Finite Math Examples

$162$ , $164$ , $157$ , $168$ , $154$ , $173$ , $177$

Step 1

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

$\overline{x} = \frac{162 + 164 + 157 + 168 + 154 + 173 + 177}{7}$

Step 2

Simplify the numerator.

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Step 2.1

Add $162$ and $164$ .

$\overline{x} = \frac{326 + 157 + 168 + 154 + 173 + 177}{7}$

Step 2.2

Add $326$ and $157$ .

$\overline{x} = \frac{483 + 168 + 154 + 173 + 177}{7}$

Step 2.3

Add $483$ and $168$ .

$\overline{x} = \frac{651 + 154 + 173 + 177}{7}$

Step 2.4

Add $651$ and $154$ .

$\overline{x} = \frac{805 + 173 + 177}{7}$

Step 2.5

Add $805$ and $173$ .

$\overline{x} = \frac{978 + 177}{7}$

Step 2.6

Add $978$ and $177$ .

$\overline{x} = \frac{1155}{7}$

Step 3

Divide $1155$ by $7$ .

$\overline{x} = 165$

Step 4

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

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

Step 5

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

$s = \frac{{(162 - 165)}^{2} + {(164 - 165)}^{2} + {(157 - 165)}^{2} + {(168 - 165)}^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $165$ from $162$ .

$s = \frac{{(- 3)}^{2} + {(164 - 165)}^{2} + {(157 - 165)}^{2} + {(168 - 165)}^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.2

Raise $- 3$ to the power of $2$ .

$s = \frac{9 + {(164 - 165)}^{2} + {(157 - 165)}^{2} + {(168 - 165)}^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.3

Subtract $165$ from $164$ .

$s = \frac{9 + {(- 1)}^{2} + {(157 - 165)}^{2} + {(168 - 165)}^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.4

Raise $- 1$ to the power of $2$ .

$s = \frac{9 + 1 + {(157 - 165)}^{2} + {(168 - 165)}^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.5

Subtract $165$ from $157$ .

$s = \frac{9 + 1 + {(- 8)}^{2} + {(168 - 165)}^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.6

Raise $- 8$ to the power of $2$ .

$s = \frac{9 + 1 + 64 + {(168 - 165)}^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.7

Subtract $165$ from $168$ .

$s = \frac{9 + 1 + 64 + 3^{2} + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.8

Raise $3$ to the power of $2$ .

$s = \frac{9 + 1 + 64 + 9 + {(154 - 165)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.9

Subtract $165$ from $154$ .

$s = \frac{9 + 1 + 64 + 9 + {(- 11)}^{2} + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.10

Raise $- 11$ to the power of $2$ .

$s = \frac{9 + 1 + 64 + 9 + 121 + {(173 - 165)}^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.11

Subtract $165$ from $173$ .

$s = \frac{9 + 1 + 64 + 9 + 121 + 8^{2} + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.12

Raise $8$ to the power of $2$ .

$s = \frac{9 + 1 + 64 + 9 + 121 + 64 + {(177 - 165)}^{2}}{7 - 1}$

Step 6.1.13

Subtract $165$ from $177$ .

$s = \frac{9 + 1 + 64 + 9 + 121 + 64 + 12^{2}}{7 - 1}$

Step 6.1.14

Raise $12$ to the power of $2$ .

$s = \frac{9 + 1 + 64 + 9 + 121 + 64 + 144}{7 - 1}$

Step 6.1.15

Add $9$ and $1$ .

$s = \frac{10 + 64 + 9 + 121 + 64 + 144}{7 - 1}$

Step 6.1.16

Add $10$ and $64$ .

$s = \frac{74 + 9 + 121 + 64 + 144}{7 - 1}$

Step 6.1.17

Add $74$ and $9$ .

$s = \frac{83 + 121 + 64 + 144}{7 - 1}$

Step 6.1.18

Add $83$ and $121$ .

$s = \frac{204 + 64 + 144}{7 - 1}$

Step 6.1.19

Add $204$ and $64$ .

$s = \frac{268 + 144}{7 - 1}$

Step 6.1.20

Add $268$ and $144$ .

$s = \frac{412}{7 - 1}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Subtract $1$ from $7$ .

$s = \frac{412}{6}$

Step 6.2.2

Cancel the common factor of $412$ and $6$ .

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Step 6.2.2.1

Factor $2$ out of $412$ .

$s = \frac{2 (206)}{6}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $2$ out of $6$ .

$s = \frac{2 \cdot 206}{2 \cdot 3}$

Step 6.2.2.2.2

Cancel the common factor.

$s = \frac{2 \cdot 206}{2 \cdot 3}$

Step 6.2.2.2.3

Rewrite the expression.

$s = \frac{206}{3}$

Step 7

Approximate the result.

$s^{2} \approx 68.6667$