Finite Math Examples

17

$17$ ,

18

$18$ ,

19

$19$ ,

20

$20$ ,

21

$21$

Step 1

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

¯ x = \frac{17 + 18 + 19 + 20 + 21}{5}

$\overline{x} = \frac{17 + 18 + 19 + 20 + 21}{5}$

Step 2

Simplify the numerator.

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Step 2.1

Add

17

$17$ and

18

$18$ .

¯ x = \frac{35 + 19 + 20 + 21}{5}

$\overline{x} = \frac{35 + 19 + 20 + 21}{5}$

Step 2.2

Add

35

$35$ and

19

$19$ .

¯ x = \frac{54 + 20 + 21}{5}

$\overline{x} = \frac{54 + 20 + 21}{5}$

Step 2.3

Add

54

$54$ and

20

$20$ .

¯ x = \frac{74 + 21}{5}

$\overline{x} = \frac{74 + 21}{5}$

Step 2.4

Add

74

$74$ and

21

$21$ .

¯ x = \frac{95}{5}

$\overline{x} = \frac{95}{5}$

¯ x = \frac{95}{5}

$\overline{x} = \frac{95}{5}$

Step 3

Divide

95

$95$ by

5

$5$ .

¯ x = 19

$\overline{x} = 19$

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{{(17 - 19)}^{2} + {(18 - 19)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}

$s = \frac{{(17 - 19)}^{2} + {(18 - 19)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract

19

$19$ from

17

$17$ .

s = \frac{{(- 2)}^{2} + {(18 - 19)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}

$s = \frac{{(- 2)}^{2} + {(18 - 19)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.2

Raise

- 2

$- 2$ to the power of

2

$2$ .

s = \frac{4 + {(18 - 19)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}

$s = \frac{4 + {(18 - 19)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.3

Subtract

19

$19$ from

18

$18$ .

s = \frac{4 + {(- 1)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}

$s = \frac{4 + {(- 1)}^{2} + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.4

Raise

$- 1$ to the power of

$2$ .

$s = \frac{4 + 1 + {(19 - 19)}^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.5

Subtract

$19$ from

$19$ .

$s = \frac{4 + 1 + 0^{2} + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.6

Raising

$0$ to any positive power yields

$0$ .

$s = \frac{4 + 1 + 0 + {(20 - 19)}^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.7

Subtract

$19$ from

$20$ .

$s = \frac{4 + 1 + 0 + 1^{2} + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.8

One to any power is one.

$s = \frac{4 + 1 + 0 + 1 + {(21 - 19)}^{2}}{5 - 1}$

Step 6.1.9

Subtract

$19$ from

$21$ .

$s = \frac{4 + 1 + 0 + 1 + 2^{2}}{5 - 1}$

Step 6.1.10

Raise

$2$ to the power of

$2$ .

$s = \frac{4 + 1 + 0 + 1 + 4}{5 - 1}$

Step 6.1.11

Add

$4$ and

$1$ .

$s = \frac{5 + 0 + 1 + 4}{5 - 1}$

Step 6.1.12

Add

$5$ and

$0$ .

$s = \frac{5 + 1 + 4}{5 - 1}$

Step 6.1.13

Add

$5$ and

$1$ .

$s = \frac{6 + 4}{5 - 1}$

Step 6.1.14

Add

$6$ and

$4$ .

$s = \frac{10}{5 - 1}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Subtract

$1$ from

$5$ .

$s = \frac{10}{4}$

Step 6.2.2

Cancel the common factor of

$10$ and

$4$ .

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Step 6.2.2.1

Factor

$2$ out of

$10$ .

$s = \frac{2 (5)}{4}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor

$2$ out of

$4$ .

$s = \frac{2 \cdot 5}{2 \cdot 2}$

Step 6.2.2.2.2

Cancel the common factor.

$s = \frac{2 \cdot 5}{2 \cdot 2}$

Step 6.2.2.2.3

Rewrite the expression.

$s = \frac{5}{2}$

Step 7

Approximate the result.

$s^{2} \approx 2.5$