Find the Variance 63 , 54 , 62 , 59 , 52

$63$ , $54$ , $62$ , $59$ , $52$

Step 1

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

$\overline{x} = \frac{63 + 54 + 62 + 59 + 52}{5}$

Step 2

Simplify the numerator.

Add $63$ and $54$ .

$\overline{x} = \frac{117 + 62 + 59 + 52}{5}$

Add $117$ and $62$ .

$\overline{x} = \frac{179 + 59 + 52}{5}$

Add $179$ and $59$ .

$\overline{x} = \frac{238 + 52}{5}$

Add $238$ and $52$ .

$\overline{x} = \frac{290}{5}$

Step 3

Divide $290$ by $5$ .

$\overline{x} = 58$

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.

$s = \frac{{(63 - 58)}^{2} + {(54 - 58)}^{2} + {(62 - 58)}^{2} + {(59 - 58)}^{2} + {(52 - 58)}^{2}}{5 - 1}$

Step 6

Simplify the result.

Simplify the numerator.

Subtract $58$ from $63$ .

$s = \frac{5^{2} + {(54 - 58)}^{2} + {(62 - 58)}^{2} + {(59 - 58)}^{2} + {(52 - 58)}^{2}}{5 - 1}$

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

$s = \frac{25 + {(54 - 58)}^{2} + {(62 - 58)}^{2} + {(59 - 58)}^{2} + {(52 - 58)}^{2}}{5 - 1}$

Subtract $58$ from $54$ .

$s = \frac{25 + {(- 4)}^{2} + {(62 - 58)}^{2} + {(59 - 58)}^{2} + {(52 - 58)}^{2}}{5 - 1}$

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

$s = \frac{25 + 16 + {(62 - 58)}^{2} + {(59 - 58)}^{2} + {(52 - 58)}^{2}}{5 - 1}$

Subtract $58$ from $62$ .

$s = \frac{25 + 16 + 4^{2} + {(59 - 58)}^{2} + {(52 - 58)}^{2}}{5 - 1}$

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

$s = \frac{25 + 16 + 16 + {(59 - 58)}^{2} + {(52 - 58)}^{2}}{5 - 1}$

Subtract $58$ from $59$ .

$s = \frac{25 + 16 + 16 + 1^{2} + {(52 - 58)}^{2}}{5 - 1}$

One to any power is one.

$s = \frac{25 + 16 + 16 + 1 + {(52 - 58)}^{2}}{5 - 1}$

Subtract $58$ from $52$ .

$s = \frac{25 + 16 + 16 + 1 + {(- 6)}^{2}}{5 - 1}$

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

$s = \frac{25 + 16 + 16 + 1 + 36}{5 - 1}$

Add $25$ and $16$ .

$s = \frac{41 + 16 + 1 + 36}{5 - 1}$

Add $41$ and $16$ .

$s = \frac{57 + 1 + 36}{5 - 1}$

Add $57$ and $1$ .

$s = \frac{58 + 36}{5 - 1}$

Add $58$ and $36$ .

$s = \frac{94}{5 - 1}$

Reduce the expression by cancelling the common factors.

Subtract $1$ from $5$ .

$s = \frac{94}{4}$

Cancel the common factor of $94$ and $4$ .

Factor $2$ out of $94$ .

$s = \frac{2 (47)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$s = \frac{2 \cdot 47}{2 \cdot 2}$

Cancel the common factor.

$s = \frac{2 \cdot 47}{2 \cdot 2}$

Rewrite the expression.

$s = \frac{47}{2}$

Step 7

Approximate the result.

$s^{2} \approx 23.5$

Find the Variance 63 , 54 , 62 , 59 , 52

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