Find the Variance 17 , 19 , 23 , 27 , 30 , 34

$17$ , $19$ , $23$ , $27$ , $30$ , $34$

Step 1

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

$\overline{x} = \frac{17 + 19 + 23 + 27 + 30 + 34}{6}$

Step 2

Simplify the numerator.

Add $17$ and $19$ .

$\overline{x} = \frac{36 + 23 + 27 + 30 + 34}{6}$

Add $36$ and $23$ .

$\overline{x} = \frac{59 + 27 + 30 + 34}{6}$

Add $59$ and $27$ .

$\overline{x} = \frac{86 + 30 + 34}{6}$

Add $86$ and $30$ .

$\overline{x} = \frac{116 + 34}{6}$

Add $116$ and $34$ .

$\overline{x} = \frac{150}{6}$

Step 3

Divide $150$ by $6$ .

$\overline{x} = 25$

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.

$s = \frac{{(17 - 25)}^{2} + {(19 - 25)}^{2} + {(23 - 25)}^{2} + {(27 - 25)}^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $25$ from $17$ .

$s = \frac{{(- 8)}^{2} + {(19 - 25)}^{2} + {(23 - 25)}^{2} + {(27 - 25)}^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

Raise $- 8$ to the power of $2$ .

$s = \frac{64 + {(19 - 25)}^{2} + {(23 - 25)}^{2} + {(27 - 25)}^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

Subtract $25$ from $19$ .

$s = \frac{64 + {(- 6)}^{2} + {(23 - 25)}^{2} + {(27 - 25)}^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

Raise $- 6$ to the power of $2$ .

$s = \frac{64 + 36 + {(23 - 25)}^{2} + {(27 - 25)}^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

Subtract $25$ from $23$ .

$s = \frac{64 + 36 + {(- 2)}^{2} + {(27 - 25)}^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

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

$s = \frac{64 + 36 + 4 + {(27 - 25)}^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

Subtract $25$ from $27$ .

$s = \frac{64 + 36 + 4 + 2^{2} + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

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

$s = \frac{64 + 36 + 4 + 4 + {(30 - 25)}^{2} + {(34 - 25)}^{2}}{6 - 1}$

Subtract $25$ from $30$ .

$s = \frac{64 + 36 + 4 + 4 + 5^{2} + {(34 - 25)}^{2}}{6 - 1}$

Raise $5$ to the power of $2$ .

$s = \frac{64 + 36 + 4 + 4 + 25 + {(34 - 25)}^{2}}{6 - 1}$

Subtract $25$ from $34$ .

$s = \frac{64 + 36 + 4 + 4 + 25 + 9^{2}}{6 - 1}$

Raise $9$ to the power of $2$ .

$s = \frac{64 + 36 + 4 + 4 + 25 + 81}{6 - 1}$

Add $64$ and $36$ .

$s = \frac{100 + 4 + 4 + 25 + 81}{6 - 1}$

Add $100$ and $4$ .

$s = \frac{104 + 4 + 25 + 81}{6 - 1}$

Add $104$ and $4$ .

$s = \frac{108 + 25 + 81}{6 - 1}$

Add $108$ and $25$ .

$s = \frac{133 + 81}{6 - 1}$

Add $133$ and $81$ .

$s = \frac{214}{6 - 1}$

Subtract $1$ from $6$ .

$s = \frac{214}{5}$

Step 7

Approximate the result.

$s^{2} \approx 42.8$

Find the Variance 17 , 19 , 23 , 27 , 30 , 34

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