Find the Variance 68 , 67 , 74 , 63 , 71

$68$ , $67$ , $74$ , $63$ , $71$

Step 1

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

$\overline{x} = \frac{68 + 67 + 74 + 63 + 71}{5}$

Step 2

Simplify the numerator.

Add $68$ and $67$ .

$\overline{x} = \frac{135 + 74 + 63 + 71}{5}$

Add $135$ and $74$ .

$\overline{x} = \frac{209 + 63 + 71}{5}$

Add $209$ and $63$ .

$\overline{x} = \frac{272 + 71}{5}$

Add $272$ and $71$ .

$\overline{x} = \frac{343}{5}$

Step 3

Divide.

$\overline{x} = 68.6$

Step 4

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 5

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

$s = \frac{{(68 - 68.6)}^{2} + {(67 - 68.6)}^{2} + {(74 - 68.6)}^{2} + {(63 - 68.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $68.6$ from $68$ .

$s = \frac{{(- 0.6)}^{2} + {(67 - 68.6)}^{2} + {(74 - 68.6)}^{2} + {(63 - 68.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Raise $- 0.6$ to the power of $2$ .

$s = \frac{0.36 + {(67 - 68.6)}^{2} + {(74 - 68.6)}^{2} + {(63 - 68.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Subtract $68.6$ from $67$ .

$s = \frac{0.36 + {(- 1.6)}^{2} + {(74 - 68.6)}^{2} + {(63 - 68.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Raise $- 1.6$ to the power of $2$ .

$s = \frac{0.36 + 2.56 + {(74 - 68.6)}^{2} + {(63 - 68.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Subtract $68.6$ from $74$ .

$s = \frac{0.36 + 2.56 + {5.4}^{2} + {(63 - 68.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Raise $5.4$ to the power of $2$ .

$s = \frac{0.36 + 2.56 + 29.16 + {(63 - 68.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Subtract $68.6$ from $63$ .

$s = \frac{0.36 + 2.56 + 29.16 + {(- 5.6)}^{2} + {(71 - 68.6)}^{2}}{5 - 1}$

Raise $- 5.6$ to the power of $2$ .

$s = \frac{0.36 + 2.56 + 29.16 + 31.36 + {(71 - 68.6)}^{2}}{5 - 1}$

Subtract $68.6$ from $71$ .

$s = \frac{0.36 + 2.56 + 29.16 + 31.36 + {2.4}^{2}}{5 - 1}$

Raise $2.4$ to the power of $2$ .

$s = \frac{0.36 + 2.56 + 29.16 + 31.36 + 5.76}{5 - 1}$

Add $0.36$ and $2.56$ .

$s = \frac{2.92 + 29.16 + 31.36 + 5.76}{5 - 1}$

Add $2.92$ and $29.16$ .

$s = \frac{32.08 + 31.36 + 5.76}{5 - 1}$

Add $32.08$ and $31.36$ .

$s = \frac{63.44 + 5.76}{5 - 1}$

Add $63.44$ and $5.76$ .

$s = \frac{69.2}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$s = \frac{69.2}{4}$

Divide $69.2$ by $4$ .

$s = 17.3$

Step 7

Approximate the result.

$s^{2} \approx 17.3$

Find the Variance 68 , 67 , 74 , 63 , 71

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