Statistics Examples

1

$1$ ,

2

$2$ ,

3

$3$ ,

4

$4$ ,

5

$5$

Step 1

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

¯ x = \frac{1 + 2 + 3 + 4 + 5}{5}

$\overline{x} = \frac{1 + 2 + 3 + 4 + 5}{5}$

Step 2

Simplify the numerator.

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Step 2.1

Add

1

$1$ and

2

$2$ .

¯ x = \frac{3 + 3 + 4 + 5}{5}

$\overline{x} = \frac{3 + 3 + 4 + 5}{5}$

Step 2.2

Add

3

$3$ and

3

$3$ .

¯ x = \frac{6 + 4 + 5}{5}

$\overline{x} = \frac{6 + 4 + 5}{5}$

Step 2.3

Add

6

$6$ and

4

$4$ .

¯ x = \frac{10 + 5}{5}

$\overline{x} = \frac{10 + 5}{5}$

Step 2.4

Add

10

$10$ and

5

$5$ .

¯ x = \frac{15}{5}

$\overline{x} = \frac{15}{5}$

¯ x = \frac{15}{5}

$\overline{x} = \frac{15}{5}$

Step 3

Divide

15

$15$ by

5

$5$ .

¯ x = 3

$\overline{x} = 3$

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} = n \sum i = 1 \frac{{(x_{i} - x_{a v g})}_{i}^{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{{(1 - 3)}^{2} + {(2 - 3)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{{(1 - 3)}^{2} + {(2 - 3)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract

3

$3$ from

1

$1$ .

s = \frac{{(- 2)}^{2} + {(2 - 3)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{{(- 2)}^{2} + {(2 - 3)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.2

Raise

- 2

$- 2$ to the power of

2

$2$ .

s = \frac{4 + {(2 - 3)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{4 + {(2 - 3)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.3

Subtract

3

$3$ from

2

$2$ .

s = \frac{4 + {(- 1)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{4 + {(- 1)}^{2} + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.4

Raise

- 1

$- 1$ to the power of

2

$2$ .

s = \frac{4 + 1 + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{4 + 1 + {(3 - 3)}^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.5

Subtract

3

$3$ from

3

$3$ .

s = \frac{4 + 1 + 0^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{4 + 1 + 0^{2} + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.6

Raising

0

$0$ to any positive power yields

0

$0$ .

s = \frac{4 + 1 + 0 + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{4 + 1 + 0 + {(4 - 3)}^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.7

Subtract

3

$3$ from

4

$4$ .

s = \frac{4 + 1 + 0 + 1^{2} + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{4 + 1 + 0 + 1^{2} + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.8

One to any power is one.

s = \frac{4 + 1 + 0 + 1 + {(5 - 3)}^{2}}{5 - 1}

$s = \frac{4 + 1 + 0 + 1 + {(5 - 3)}^{2}}{5 - 1}$

Step 6.1.9

Subtract

3

$3$ from

5

$5$ .

s = \frac{4 + 1 + 0 + 1 + 2^{2}}{5 - 1}

$s = \frac{4 + 1 + 0 + 1 + 2^{2}}{5 - 1}$

Step 6.1.10

Raise

2

$2$ to the power of

2

$2$ .

s = \frac{4 + 1 + 0 + 1 + 4}{5 - 1}

$s = \frac{4 + 1 + 0 + 1 + 4}{5 - 1}$

Step 6.1.11

Add

4

$4$ and

1

$1$ .

s = \frac{5 + 0 + 1 + 4}{5 - 1}

$s = \frac{5 + 0 + 1 + 4}{5 - 1}$

Step 6.1.12

Add

5

$5$ and

0

$0$ .

s = \frac{5 + 1 + 4}{5 - 1}

$s = \frac{5 + 1 + 4}{5 - 1}$

Step 6.1.13

Add

5

$5$ and

1

$1$ .

s = \frac{6 + 4}{5 - 1}

$s = \frac{6 + 4}{5 - 1}$

Step 6.1.14

Add

6

$6$ and

4

$4$ .

s = \frac{10}{5 - 1}

$s = \frac{10}{5 - 1}$

s = \frac{10}{5 - 1}

$s = \frac{10}{5 - 1}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Subtract

1

$1$ from

5

$5$ .

s = \frac{10}{4}

$s = \frac{10}{4}$

Step 6.2.2

Cancel the common factor of

10

$10$ and

4

$4$ .

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Step 6.2.2.1

Factor

2

$2$ out of

10

$10$ .

s = \frac{2 (5)}{4}

$s = \frac{2 (5)}{4}$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor

2

$2$ out of

4

$4$ .

s = \frac{2 \cdot 5}{2 \cdot 2}

$s = \frac{2 \cdot 5}{2 \cdot 2}$

Step 6.2.2.2.2

Cancel the common factor.

$s = \frac{2 \cdot 5}{2 \cdot 2}$

Step 6.2.2.2.3

Rewrite the expression.

$s = \frac{5}{2}$

Step 7

Approximate the result.

$s^{2} \approx 2.5$

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