Pre-Algebra Examples

$23$ , $19$ , $22$ , $30$ , $28$

Step 1

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

$\overline{x} = \frac{23 + 19 + 22 + 30 + 28}{5}$

Step 2

Simplify the numerator.

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Step 2.1

Add $23$ and $19$ .

$\overline{x} = \frac{42 + 22 + 30 + 28}{5}$

Step 2.2

Add $42$ and $22$ .

$\overline{x} = \frac{64 + 30 + 28}{5}$

Step 2.3

Add $64$ and $30$ .

$\overline{x} = \frac{94 + 28}{5}$

Step 2.4

Add $94$ and $28$ .

$\overline{x} = \frac{122}{5}$

Step 3

Divide.

$\overline{x} = 24.4$

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.

$s = \frac{{(23 - 24.4)}^{2} + {(19 - 24.4)}^{2} + {(22 - 24.4)}^{2} + {(30 - 24.4)}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $24.4$ from $23$ .

$s = \frac{{(- 1.4)}^{2} + {(19 - 24.4)}^{2} + {(22 - 24.4)}^{2} + {(30 - 24.4)}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.2

Raise $- 1.4$ to the power of $2$ .

$s = \frac{1.96 + {(19 - 24.4)}^{2} + {(22 - 24.4)}^{2} + {(30 - 24.4)}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.3

Subtract $24.4$ from $19$ .

$s = \frac{1.96 + {(- 5.4)}^{2} + {(22 - 24.4)}^{2} + {(30 - 24.4)}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.4

Raise $- 5.4$ to the power of $2$ .

$s = \frac{1.96 + 29.16 + {(22 - 24.4)}^{2} + {(30 - 24.4)}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.5

Subtract $24.4$ from $22$ .

$s = \frac{1.96 + 29.16 + {(- 2.4)}^{2} + {(30 - 24.4)}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.6

Raise $- 2.4$ to the power of $2$ .

$s = \frac{1.96 + 29.16 + 5.76 + {(30 - 24.4)}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.7

Subtract $24.4$ from $30$ .

$s = \frac{1.96 + 29.16 + 5.76 + {5.6}^{2} + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.8

Raise $5.6$ to the power of $2$ .

$s = \frac{1.96 + 29.16 + 5.76 + 31.36 + {(28 - 24.4)}^{2}}{5 - 1}$

Step 6.1.9

Subtract $24.4$ from $28$ .

$s = \frac{1.96 + 29.16 + 5.76 + 31.36 + {3.6}^{2}}{5 - 1}$

Step 6.1.10

Raise $3.6$ to the power of $2$ .

$s = \frac{1.96 + 29.16 + 5.76 + 31.36 + 12.96}{5 - 1}$

Step 6.1.11

Add $1.96$ and $29.16$ .

$s = \frac{31.12 + 5.76 + 31.36 + 12.96}{5 - 1}$

Step 6.1.12

Add $31.12$ and $5.76$ .

$s = \frac{36.88 + 31.36 + 12.96}{5 - 1}$

Step 6.1.13

Add $36.88$ and $31.36$ .

$s = \frac{68.24 + 12.96}{5 - 1}$

Step 6.1.14

Add $68.24$ and $12.96$ .

$s = \frac{81.2}{5 - 1}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract $1$ from $5$ .

$s = \frac{81.2}{4}$

Step 6.2.2

Divide $81.2$ by $4$ .

$s = 20.3$

Step 7

Approximate the result.

$s^{2} \approx 20.3$