Find the Variance 100 , 140 , 130 , 180 , 80 , 160

$100$ , $140$ , $130$ , $180$ , $80$ , $160$

Step 1

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

$\overline{x} = \frac{100 + 140 + 130 + 180 + 80 + 160}{6}$

Step 2

Cancel the common factor of $100 + 140 + 130 + 180 + 80 + 160$ and $6$ .

Factor $2$ out of $100$ .

$\overline{x} = \frac{2 (50) + 140 + 130 + 180 + 80 + 160}{6}$

Factor $2$ out of $140$ .

$\overline{x} = \frac{2 \cdot 50 + 2 \cdot 70 + 130 + 180 + 80 + 160}{6}$

Factor $2$ out of $2 \cdot 50 + 2 \cdot 70$ .

$\overline{x} = \frac{2 \cdot (50 + 70) + 130 + 180 + 80 + 160}{6}$

Factor $2$ out of $130$ .

$\overline{x} = \frac{2 \cdot (50 + 70) + 2 (65) + 180 + 80 + 160}{6}$

Factor $2$ out of $2 \cdot (50 + 70) + 2 (65)$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65) + 180 + 80 + 160}{6}$

Factor $2$ out of $180$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65) + 2 (90) + 80 + 160}{6}$

Factor $2$ out of $2 \cdot (50 + 70 + 65) + 2 (90)$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65 + 90) + 80 + 160}{6}$

Factor $2$ out of $80$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65 + 90) + 2 (40) + 160}{6}$

Factor $2$ out of $2 \cdot (50 + 70 + 65 + 90) + 2 (40)$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65 + 90 + 40) + 160}{6}$

Factor $2$ out of $160$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65 + 90 + 40) + 2 (80)}{6}$

Factor $2$ out of $2 \cdot (50 + 70 + 65 + 90 + 40) + 2 (80)$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65 + 90 + 40 + 80)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\overline{x} = \frac{2 \cdot (50 + 70 + 65 + 90 + 40 + 80)}{2 (3)}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot (50 + 70 + 65 + 90 + 40 + 80)}{2 \cdot 3}$

Rewrite the expression.

$\overline{x} = \frac{50 + 70 + 65 + 90 + 40 + 80}{3}$

Step 3

Simplify the numerator.

Add $50$ and $70$ .

$\overline{x} = \frac{120 + 65 + 90 + 40 + 80}{3}$

Add $120$ and $65$ .

$\overline{x} = \frac{185 + 90 + 40 + 80}{3}$

Add $185$ and $90$ .

$\overline{x} = \frac{275 + 40 + 80}{3}$

Add $275$ and $40$ .

$\overline{x} = \frac{315 + 80}{3}$

Add $315$ and $80$ .

$\overline{x} = \frac{395}{3}$

Step 4

Divide.

$\overline{x} = 131.\overline{6}$

Step 5

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.

$\overline{x} = 131.7$

Step 6

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

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

Step 7

Set up the formula for variance for this set of numbers.

$s = \frac{{(100 - 131.7)}^{2} + {(140 - 131.7)}^{2} + {(130 - 131.7)}^{2} + {(180 - 131.7)}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Step 8

Simplify the result.

Simplify the numerator.

Subtract $131.7$ from $100$ .

$s = \frac{{(- 31.7)}^{2} + {(140 - 131.7)}^{2} + {(130 - 131.7)}^{2} + {(180 - 131.7)}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Raise $- 31.7$ to the power of $2$ .

$s = \frac{1004.89 + {(140 - 131.7)}^{2} + {(130 - 131.7)}^{2} + {(180 - 131.7)}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Subtract $131.7$ from $140$ .

$s = \frac{1004.89 + {8.3}^{2} + {(130 - 131.7)}^{2} + {(180 - 131.7)}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Raise $8.3$ to the power of $2$ .

$s = \frac{1004.89 + 68.89 + {(130 - 131.7)}^{2} + {(180 - 131.7)}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Subtract $131.7$ from $130$ .

$s = \frac{1004.89 + 68.89 + {(- 1.7)}^{2} + {(180 - 131.7)}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Raise $- 1.7$ to the power of $2$ .

$s = \frac{1004.89 + 68.89 + 2.89 + {(180 - 131.7)}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Subtract $131.7$ from $180$ .

$s = \frac{1004.89 + 68.89 + 2.89 + {48.3}^{2} + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Raise $48.3$ to the power of $2$ .

$s = \frac{1004.89 + 68.89 + 2.89 + 2332.89 + {(80 - 131.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Subtract $131.7$ from $80$ .

$s = \frac{1004.89 + 68.89 + 2.89 + 2332.89 + {(- 51.7)}^{2} + {(160 - 131.7)}^{2}}{6 - 1}$

Raise $- 51.7$ to the power of $2$ .

$s = \frac{1004.89 + 68.89 + 2.89 + 2332.89 + 2672.89 + {(160 - 131.7)}^{2}}{6 - 1}$

Subtract $131.7$ from $160$ .

$s = \frac{1004.89 + 68.89 + 2.89 + 2332.89 + 2672.89 + {28.3}^{2}}{6 - 1}$

Raise $28.3$ to the power of $2$ .

$s = \frac{1004.89 + 68.89 + 2.89 + 2332.89 + 2672.89 + 800.89}{6 - 1}$

Add $1004.89$ and $68.89$ .

$s = \frac{1073.78 + 2.89 + 2332.89 + 2672.89 + 800.89}{6 - 1}$

Add $1073.78$ and $2.89$ .

$s = \frac{1076.67 + 2332.89 + 2672.89 + 800.89}{6 - 1}$

Add $1076.67$ and $2332.89$ .

$s = \frac{3409.56 + 2672.89 + 800.89}{6 - 1}$

Add $3409.56$ and $2672.89$ .

$s = \frac{6082.45 + 800.89}{6 - 1}$

Add $6082.45$ and $800.89$ .

$s = \frac{6883.34}{6 - 1}$

Simplify the expression.

Subtract $1$ from $6$ .

$s = \frac{6883.34}{5}$

Divide $6883.34$ by $5$ .

$s = 1376.668$

Step 9

Approximate the result.

$s^{2} \approx 1376.668$

Find the Variance 100 , 140 , 130 , 180 , 80 , 160

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