Finite Math Examples

$415$ , $457$ , $409$ , $237$

Step 1

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

$\overline{x} = \frac{415 + 457 + 409 + 237}{4}$

Step 2

Simplify the numerator.

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Step 2.1

Add $415$ and $457$ .

$\overline{x} = \frac{872 + 409 + 237}{4}$

Step 2.2

Add $872$ and $409$ .

$\overline{x} = \frac{1281 + 237}{4}$

Step 2.3

Add $1281$ and $237$ .

$\overline{x} = \frac{1518}{4}$

Step 3

Cancel the common factor of $1518$ and $4$ .

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Step 3.1

Factor $2$ out of $1518$ .

$\overline{x} = \frac{2 (759)}{4}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $4$ .

$\overline{x} = \frac{2 \cdot 759}{2 \cdot 2}$

Step 3.2.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 759}{2 \cdot 2}$

Step 3.2.3

Rewrite the expression.

$\overline{x} = \frac{759}{2}$

Step 4

Divide.

$\overline{x} = 379.5$

Step 5

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 6

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

$s = \frac{{(415 - 379.5)}^{2} + {(457 - 379.5)}^{2} + {(409 - 379.5)}^{2} + {(237 - 379.5)}^{2}}{4 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract $379.5$ from $415$ .

$s = \frac{{35.5}^{2} + {(457 - 379.5)}^{2} + {(409 - 379.5)}^{2} + {(237 - 379.5)}^{2}}{4 - 1}$

Step 7.1.2

Raise $35.5$ to the power of $2$ .

$s = \frac{1260.25 + {(457 - 379.5)}^{2} + {(409 - 379.5)}^{2} + {(237 - 379.5)}^{2}}{4 - 1}$

Step 7.1.3

Subtract $379.5$ from $457$ .

$s = \frac{1260.25 + {77.5}^{2} + {(409 - 379.5)}^{2} + {(237 - 379.5)}^{2}}{4 - 1}$

Step 7.1.4

Raise $77.5$ to the power of $2$ .

$s = \frac{1260.25 + 6006.25 + {(409 - 379.5)}^{2} + {(237 - 379.5)}^{2}}{4 - 1}$

Step 7.1.5

Subtract $379.5$ from $409$ .

$s = \frac{1260.25 + 6006.25 + {29.5}^{2} + {(237 - 379.5)}^{2}}{4 - 1}$

Step 7.1.6

Raise $29.5$ to the power of $2$ .

$s = \frac{1260.25 + 6006.25 + 870.25 + {(237 - 379.5)}^{2}}{4 - 1}$

Step 7.1.7

Subtract $379.5$ from $237$ .

$s = \frac{1260.25 + 6006.25 + 870.25 + {(- 142.5)}^{2}}{4 - 1}$

Step 7.1.8

Raise $- 142.5$ to the power of $2$ .

$s = \frac{1260.25 + 6006.25 + 870.25 + 20306.25}{4 - 1}$

Step 7.1.9

Add $1260.25$ and $6006.25$ .

$s = \frac{7266.5 + 870.25 + 20306.25}{4 - 1}$

Step 7.1.10

Add $7266.5$ and $870.25$ .

$s = \frac{8136.75 + 20306.25}{4 - 1}$

Step 7.1.11

Add $8136.75$ and $20306.25$ .

$s = \frac{28443}{4 - 1}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Subtract $1$ from $4$ .

$s = \frac{28443}{3}$

Step 7.2.2

Divide $28443$ by $3$ .

$s = 9481$

Step 8

Approximate the result.

$s^{2} \approx 9481$