .

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Step 11.2.2.1

Factor

$2$ out of

$158330$ .

$s = \frac{2 (79165)}{4}$

Step 11.2.2.2

Cancel the common factors.

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Step 11.2.2.2.1

Factor

$2$ out of

$4$ .

$s = \frac{2 \cdot 79165}{2 \cdot 2}$

Step 11.2.2.2.2

Cancel the common factor.

$s = \frac{2 \cdot 79165}{2 \cdot 2}$

Step 11.2.2.2.3

Rewrite the expression.

$s = \frac{79165}{2}$

Step 12

Approximate the result.

$s^{2} \approx 39582.5$