Find the Variance 3 , 4 , 5 , 7 , 10 , 12 , 15

$3$ , $4$ , $5$ , $7$ , $10$ , $12$ , $15$

Step 1

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

$\overline{x} = \frac{3 + 4 + 5 + 7 + 10 + 12 + 15}{7}$

Step 2

Simplify the numerator.

Add $3$ and $4$ .

$\overline{x} = \frac{7 + 5 + 7 + 10 + 12 + 15}{7}$

Add $7$ and $5$ .

$\overline{x} = \frac{12 + 7 + 10 + 12 + 15}{7}$

Add $12$ and $7$ .

$\overline{x} = \frac{19 + 10 + 12 + 15}{7}$

Add $19$ and $10$ .

$\overline{x} = \frac{29 + 12 + 15}{7}$

Add $29$ and $12$ .

$\overline{x} = \frac{41 + 15}{7}$

Add $41$ and $15$ .

$\overline{x} = \frac{56}{7}$

Step 3

Divide $56$ by $7$ .

$\overline{x} = 8$

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.

$s = \frac{{(3 - 8)}^{2} + {(4 - 8)}^{2} + {(5 - 8)}^{2} + {(7 - 8)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $8$ from $3$ .

$s = \frac{{(- 5)}^{2} + {(4 - 8)}^{2} + {(5 - 8)}^{2} + {(7 - 8)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

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

$s = \frac{25 + {(4 - 8)}^{2} + {(5 - 8)}^{2} + {(7 - 8)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Subtract $8$ from $4$ .

$s = \frac{25 + {(- 4)}^{2} + {(5 - 8)}^{2} + {(7 - 8)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Raise $- 4$ to the power of $2$ .

$s = \frac{25 + 16 + {(5 - 8)}^{2} + {(7 - 8)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Subtract $8$ from $5$ .

$s = \frac{25 + 16 + {(- 3)}^{2} + {(7 - 8)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Raise $- 3$ to the power of $2$ .

$s = \frac{25 + 16 + 9 + {(7 - 8)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Subtract $8$ from $7$ .

$s = \frac{25 + 16 + 9 + {(- 1)}^{2} + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Raise $- 1$ to the power of $2$ .

$s = \frac{25 + 16 + 9 + 1 + {(10 - 8)}^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Subtract $8$ from $10$ .

$s = \frac{25 + 16 + 9 + 1 + 2^{2} + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

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

$s = \frac{25 + 16 + 9 + 1 + 4 + {(12 - 8)}^{2} + {(15 - 8)}^{2}}{7 - 1}$

Subtract $8$ from $12$ .

$s = \frac{25 + 16 + 9 + 1 + 4 + 4^{2} + {(15 - 8)}^{2}}{7 - 1}$

Raise $4$ to the power of $2$ .

$s = \frac{25 + 16 + 9 + 1 + 4 + 16 + {(15 - 8)}^{2}}{7 - 1}$

Subtract $8$ from $15$ .

$s = \frac{25 + 16 + 9 + 1 + 4 + 16 + 7^{2}}{7 - 1}$

Raise $7$ to the power of $2$ .

$s = \frac{25 + 16 + 9 + 1 + 4 + 16 + 49}{7 - 1}$

Add $25$ and $16$ .

$s = \frac{41 + 9 + 1 + 4 + 16 + 49}{7 - 1}$

Add $41$ and $9$ .

$s = \frac{50 + 1 + 4 + 16 + 49}{7 - 1}$

Add $50$ and $1$ .

$s = \frac{51 + 4 + 16 + 49}{7 - 1}$

Add $51$ and $4$ .

$s = \frac{55 + 16 + 49}{7 - 1}$

Add $55$ and $16$ .

$s = \frac{71 + 49}{7 - 1}$

Add $71$ and $49$ .

$s = \frac{120}{7 - 1}$

Simplify the expression.

Subtract $1$ from $7$ .

$s = \frac{120}{6}$

Divide $120$ by $6$ .

$s = 20$

Step 7

Approximate the result.

$s^{2} \approx 20$

Find the Variance 3 , 4 , 5 , 7 , 10 , 12 , 15

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