Pre-Algebra Examples

\frac{1}{10}

$\frac{1}{10}$ ,

\frac{1}{16}

$\frac{1}{16}$ ,

\frac{1}{22}

$\frac{1}{22}$ ,

\frac{1}{28}

$\frac{1}{28}$ ,

\frac{1}{34}

$\frac{1}{34}$

Step 1

The five-number summary is a descriptive statistic that provides information about a set of observations. It consists of the following statistics:

1. Minimum (Min) - the smallest observation

2. Maximum (Max) - the largest observation

3. Median

M

$M$ - the middle term

4. First Quartile

Q_{1}

$Q_{1}$ - the middle term of values below the median

5. Third Quartile

Q_{3}

$Q_{3}$ - the middle term of values above the median

Step 2

Arrange the terms in ascending order.

\frac{1}{34}, \frac{1}{28}, \frac{1}{22}, \frac{1}{16}, \frac{1}{10}

$\frac{1}{34}, \frac{1}{28}, \frac{1}{22}, \frac{1}{16}, \frac{1}{10}$

Step 3

The minimum value is the smallest value in the arranged data set.

\frac{1}{34}

$\frac{1}{34}$

Step 4

The maximum value is the largest value in the arranged data set.

\frac{1}{10}

$\frac{1}{10}$

Step 5

The median is the middle term in the arranged data set.

\frac{1}{22}

$\frac{1}{22}$

Step 6

Find the first quartile by finding the median of the set of values to the left of the median.

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Step 6.1

The lower half of data is the set below the median.

\frac{1}{34}, \frac{1}{28}

$\frac{1}{34}, \frac{1}{28}$

Step 6.2

The median for the lower half of data

\frac{1}{34}, \frac{1}{28}

$\frac{1}{34}, \frac{1}{28}$ is the lower or first quartile. In this case, the first quartile is

0.03256302

$0.03256302$ .

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Step 6.2.1

The median is the middle term in the arranged data set. In the case of an even number of terms, the median is the average of the two middle terms.

\frac{\frac{1}{34} + \frac{1}{28}}{2}

$\frac{\frac{1}{34} + \frac{1}{28}}{2}$

Step 6.2.2

Remove parentheses.

\frac{\frac{1}{34} + \frac{1}{28}}{2}

$\frac{\frac{1}{34} + \frac{1}{28}}{2}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

To write

\frac{1}{34}

$\frac{1}{34}$ as a fraction with a common denominator, multiply by

\frac{14}{14}

$\frac{14}{14}$ .

\frac{\frac{1}{34} \cdot \frac{14}{14} + \frac{1}{28}}{2}

$\frac{\frac{1}{34} \cdot \frac{14}{14} + \frac{1}{28}}{2}$

Step 6.2.3.2

To write

\frac{1}{28}

$\frac{1}{28}$ as a fraction with a common denominator, multiply by

\frac{17}{17}

$\frac{17}{17}$ .

\frac{\frac{1}{34} \cdot \frac{14}{14} + \frac{1}{28} \cdot \frac{17}{17}}{2}

$\frac{\frac{1}{34} \cdot \frac{14}{14} + \frac{1}{28} \cdot \frac{17}{17}}{2}$

Step 6.2.3.3

Write each expression with a common denominator of

476

$476$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 6.2.3.3.1

Multiply

\frac{1}{34}

$\frac{1}{34}$ by

\frac{14}{14}

$\frac{14}{14}$ .

\frac{\frac{14}{34 \cdot 14} + \frac{1}{28} \cdot \frac{17}{17}}{2}

$\frac{\frac{14}{34 \cdot 14} + \frac{1}{28} \cdot \frac{17}{17}}{2}$

Step 6.2.3.3.2

Multiply

34

$34$ by

14

$14$ .

\frac{\frac{14}{476} + \frac{1}{28} \cdot \frac{17}{17}}{2}

$\frac{\frac{14}{476} + \frac{1}{28} \cdot \frac{17}{17}}{2}$

Step 6.2.3.3.3

Multiply

\frac{1}{28}

$\frac{1}{28}$ by

\frac{17}{17}

$\frac{17}{17}$ .

\frac{\frac{14}{476} + \frac{17}{28 \cdot 17}}{2}

$\frac{\frac{14}{476} + \frac{17}{28 \cdot 17}}{2}$

Step 6.2.3.3.4

Multiply

28

$28$ by

17

$17$ .

\frac{\frac{14}{476} + \frac{17}{476}}{2}

$\frac{\frac{14}{476} + \frac{17}{476}}{2}$

\frac{\frac{14}{476} + \frac{17}{476}}{2}

$\frac{\frac{14}{476} + \frac{17}{476}}{2}$

Step 6.2.3.4

Combine the numerators over the common denominator.

\frac{\frac{14 + 17}{476}}{2}

$\frac{\frac{14 + 17}{476}}{2}$

Step 6.2.3.5

Add

14

$14$ and

17

$17$ .

\frac{\frac{31}{476}}{2}

$\frac{\frac{31}{476}}{2}$

\frac{\frac{31}{476}}{2}

$\frac{\frac{31}{476}}{2}$

Step 6.2.4

Multiply the numerator by the reciprocal of the denominator.

\frac{31}{476} \cdot \frac{1}{2}

$\frac{31}{476} \cdot \frac{1}{2}$

Step 6.2.5

Multiply

\frac{31}{476} \cdot \frac{1}{2}

$\frac{31}{476} \cdot \frac{1}{2}$ .

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Step 6.2.5.1

Multiply

\frac{31}{476}

$\frac{31}{476}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\frac{31}{476 \cdot 2}

$\frac{31}{476 \cdot 2}$

Step 6.2.5.2

Multiply

476

$476$ by

2

$2$ .

\frac{31}{952}

$\frac{31}{952}$

\frac{31}{952}

$\frac{31}{952}$

Step 6.2.6

Convert the median

\frac{31}{952}

$\frac{31}{952}$ to decimal.

0.03256302

$0.03256302$

0.03256302

$0.03256302$

0.03256302

$0.03256302$

Step 7

Find the third quartile by finding the median of the set of values to the right of the median.

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Step 7.1

The upper half of data is the set above the median.

\frac{1}{16}, \frac{1}{10}

$\frac{1}{16}, \frac{1}{10}$

Step 7.2

The median for the upper half of data

\frac{1}{16}, \frac{1}{10}

$\frac{1}{16}, \frac{1}{10}$ is the upper or third quartile. In this case, the third quartile is

0.08125

$0.08125$ .

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Step 7.2.1

The median is the middle term in the arranged data set. In the case of an even number of terms, the median is the average of the two middle terms.

\frac{\frac{1}{16} + \frac{1}{10}}{2}

$\frac{\frac{1}{16} + \frac{1}{10}}{2}$

Step 7.2.2

Remove parentheses.

\frac{\frac{1}{16} + \frac{1}{10}}{2}

$\frac{\frac{1}{16} + \frac{1}{10}}{2}$

Step 7.2.3

Simplify the numerator.

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Step 7.2.3.1

To write

\frac{1}{16}

$\frac{1}{16}$ as a fraction with a common denominator, multiply by

\frac{5}{5}

$\frac{5}{5}$ .

\frac{\frac{1}{16} \cdot \frac{5}{5} + \frac{1}{10}}{2}

$\frac{\frac{1}{16} \cdot \frac{5}{5} + \frac{1}{10}}{2}$

Step 7.2.3.2

To write

\frac{1}{10}

$\frac{1}{10}$ as a fraction with a common denominator, multiply by

\frac{8}{8}

$\frac{8}{8}$ .

\frac{\frac{1}{16} \cdot \frac{5}{5} + \frac{1}{10} \cdot \frac{8}{8}}{2}

$\frac{\frac{1}{16} \cdot \frac{5}{5} + \frac{1}{10} \cdot \frac{8}{8}}{2}$

Step 7.2.3.3

Write each expression with a common denominator of

80

$80$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 7.2.3.3.1

Multiply

\frac{1}{16}

$\frac{1}{16}$ by

\frac{5}{5}

$\frac{5}{5}$ .

\frac{\frac{5}{16 \cdot 5} + \frac{1}{10} \cdot \frac{8}{8}}{2}

$\frac{\frac{5}{16 \cdot 5} + \frac{1}{10} \cdot \frac{8}{8}}{2}$

Step 7.2.3.3.2

Multiply

16

$16$ by

5

$5$ .

\frac{\frac{5}{80} + \frac{1}{10} \cdot \frac{8}{8}}{2}

$\frac{\frac{5}{80} + \frac{1}{10} \cdot \frac{8}{8}}{2}$

Step 7.2.3.3.3

Multiply

\frac{1}{10}

$\frac{1}{10}$ by

\frac{8}{8}

$\frac{8}{8}$ .

\frac{\frac{5}{80} + \frac{8}{10 \cdot 8}}{2}

$\frac{\frac{5}{80} + \frac{8}{10 \cdot 8}}{2}$

Step 7.2.3.3.4

Multiply

10

$10$ by

8

$8$ .

\frac{\frac{5}{80} + \frac{8}{80}}{2}

$\frac{\frac{5}{80} + \frac{8}{80}}{2}$

\frac{\frac{5}{80} + \frac{8}{80}}{2}

$\frac{\frac{5}{80} + \frac{8}{80}}{2}$

Step 7.2.3.4

Combine the numerators over the common denominator.

\frac{\frac{5 + 8}{80}}{2}

$\frac{\frac{5 + 8}{80}}{2}$

Step 7.2.3.5

Add

5

$5$ and

8

$8$ .

\frac{\frac{13}{80}}{2}

$\frac{\frac{13}{80}}{2}$

\frac{\frac{13}{80}}{2}

$\frac{\frac{13}{80}}{2}$

Step 7.2.4

Multiply the numerator by the reciprocal of the denominator.

\frac{13}{80} \cdot \frac{1}{2}

$\frac{13}{80} \cdot \frac{1}{2}$

Step 7.2.5

Multiply

\frac{13}{80} \cdot \frac{1}{2}

$\frac{13}{80} \cdot \frac{1}{2}$ .

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Step 7.2.5.1

Multiply

\frac{13}{80}

$\frac{13}{80}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\frac{13}{80 \cdot 2}

$\frac{13}{80 \cdot 2}$

Step 7.2.5.2

Multiply

80

$80$ by

2

$2$ .

\frac{13}{160}

$\frac{13}{160}$

\frac{13}{160}

$\frac{13}{160}$

Step 7.2.6

Convert the median

\frac{13}{160}

$\frac{13}{160}$ to decimal.

0.08125

$0.08125$

0.08125

$0.08125$

0.08125

$0.08125$

Step 8

The five most important sample values are sample minimum, sample maximum, median, lower quartile, and upper quartile.

Min = \frac{1}{34}

$Min = \frac{1}{34}$

Max = \frac{1}{10}

$Max = \frac{1}{10}$

M = \frac{1}{22}

$M = \frac{1}{22}$

Q_{1} = 0.03256302

$Q_{1} = 0.03256302$

Q_{3} = 0.08125

$Q_{3} = 0.08125$