Statistics Examples

$3$ , $5$ , $12$ , $14$ , $18$

Step 1

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

$\overline{x} = \frac{3 + 5 + 12 + 14 + 18}{5}$

Step 2

Simplify the numerator.

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Step 2.1

Add $3$ and $5$ .

$\overline{x} = \frac{8 + 12 + 14 + 18}{5}$

Step 2.2

Add $8$ and $12$ .

$\overline{x} = \frac{20 + 14 + 18}{5}$

Step 2.3

Add $20$ and $14$ .

$\overline{x} = \frac{34 + 18}{5}$

Step 2.4

Add $34$ and $18$ .

$\overline{x} = \frac{52}{5}$

Step 3

Divide.

$\overline{x} = 10.4$

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.

$s = \frac{{(3 - 10.4)}^{2} + {(5 - 10.4)}^{2} + {(12 - 10.4)}^{2} + {(14 - 10.4)}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract $10.4$ from $3$ .

$s = \frac{{(- 7.4)}^{2} + {(5 - 10.4)}^{2} + {(12 - 10.4)}^{2} + {(14 - 10.4)}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.2

Raise $- 7.4$ to the power of $2$ .

$s = \frac{54.76 + {(5 - 10.4)}^{2} + {(12 - 10.4)}^{2} + {(14 - 10.4)}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.3

Subtract $10.4$ from $5$ .

$s = \frac{54.76 + {(- 5.4)}^{2} + {(12 - 10.4)}^{2} + {(14 - 10.4)}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.4

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

$s = \frac{54.76 + 29.16 + {(12 - 10.4)}^{2} + {(14 - 10.4)}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.5

Subtract $10.4$ from $12$ .

$s = \frac{54.76 + 29.16 + {1.6}^{2} + {(14 - 10.4)}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.6

Raise $1.6$ to the power of $2$ .

$s = \frac{54.76 + 29.16 + 2.56 + {(14 - 10.4)}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.7

Subtract $10.4$ from $14$ .

$s = \frac{54.76 + 29.16 + 2.56 + {3.6}^{2} + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.8

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

$s = \frac{54.76 + 29.16 + 2.56 + 12.96 + {(18 - 10.4)}^{2}}{5 - 1}$

Step 6.1.9

Subtract $10.4$ from $18$ .

$s = \frac{54.76 + 29.16 + 2.56 + 12.96 + {7.6}^{2}}{5 - 1}$

Step 6.1.10

Raise $7.6$ to the power of $2$ .

$s = \frac{54.76 + 29.16 + 2.56 + 12.96 + 57.76}{5 - 1}$

Step 6.1.11

Add $54.76$ and $29.16$ .

$s = \frac{83.92 + 2.56 + 12.96 + 57.76}{5 - 1}$

Step 6.1.12

Add $83.92$ and $2.56$ .

$s = \frac{86.48 + 12.96 + 57.76}{5 - 1}$

Step 6.1.13

Add $86.48$ and $12.96$ .

$s = \frac{99.44 + 57.76}{5 - 1}$

Step 6.1.14

Add $99.44$ and $57.76$ .

$s = \frac{157.2}{5 - 1}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract $1$ from $5$ .

$s = \frac{157.2}{4}$

Step 6.2.2

Divide $157.2$ by $4$ .

$s = 39.3$

Step 7

Approximate the result.

$s^{2} \approx 39.3$

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