Statistics Examples

12

$12$ ,

15

$15$ ,

45

$45$ ,

65

$65$ ,

78

$78$

Step 1

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

¯ x = \frac{12 + 15 + 45 + 65 + 78}{5}

$\overline{x} = \frac{12 + 15 + 45 + 65 + 78}{5}$

Step 2

Simplify the numerator.

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Step 2.1

Add

12

$12$ and

15

$15$ .

¯ x = \frac{27 + 45 + 65 + 78}{5}

$\overline{x} = \frac{27 + 45 + 65 + 78}{5}$

Step 2.2

Add

27

$27$ and

45

$45$ .

¯ x = \frac{72 + 65 + 78}{5}

$\overline{x} = \frac{72 + 65 + 78}{5}$

Step 2.3

Add

72

$72$ and

65

$65$ .

¯ x = \frac{137 + 78}{5}

$\overline{x} = \frac{137 + 78}{5}$

Step 2.4

Add

137

$137$ and

78

$78$ .

¯ x = \frac{215}{5}

$\overline{x} = \frac{215}{5}$

¯ x = \frac{215}{5}

$\overline{x} = \frac{215}{5}$

Step 3

Divide

215

$215$ by

5

$5$ .

¯ x = 43

$\overline{x} = 43$

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{{(12 - 43)}^{2} + {(15 - 43)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{{(12 - 43)}^{2} + {(15 - 43)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6

Simplify the result.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract

43

$43$ from

12

$12$ .

s = \frac{{(- 31)}^{2} + {(15 - 43)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{{(- 31)}^{2} + {(15 - 43)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.2

Raise

- 31

$- 31$ to the power of

2

$2$ .

s = \frac{961 + {(15 - 43)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{961 + {(15 - 43)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.3

Subtract

43

$43$ from

15

$15$ .

s = \frac{961 + {(- 28)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{961 + {(- 28)}^{2} + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.4

Raise

- 28

$- 28$ to the power of

2

$2$ .

s = \frac{961 + 784 + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{961 + 784 + {(45 - 43)}^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.5

Subtract

43

$43$ from

45

$45$ .

s = \frac{961 + 784 + 2^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{961 + 784 + 2^{2} + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.6

Raise

2

$2$ to the power of

2

$2$ .

s = \frac{961 + 784 + 4 + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{961 + 784 + 4 + {(65 - 43)}^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.7

Subtract

43

$43$ from

65

$65$ .

s = \frac{961 + 784 + 4 + 22^{2} + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{961 + 784 + 4 + 22^{2} + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.8

Raise

22

$22$ to the power of

2

$2$ .

s = \frac{961 + 784 + 4 + 484 + {(78 - 43)}^{2}}{5 - 1}

$s = \frac{961 + 784 + 4 + 484 + {(78 - 43)}^{2}}{5 - 1}$

Step 6.1.9

Subtract

43

$43$ from

78

$78$ .

s = \frac{961 + 784 + 4 + 484 + 35^{2}}{5 - 1}

$s = \frac{961 + 784 + 4 + 484 + 35^{2}}{5 - 1}$

Step 6.1.10

Raise

35

$35$ to the power of

2

$2$ .

s = \frac{961 + 784 + 4 + 484 + 1225}{5 - 1}

$s = \frac{961 + 784 + 4 + 484 + 1225}{5 - 1}$

Step 6.1.11

Add

961

$961$ and

784

$784$ .

s = \frac{1745 + 4 + 484 + 1225}{5 - 1}

$s = \frac{1745 + 4 + 484 + 1225}{5 - 1}$

Step 6.1.12

Add

1745

$1745$ and

4

$4$ .

s = \frac{1749 + 484 + 1225}{5 - 1}

$s = \frac{1749 + 484 + 1225}{5 - 1}$

Step 6.1.13

Add

1749

$1749$ and

484

$484$ .

s = \frac{2233 + 1225}{5 - 1}

$s = \frac{2233 + 1225}{5 - 1}$

Step 6.1.14

Add

2233

$2233$ and

1225

$1225$ .

s = \frac{3458}{5 - 1}

$s = \frac{3458}{5 - 1}$

s = \frac{3458}{5 - 1}

$s = \frac{3458}{5 - 1}$

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{3458}{4}

$s = \frac{3458}{4}$

Step 6.2.2

Cancel the common factor of

3458

$3458$ and

4

$4$ .

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Step 6.2.2.1

Factor

2

$2$ out of

3458

$3458$ .

s = \frac{2 (1729)}{4}

$s = \frac{2 (1729)}{4}$

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 1729}{2 \cdot 2}

$s = \frac{2 \cdot 1729}{2 \cdot 2}$

Step 6.2.2.2.2

Cancel the common factor.

$s = \frac{2 \cdot 1729}{2 \cdot 2}$

Step 6.2.2.2.3

Rewrite the expression.

$s = \frac{1729}{2}$

Step 7

Approximate the result.

$s^{2} \approx 864.5$

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