Basic Math Examples

5

$5$ ,

10

$10$ ,

15

$15$ ,

20

$20$

Step 1

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

¯ x = \frac{5 + 10 + 15 + 20}{4}

$\overline{x} = \frac{5 + 10 + 15 + 20}{4}$

Step 2

Simplify the numerator.

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Step 2.1

Add

5

$5$ and

10

$10$ .

¯ x = \frac{15 + 15 + 20}{4}

$\overline{x} = \frac{15 + 15 + 20}{4}$

Step 2.2

Add

15

$15$ and

15

$15$ .

¯ x = \frac{30 + 20}{4}

$\overline{x} = \frac{30 + 20}{4}$

Step 2.3

Add

30

$30$ and

20

$20$ .

¯ x = \frac{50}{4}

$\overline{x} = \frac{50}{4}$

¯ x = \frac{50}{4}

$\overline{x} = \frac{50}{4}$

Step 3

Cancel the common factor of

50

$50$ and

4

$4$ .

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Step 3.1

Factor

2

$2$ out of

50

$50$ .

¯ x = \frac{2 (25)}{4}

$\overline{x} = \frac{2 (25)}{4}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor

2

$2$ out of

4

$4$ .

¯ x = \frac{2 \cdot 25}{2 \cdot 2}

$\overline{x} = \frac{2 \cdot 25}{2 \cdot 2}$

Step 3.2.2

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 25}{2 \cdot 2}$

Step 3.2.3

Rewrite the expression.

$\overline{x} = \frac{25}{2}$

Step 4

Divide.

$\overline{x} = 12.5$

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.

$s = \frac{{(5 - 12.5)}^{2} + {(10 - 12.5)}^{2} + {(15 - 12.5)}^{2} + {(20 - 12.5)}^{2}}{4 - 1}$

Step 7

Simplify the result.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Subtract

$12.5$ from

$5$ .

$s = \frac{{(- 7.5)}^{2} + {(10 - 12.5)}^{2} + {(15 - 12.5)}^{2} + {(20 - 12.5)}^{2}}{4 - 1}$

Step 7.1.2

Raise

$- 7.5$ to the power of

$2$ .

$s = \frac{56.25 + {(10 - 12.5)}^{2} + {(15 - 12.5)}^{2} + {(20 - 12.5)}^{2}}{4 - 1}$

Step 7.1.3

Subtract

$12.5$ from

$10$ .

$s = \frac{56.25 + {(- 2.5)}^{2} + {(15 - 12.5)}^{2} + {(20 - 12.5)}^{2}}{4 - 1}$

Step 7.1.4

Raise

$- 2.5$ to the power of

$2$ .

$s = \frac{56.25 + 6.25 + {(15 - 12.5)}^{2} + {(20 - 12.5)}^{2}}{4 - 1}$

Step 7.1.5

Subtract

$12.5$ from

$15$ .

$s = \frac{56.25 + 6.25 + {2.5}^{2} + {(20 - 12.5)}^{2}}{4 - 1}$

Step 7.1.6

Raise

$2.5$ to the power of

$2$ .

$s = \frac{56.25 + 6.25 + 6.25 + {(20 - 12.5)}^{2}}{4 - 1}$

Step 7.1.7

Subtract

$12.5$ from

$20$ .

$s = \frac{56.25 + 6.25 + 6.25 + {7.5}^{2}}{4 - 1}$

Step 7.1.8

Raise

$7.5$ to the power of

$2$ .

$s = \frac{56.25 + 6.25 + 6.25 + 56.25}{4 - 1}$

Step 7.1.9

Add

$56.25$ and

$6.25$ .

$s = \frac{62.5 + 6.25 + 56.25}{4 - 1}$

Step 7.1.10

Add

$62.5$ and

$6.25$ .

$s = \frac{68.75 + 56.25}{4 - 1}$

Step 7.1.11

Add

$68.75$ and

$56.25$ .

$s = \frac{125}{4 - 1}$

Step 7.2

Reduce the expression by cancelling the common factors.

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Step 7.2.1

Subtract

$1$ from

$4$ .

$s = \frac{125}{3}$

Step 7.2.2

Cancel the common factor of

$125$ and

$3$ .

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Step 7.2.2.1

Rewrite

$125$ as

$1 (125)$ .

$s = \frac{1 (125)}{3}$

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Rewrite

$3$ as

$1 (3)$ .

$s = \frac{1 \cdot 125}{1 \cdot 3}$

Step 7.2.2.2.2

Cancel the common factor.

$s = \frac{1 \cdot 125}{1 \cdot 3}$

Step 7.2.2.2.3

Rewrite the expression.

$s = \frac{125}{3}$

Step 8

Approximate the result.

$s^{2} \approx 41.6667$