Statistics Examples

3

$3$ ,

5

$5$ ,

12

$12$ ,

14

$14$ ,

18

$18$

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}

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

Step 2

Simplify the numerator.

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Step 2.1

Add

3

$3$ and

5

$5$ .

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

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

Step 2.2

Add

8

$8$ and

12

$12$ .

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

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

Step 2.3

Add

20

$20$ and

14

$14$ .

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

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

Step 2.4

Add

34

$34$ and

18

$18$ .

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

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

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

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

Step 3

Divide.

\overline{x} = 10.4

$\overline{x} = 10.4$

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}

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

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}

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

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

$10.4$ from

3

$3$ .

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

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

Step 6.1.2

Raise

- 7.4

$- 7.4$ to the power of

2

$2$ .

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

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

Step 6.1.3

Subtract

10.4

$10.4$ from

5

$5$ .

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

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

Step 6.1.4

Raise

- 5.4

$- 5.4$ to the power of

2

$2$ .

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

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

Step 6.1.5

Subtract

10.4

$10.4$ from

12

$12$ .

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

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

Step 6.1.6

Raise

1.6

$1.6$ to the power of

2

$2$ .

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

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

Step 6.1.7

Subtract

10.4

$10.4$ from

14

$14$ .

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

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

Step 6.1.8

Raise

3.6

$3.6$ to the power of

2

$2$ .

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

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

Step 6.1.9

Subtract

10.4

$10.4$ from

18

$18$ .

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

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

Step 6.1.10

Raise

7.6

$7.6$ to the power of

2

$2$ .

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

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

Step 6.1.11

Add

54.76

$54.76$ and

29.16

$29.16$ .

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

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

Step 6.1.12

Add

83.92

$83.92$ and

2.56

$2.56$ .

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

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

Step 6.1.13

Add

86.48

$86.48$ and

12.96

$12.96$ .

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

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

Step 6.1.14

Add

99.44

$99.44$ and

57.76

$57.76$ .

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

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

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

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

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract

1

$1$ from

5

$5$ .

s = \frac{157.2}{4}

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

Step 6.2.2

Divide

157.2

$157.2$ by

4

$4$ .

s = 39.3

$s = 39.3$

s = 39.3

$s = 39.3$

s = 39.3

$s = 39.3$

Step 7

Approximate the result.

s^{2} \approx 39.3

$s^{2} \approx 39.3$

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