Statistics Examples

2

$2$ ,

8

$8$ ,

8

$8$ ,

12

$12$

Step 1

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

μ = \frac{2 + 8 + 8 + 12}{4}

$μ = \frac{2 + 8 + 8 + 12}{4}$

Step 2

Cancel the common factor of

2 + 8 + 8 + 12

$2 + 8 + 8 + 12$ and

4

$4$ .

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Step 2.1

Factor

2

$2$ out of

2

$2$ .

μ = \frac{2 (1) + 8 + 8 + 12}{4}

$μ = \frac{2 (1) + 8 + 8 + 12}{4}$

Step 2.2

Factor

2

$2$ out of

8

$8$ .

μ = \frac{2 \cdot 1 + 2 \cdot 4 + 8 + 12}{4}

$μ = \frac{2 \cdot 1 + 2 \cdot 4 + 8 + 12}{4}$

Step 2.3

Factor

2

$2$ out of

2 \cdot 1 + 2 \cdot 4

$2 \cdot 1 + 2 \cdot 4$ .

μ = \frac{2 \cdot (1 + 4) + 8 + 12}{4}

$μ = \frac{2 \cdot (1 + 4) + 8 + 12}{4}$

Step 2.4

Factor

2

$2$ out of

8

$8$ .

μ = \frac{2 \cdot (1 + 4) + 2 (4) + 12}{4}

$μ = \frac{2 \cdot (1 + 4) + 2 (4) + 12}{4}$

Step 2.5

Factor

2

$2$ out of

2 \cdot (1 + 4) + 2 (4)

$2 \cdot (1 + 4) + 2 (4)$ .

μ = \frac{2 \cdot (1 + 4 + 4) + 12}{4}

$μ = \frac{2 \cdot (1 + 4 + 4) + 12}{4}$

Step 2.6

Factor

2

$2$ out of

12

$12$ .

μ = \frac{2 \cdot (1 + 4 + 4) + 2 (6)}{4}

$μ = \frac{2 \cdot (1 + 4 + 4) + 2 (6)}{4}$

Step 2.7

Factor

2

$2$ out of

2 \cdot (1 + 4 + 4) + 2 (6)

$2 \cdot (1 + 4 + 4) + 2 (6)$ .

μ = \frac{2 \cdot (1 + 4 + 4 + 6)}{4}

$μ = \frac{2 \cdot (1 + 4 + 4 + 6)}{4}$

Step 2.8

Cancel the common factors.

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Step 2.8.1

Factor

2

$2$ out of

4

$4$ .

μ = \frac{2 \cdot (1 + 4 + 4 + 6)}{2 (2)}

$μ = \frac{2 \cdot (1 + 4 + 4 + 6)}{2 (2)}$

Step 2.8.2

Cancel the common factor.

$μ = \frac{2 \cdot (1 + 4 + 4 + 6)}{2 \cdot 2}$

Step 2.8.3

Rewrite the expression.

$μ = \frac{1 + 4 + 4 + 6}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Add

$1$ and

$4$ .

$μ = \frac{5 + 4 + 6}{2}$

Step 3.2

Add

$5$ and

$4$ .

$μ = \frac{9 + 6}{2}$

Step 3.3

Add

$9$ and

$6$ .

$μ = \frac{15}{2}$

Step 4

Divide.

$μ = 7.5$

Step 5

Arrange the terms in ascending order.

$M e d i a n = 2, 8, 8, 12$

Step 6

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.

$M e d i a n = \frac{8 + 8}{2}$

Step 7

Remove parentheses.

$M e d i a n = \frac{8 + 8}{2}$

Step 8

Cancel the common factor of

$8 + 8$ and

$2$ .

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Step 8.1

Factor

$2$ out of

$8$ .

$M e d i a n = \frac{2 \cdot 4 + 8}{2}$

Step 8.2

Factor

$2$ out of

$8$ .

$M e d i a n = \frac{2 \cdot 4 + 2 \cdot 4}{2}$

Step 8.3

Factor

$2$ out of

$2 \cdot 4 + 2 \cdot 4$ .

$M e d i a n = \frac{2 \cdot (4 + 4)}{2}$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor

$2$ out of

$2$ .

$M e d i a n = \frac{2 \cdot (4 + 4)}{2 (1)}$

Step 8.4.2

Cancel the common factor.

$M e d i a n = \frac{2 \cdot (4 + 4)}{2 \cdot 1}$

Step 8.4.3

Rewrite the expression.

$M e d i a n = \frac{4 + 4}{1}$

Step 8.4.4

Divide

$4 + 4$ by

$1$ .

$M e d i a n = 4 + 4$

Step 9

Add

$4$ and

$4$ .

$M e d i a n = 8$

Step 10

Convert the median

$8$ to decimal.

$M e d i a n = 8$

Step 11

Since the mean is less than the median, the data set is negatively skewed.

Negatively Skewed

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