Algebra Examples

92

$92$ ,

97

$97$ ,

53

$53$ ,

90

$90$ ,

95

$95$ ,

98

$98$

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.

\overline{x} = \frac{92 + 97 + 53 + 90 + 95 + 98}{6}

$\overline{x} = \frac{92 + 97 + 53 + 90 + 95 + 98}{6}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Add

92

$92$ and

97

$97$ .

\overline{x} = \frac{189 + 53 + 90 + 95 + 98}{6}

$\overline{x} = \frac{189 + 53 + 90 + 95 + 98}{6}$

Step 1.2.2

Add

189

$189$ and

53

$53$ .

\overline{x} = \frac{242 + 90 + 95 + 98}{6}

$\overline{x} = \frac{242 + 90 + 95 + 98}{6}$

Step 1.2.3

Add

242

$242$ and

90

$90$ .

\overline{x} = \frac{332 + 95 + 98}{6}

$\overline{x} = \frac{332 + 95 + 98}{6}$

Step 1.2.4

Add

332

$332$ and

95

$95$ .

\overline{x} = \frac{427 + 98}{6}

$\overline{x} = \frac{427 + 98}{6}$

Step 1.2.5

Add

427

$427$ and

98

$98$ .

\overline{x} = \frac{525}{6}

$\overline{x} = \frac{525}{6}$

\overline{x} = \frac{525}{6}

$\overline{x} = \frac{525}{6}$

Step 1.3

Cancel the common factor of

525

$525$ and

6

$6$ .

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Step 1.3.1

Factor

3

$3$ out of

525

$525$ .

\overline{x} = \frac{3 (175)}{6}

$\overline{x} = \frac{3 (175)}{6}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor

3

$3$ out of

6

$6$ .

\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}

$\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}$

Step 1.3.2.2

Cancel the common factor.

\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}

$\overline{x} = \frac{3 \cdot 175}{3 \cdot 2}$

Step 1.3.2.3

Rewrite the expression.

\overline{x} = \frac{175}{2}

$\overline{x} = \frac{175}{2}$

\overline{x} = \frac{175}{2}

$\overline{x} = \frac{175}{2}$

\overline{x} = \frac{175}{2}

$\overline{x} = \frac{175}{2}$

Step 1.4

Divide.

\overline{x} = 87.5

$\overline{x} = 87.5$

\overline{x} = 87.5

$\overline{x} = 87.5$

Step 2

Simplify each value in the list.

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Step 2.1

Convert

92

$92$ to a decimal value.

92

$92$

Step 2.2

Convert

97

$97$ to a decimal value.

97

$97$

Step 2.3

Convert

53

$53$ to a decimal value.

53

$53$

Step 2.4

Convert

90

$90$ to a decimal value.

90

$90$

Step 2.5

Convert

95

$95$ to a decimal value.

95

$95$

Step 2.6

Convert

98

$98$ to a decimal value.

98

$98$

Step 2.7

The simplified values are

92, 97, 53, 90, 95, 98

$92, 97, 53, 90, 95, 98$ .

92, 97, 53, 90, 95, 98

$92, 97, 53, 90, 95, 98$

92, 97, 53, 90, 95, 98

$92, 97, 53, 90, 95, 98$

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 = \sum_{i = 1}^{n} \sqrt{\frac{{(x_{i} - x_{a v g})}^{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{{(92 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{{(92 - 87.5)}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5

Simplify the result.

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Step 5.1

Subtract

87.5

$87.5$ from

92

$92$ .

s = \sqrt{\frac{{4.5}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{{4.5}^{2} + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.2

Raise

4.5

$4.5$ to the power of

2

$2$ .

s = \sqrt{\frac{20.25 + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + {(97 - 87.5)}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.3

Subtract

87.5

$87.5$ from

97

$97$ .

s = \sqrt{\frac{20.25 + {9.5}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + {9.5}^{2} + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.4

Raise

9.5

$9.5$ to the power of

2

$2$ .

s = \sqrt{\frac{20.25 + 90.25 + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + {(53 - 87.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.5

Subtract

87.5

$87.5$ from

53

$53$ .

s = \sqrt{\frac{20.25 + 90.25 + {(- 34.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + {(- 34.5)}^{2} + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.6

Raise

- 34.5

$- 34.5$ to the power of

2

$2$ .

s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + {(90 - 87.5)}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.7

Subtract

87.5

$87.5$ from

90

$90$ .

s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + {2.5}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + {2.5}^{2} + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.8

Raise

2.5

$2.5$ to the power of

2

$2$ .

s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + {(95 - 87.5)}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.9

Subtract

87.5

$87.5$ from

95

$95$ .

s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + {7.5}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + {7.5}^{2} + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.10

Raise

7.5

$7.5$ to the power of

2

$2$ .

s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {(98 - 87.5)}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {(98 - 87.5)}^{2}}{6 - 1}}$

Step 5.11

Subtract

87.5

$87.5$ from

98

$98$ .

s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {10.5}^{2}}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + {10.5}^{2}}{6 - 1}}$

Step 5.12

Raise

10.5

$10.5$ to the power of

2

$2$ .

s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}}

$s = \sqrt{\frac{20.25 + 90.25 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}}$

Step 5.13

Add

20.25

$20.25$ and

90.25

$90.25$ .

s = \sqrt{\frac{110.5 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}}

$s = \sqrt{\frac{110.5 + 1190.25 + 6.25 + 56.25 + 110.25}{6 - 1}}$

Step 5.14

Add

110.5

$110.5$ and

1190.25

$1190.25$ .

s = \sqrt{\frac{1300.75 + 6.25 + 56.25 + 110.25}{6 - 1}}

$s = \sqrt{\frac{1300.75 + 6.25 + 56.25 + 110.25}{6 - 1}}$

Step 5.15

Add

1300.75

$1300.75$ and

6.25

$6.25$ .

s = \sqrt{\frac{1307 + 56.25 + 110.25}{6 - 1}}

$s = \sqrt{\frac{1307 + 56.25 + 110.25}{6 - 1}}$

Step 5.16

Add

1307

$1307$ and

56.25

$56.25$ .

s = \sqrt{\frac{1363.25 + 110.25}{6 - 1}}

$s = \sqrt{\frac{1363.25 + 110.25}{6 - 1}}$

Step 5.17

Add

1363.25

$1363.25$ and

110.25

$110.25$ .

s = \sqrt{\frac{1473.5}{6 - 1}}

$s = \sqrt{\frac{1473.5}{6 - 1}}$

Step 5.18

Subtract

1

$1$ from

6

$6$ .

s = \sqrt{\frac{1473.5}{5}}

$s = \sqrt{\frac{1473.5}{5}}$

Step 5.19

Divide

1473.5

$1473.5$ by

5

$5$ .

s = \sqrt{294.7}

$s = \sqrt{294.7}$

s = \sqrt{294.7}

$s = \sqrt{294.7}$

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.

17.2

$17.2$