Statistics Examples

22

$22$ ,

25

$25$ ,

63

$63$ ,

65

$65$

Step 1

Find the mean.

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Step 1.1

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

¯ x = \frac{22 + 25 + 63 + 65}{4}

$\overline{x} = \frac{22 + 25 + 63 + 65}{4}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Add

22

$22$ and

25

$25$ .

¯ x = \frac{47 + 63 + 65}{4}

$\overline{x} = \frac{47 + 63 + 65}{4}$

Step 1.2.2

Add

47

$47$ and

63

$63$ .

¯ x = \frac{110 + 65}{4}

$\overline{x} = \frac{110 + 65}{4}$

Step 1.2.3

Add

110

$110$ and

65

$65$ .

¯ x = \frac{175}{4}

$\overline{x} = \frac{175}{4}$

¯ x = \frac{175}{4}

$\overline{x} = \frac{175}{4}$

Step 1.3

Divide.

¯ x = 43.75

$\overline{x} = 43.75$

Step 1.4

The mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

¯ x = 43.8

$\overline{x} = 43.8$

¯ x = 43.8

$\overline{x} = 43.8$

Step 2

Simplify each value in the list.

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Step 2.1

Convert

22

$22$ to a decimal value.

22

$22$

Step 2.2

Convert

25

$25$ to a decimal value.

25

$25$

Step 2.3

Convert

63

$63$ to a decimal value.

63

$63$

Step 2.4

Convert

65

$65$ to a decimal value.

65

$65$

Step 2.5

The simplified values are

22, 25, 63, 65

$22, 25, 63, 65$ .

22, 25, 63, 65

$22, 25, 63, 65$

22, 25, 63, 65

$22, 25, 63, 65$

Step 3

Set up the formula for sample standard deviation. The standard deviation of a set of values is a measure of the spread of its values.

s = n \sum i = 1 \sqrt{\frac{{(x_{i} - x_{a v g})}_{i}^{2}}{n - 1}}

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

Step 4

Set up the formula for standard deviation for this set of numbers.

$s = \sqrt{\frac{{(22 - 43.8)}^{2} + {(25 - 43.8)}^{2} + {(63 - 43.8)}^{2} + {(65 - 43.8)}^{2}}{4 - 1}}$

Step 5

Simplify the result.

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Step 5.1

Subtract

$43.8$ from

$22$ .

$s = \sqrt{\frac{{(- 21.8)}^{2} + {(25 - 43.8)}^{2} + {(63 - 43.8)}^{2} + {(65 - 43.8)}^{2}}{4 - 1}}$

Step 5.2

Raise

$- 21.8$ to the power of

$2$ .

$s = \sqrt{\frac{475.24 + {(25 - 43.8)}^{2} + {(63 - 43.8)}^{2} + {(65 - 43.8)}^{2}}{4 - 1}}$

Step 5.3

Subtract

$43.8$ from

$25$ .

$s = \sqrt{\frac{475.24 + {(- 18.8)}^{2} + {(63 - 43.8)}^{2} + {(65 - 43.8)}^{2}}{4 - 1}}$

Step 5.4

Raise

$- 18.8$ to the power of

$2$ .

$s = \sqrt{\frac{475.24 + 353.44 + {(63 - 43.8)}^{2} + {(65 - 43.8)}^{2}}{4 - 1}}$

Step 5.5

Subtract

$43.8$ from

$63$ .

$s = \sqrt{\frac{475.24 + 353.44 + {19.2}^{2} + {(65 - 43.8)}^{2}}{4 - 1}}$

Step 5.6

Raise

$19.2$ to the power of

$2$ .

$s = \sqrt{\frac{475.24 + 353.44 + 368.64 + {(65 - 43.8)}^{2}}{4 - 1}}$

Step 5.7

Subtract

$43.8$ from

$65$ .

$s = \sqrt{\frac{475.24 + 353.44 + 368.64 + {21.2}^{2}}{4 - 1}}$

Step 5.8

Raise

$21.2$ to the power of

$2$ .

$s = \sqrt{\frac{475.24 + 353.44 + 368.64 + 449.44}{4 - 1}}$

Step 5.9

Add

$475.24$ and

$353.44$ .

$s = \sqrt{\frac{828.68 + 368.64 + 449.44}{4 - 1}}$

Step 5.10

Add

$828.68$ and

$368.64$ .

$s = \sqrt{\frac{1197.32 + 449.44}{4 - 1}}$

Step 5.11

Add

$1197.32$ and

$449.44$ .

$s = \sqrt{\frac{1646.76}{4 - 1}}$

Step 5.12

Subtract

$1$ from

$4$ .

$s = \sqrt{\frac{1646.76}{3}}$

Step 5.13

Divide

$1646.76$ by

$3$ .

$s = \sqrt{548.92}$

Step 6

The standard deviation should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$23.4$

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