Statistics Examples

23

$23$ ,

28

$28$ ,

45

$45$ ,

56

$56$ ,

78

$78$

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{23 + 28 + 45 + 56 + 78}{5}

$\overline{x} = \frac{23 + 28 + 45 + 56 + 78}{5}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Add

23

$23$ and

28

$28$ .

¯ x = \frac{51 + 45 + 56 + 78}{5}

$\overline{x} = \frac{51 + 45 + 56 + 78}{5}$

Step 1.2.2

Add

51

$51$ and

45

$45$ .

¯ x = \frac{96 + 56 + 78}{5}

$\overline{x} = \frac{96 + 56 + 78}{5}$

Step 1.2.3

Add

96

$96$ and

56

$56$ .

¯ x = \frac{152 + 78}{5}

$\overline{x} = \frac{152 + 78}{5}$

Step 1.2.4

Add

152

$152$ and

78

$78$ .

¯ x = \frac{230}{5}

$\overline{x} = \frac{230}{5}$

¯ x = \frac{230}{5}

$\overline{x} = \frac{230}{5}$

Step 1.3

Divide

230

$230$ by

5

$5$ .

¯ x = 46

$\overline{x} = 46$

¯ x = 46

$\overline{x} = 46$

Step 2

Simplify each value in the list.

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Step 2.1

Convert

23

$23$ to a decimal value.

23

$23$

Step 2.2

Convert

28

$28$ to a decimal value.

28

$28$

Step 2.3

Convert

45

$45$ to a decimal value.

45

$45$

Step 2.4

Convert

56

$56$ to a decimal value.

56

$56$

Step 2.5

Convert

78

$78$ to a decimal value.

78

$78$

Step 2.6

The simplified values are

23, 28, 45, 56, 78

$23, 28, 45, 56, 78$ .

23, 28, 45, 56, 78

$23, 28, 45, 56, 78$

23, 28, 45, 56, 78

$23, 28, 45, 56, 78$

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{{(23 - 46)}^{2} + {(28 - 46)}^{2} + {(45 - 46)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}

$s = \sqrt{\frac{{(23 - 46)}^{2} + {(28 - 46)}^{2} + {(45 - 46)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5

Simplify the result.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Subtract

46

$46$ from

23

$23$ .

s = \sqrt{\frac{{(- 23)}^{2} + {(28 - 46)}^{2} + {(45 - 46)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}

$s = \sqrt{\frac{{(- 23)}^{2} + {(28 - 46)}^{2} + {(45 - 46)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.2

Raise

- 23

$- 23$ to the power of

2

$2$ .

$s = \sqrt{\frac{529 + {(28 - 46)}^{2} + {(45 - 46)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.3

Subtract

$46$ from

$28$ .

$s = \sqrt{\frac{529 + {(- 18)}^{2} + {(45 - 46)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.4

Raise

$- 18$ to the power of

$2$ .

$s = \sqrt{\frac{529 + 324 + {(45 - 46)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.5

Subtract

$46$ from

$45$ .

$s = \sqrt{\frac{529 + 324 + {(- 1)}^{2} + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.6

Raise

$- 1$ to the power of

$2$ .

$s = \sqrt{\frac{529 + 324 + 1 + {(56 - 46)}^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.7

Subtract

$46$ from

$56$ .

$s = \sqrt{\frac{529 + 324 + 1 + 10^{2} + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.8

Raise

$10$ to the power of

$2$ .

$s = \sqrt{\frac{529 + 324 + 1 + 100 + {(78 - 46)}^{2}}{5 - 1}}$

Step 5.1.9

Subtract

$46$ from

$78$ .

$s = \sqrt{\frac{529 + 324 + 1 + 100 + 32^{2}}{5 - 1}}$

Step 5.1.10

Raise

$32$ to the power of

$2$ .

$s = \sqrt{\frac{529 + 324 + 1 + 100 + 1024}{5 - 1}}$

Step 5.1.11

Add

$529$ and

$324$ .

$s = \sqrt{\frac{853 + 1 + 100 + 1024}{5 - 1}}$

Step 5.1.12

Add

$853$ and

$1$ .

$s = \sqrt{\frac{854 + 100 + 1024}{5 - 1}}$

Step 5.1.13

Add

$854$ and

$100$ .

$s = \sqrt{\frac{954 + 1024}{5 - 1}}$

Step 5.1.14

Add

$954$ and

$1024$ .

$s = \sqrt{\frac{1978}{5 - 1}}$

Step 5.1.15

Subtract

$1$ from

$5$ .

$s = \sqrt{\frac{1978}{4}}$

Step 5.2

Cancel the common factor of

$1978$ and

$4$ .

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Step 5.2.1

Factor

$2$ out of

$1978$ .

$s = \sqrt{\frac{2 (989)}{4}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor

$2$ out of

$4$ .

$s = \sqrt{\frac{2 \cdot 989}{2 \cdot 2}}$

Step 5.2.2.2

Cancel the common factor.

$s = \sqrt{\frac{2 \cdot 989}{2 \cdot 2}}$

Step 5.2.2.3

Rewrite the expression.

$s = \sqrt{\frac{989}{2}}$

Step 5.3

Rewrite

$\sqrt{\frac{989}{2}}$ as

$\frac{\sqrt{989}}{\sqrt{2}}$ .

$s = \frac{\sqrt{989}}{\sqrt{2}}$

Step 5.4

Multiply

$\frac{\sqrt{989}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$s = \frac{\sqrt{989}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.5

Combine and simplify the denominator.

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Step 5.5.1

Multiply

$\frac{\sqrt{989}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$s = \frac{\sqrt{989} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.5.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$s = \frac{\sqrt{989} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.5.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$s = \frac{\sqrt{989} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.5.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$s = \frac{\sqrt{989} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.5.5

Add

$1$ and

$1$ .

$s = \frac{\sqrt{989} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.5.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 5.5.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$s = \frac{\sqrt{989} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.5.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$s = \frac{\sqrt{989} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$s = \frac{\sqrt{989} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.5.6.4

Cancel the common factor of

$2$ .

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Step 5.5.6.4.1

Cancel the common factor.

$s = \frac{\sqrt{989} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.5.6.4.2

Rewrite the expression.

$s = \frac{\sqrt{989} \sqrt{2}}{2}$

Step 5.5.6.5

Evaluate the exponent.

$s = \frac{\sqrt{989} \sqrt{2}}{2}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Combine using the product rule for radicals.

$s = \frac{\sqrt{989 \cdot 2}}{2}$

Step 5.6.2

Multiply

$989$ by

$2$ .

$s = \frac{\sqrt{1978}}{2}$

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.

$22.2$

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