Find the Variance 35 , 37 , 42 , 57 , 61 , 70

$35$ , $37$ , $42$ , $57$ , $61$ , $70$

Step 1

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

$\overline{x} = \frac{35 + 37 + 42 + 57 + 61 + 70}{6}$

Step 2

Simplify the numerator.

Add $35$ and $37$ .

$\overline{x} = \frac{72 + 42 + 57 + 61 + 70}{6}$

Add $72$ and $42$ .

$\overline{x} = \frac{114 + 57 + 61 + 70}{6}$

Add $114$ and $57$ .

$\overline{x} = \frac{171 + 61 + 70}{6}$

Add $171$ and $61$ .

$\overline{x} = \frac{232 + 70}{6}$

Add $232$ and $70$ .

$\overline{x} = \frac{302}{6}$

Step 3

Cancel the common factor of $302$ and $6$ .

Factor $2$ out of $302$ .

$\overline{x} = \frac{2 (151)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\overline{x} = \frac{2 \cdot 151}{2 \cdot 3}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot 151}{2 \cdot 3}$

Rewrite the expression.

$\overline{x} = \frac{151}{3}$

Step 4

Divide.

$\overline{x} = 50.\overline{3}$

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.

$\overline{x} = 50.3$

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.

$s = \frac{{(35 - 50.3)}^{2} + {(37 - 50.3)}^{2} + {(42 - 50.3)}^{2} + {(57 - 50.3)}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Step 8

Simplify the result.

Simplify the numerator.

Subtract $50.3$ from $35$ .

$s = \frac{{(- 15.3)}^{2} + {(37 - 50.3)}^{2} + {(42 - 50.3)}^{2} + {(57 - 50.3)}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Raise $- 15.3$ to the power of $2$ .

$s = \frac{234.09 + {(37 - 50.3)}^{2} + {(42 - 50.3)}^{2} + {(57 - 50.3)}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Subtract $50.3$ from $37$ .

$s = \frac{234.09 + {(- 13.3)}^{2} + {(42 - 50.3)}^{2} + {(57 - 50.3)}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Raise $- 13.3$ to the power of $2$ .

$s = \frac{234.09 + 176.89 + {(42 - 50.3)}^{2} + {(57 - 50.3)}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Subtract $50.3$ from $42$ .

$s = \frac{234.09 + 176.89 + {(- 8.3)}^{2} + {(57 - 50.3)}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

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

$s = \frac{234.09 + 176.89 + 68.89 + {(57 - 50.3)}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Subtract $50.3$ from $57$ .

$s = \frac{234.09 + 176.89 + 68.89 + {6.7}^{2} + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Raise $6.7$ to the power of $2$ .

$s = \frac{234.09 + 176.89 + 68.89 + 44.89 + {(61 - 50.3)}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Subtract $50.3$ from $61$ .

$s = \frac{234.09 + 176.89 + 68.89 + 44.89 + {10.7}^{2} + {(70 - 50.3)}^{2}}{6 - 1}$

Raise $10.7$ to the power of $2$ .

$s = \frac{234.09 + 176.89 + 68.89 + 44.89 + 114.49 + {(70 - 50.3)}^{2}}{6 - 1}$

Subtract $50.3$ from $70$ .

$s = \frac{234.09 + 176.89 + 68.89 + 44.89 + 114.49 + {19.7}^{2}}{6 - 1}$

Raise $19.7$ to the power of $2$ .

$s = \frac{234.09 + 176.89 + 68.89 + 44.89 + 114.49 + 388.09}{6 - 1}$

Add $234.09$ and $176.89$ .

$s = \frac{410.98 + 68.89 + 44.89 + 114.49 + 388.09}{6 - 1}$

Add $410.98$ and $68.89$ .

$s = \frac{479.87 + 44.89 + 114.49 + 388.09}{6 - 1}$

Add $479.87$ and $44.89$ .

$s = \frac{524.76 + 114.49 + 388.09}{6 - 1}$

Add $524.76$ and $114.49$ .

$s = \frac{639.25 + 388.09}{6 - 1}$

Add $639.25$ and $388.09$ .

$s = \frac{1027.34}{6 - 1}$

Simplify the expression.

Subtract $1$ from $6$ .

$s = \frac{1027.34}{5}$

Divide $1027.34$ by $5$ .

$s = 205.468$

Step 9

Approximate the result.

$s^{2} \approx 205.468$

Find the Variance 35 , 37 , 42 , 57 , 61 , 70

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