Find the Variance 14 , 26 , 40 , 52

$14$ , $26$ , $40$ , $52$

Step 1

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

$\overline{x} = \frac{14 + 26 + 40 + 52}{4}$

Step 2

Cancel the common factor of $14 + 26 + 40 + 52$ and $4$ .

Factor $2$ out of $14$ .

$\overline{x} = \frac{2 (7) + 26 + 40 + 52}{4}$

Factor $2$ out of $26$ .

$\overline{x} = \frac{2 \cdot 7 + 2 \cdot 13 + 40 + 52}{4}$

Factor $2$ out of $2 \cdot 7 + 2 \cdot 13$ .

$\overline{x} = \frac{2 \cdot (7 + 13) + 40 + 52}{4}$

Factor $2$ out of $40$ .

$\overline{x} = \frac{2 \cdot (7 + 13) + 2 (20) + 52}{4}$

Factor $2$ out of $2 \cdot (7 + 13) + 2 (20)$ .

$\overline{x} = \frac{2 \cdot (7 + 13 + 20) + 52}{4}$

Factor $2$ out of $52$ .

$\overline{x} = \frac{2 \cdot (7 + 13 + 20) + 2 (26)}{4}$

Factor $2$ out of $2 \cdot (7 + 13 + 20) + 2 (26)$ .

$\overline{x} = \frac{2 \cdot (7 + 13 + 20 + 26)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\overline{x} = \frac{2 \cdot (7 + 13 + 20 + 26)}{2 (2)}$

Cancel the common factor.

$\overline{x} = \frac{2 \cdot (7 + 13 + 20 + 26)}{2 \cdot 2}$

Rewrite the expression.

$\overline{x} = \frac{7 + 13 + 20 + 26}{2}$

Step 3

Simplify the numerator.

Add $7$ and $13$ .

$\overline{x} = \frac{20 + 20 + 26}{2}$

Add $20$ and $20$ .

$\overline{x} = \frac{40 + 26}{2}$

Add $40$ and $26$ .

$\overline{x} = \frac{66}{2}$

Step 4

Divide $66$ by $2$ .

$\overline{x} = 33$

Step 5

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 6

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

$s = \frac{{(14 - 33)}^{2} + {(26 - 33)}^{2} + {(40 - 33)}^{2} + {(52 - 33)}^{2}}{4 - 1}$

Step 7

Simplify the result.

Simplify the numerator.

Subtract $33$ from $14$ .

$s = \frac{{(- 19)}^{2} + {(26 - 33)}^{2} + {(40 - 33)}^{2} + {(52 - 33)}^{2}}{4 - 1}$

Raise $- 19$ to the power of $2$ .

$s = \frac{361 + {(26 - 33)}^{2} + {(40 - 33)}^{2} + {(52 - 33)}^{2}}{4 - 1}$

Subtract $33$ from $26$ .

$s = \frac{361 + {(- 7)}^{2} + {(40 - 33)}^{2} + {(52 - 33)}^{2}}{4 - 1}$

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

$s = \frac{361 + 49 + {(40 - 33)}^{2} + {(52 - 33)}^{2}}{4 - 1}$

Subtract $33$ from $40$ .

$s = \frac{361 + 49 + 7^{2} + {(52 - 33)}^{2}}{4 - 1}$

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

$s = \frac{361 + 49 + 49 + {(52 - 33)}^{2}}{4 - 1}$

Subtract $33$ from $52$ .

$s = \frac{361 + 49 + 49 + 19^{2}}{4 - 1}$

Raise $19$ to the power of $2$ .

$s = \frac{361 + 49 + 49 + 361}{4 - 1}$

Add $361$ and $49$ .

$s = \frac{410 + 49 + 361}{4 - 1}$

Add $410$ and $49$ .

$s = \frac{459 + 361}{4 - 1}$

Add $459$ and $361$ .

$s = \frac{820}{4 - 1}$

Subtract $1$ from $4$ .

$s = \frac{820}{3}$

Step 8

Approximate the result.

$s^{2} \approx 273.3333$

Find the Variance 14 , 26 , 40 , 52

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