Find the Variance 35 , 67 , 21 , 16 , 24 , 51 , 18 , 32

$35$ , $67$ , $21$ , $16$ , $24$ , $51$ , $18$ , $32$

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

$\overline{x} = \frac{35 + 67 + 21 + 16 + 24 + 51 + 18 + 32}{8}$

Simplify the numerator.

Add $35$ and $67$ .

$\overline{x} = \frac{102 + 21 + 16 + 24 + 51 + 18 + 32}{8}$

Add $102$ and $21$ .

$\overline{x} = \frac{123 + 16 + 24 + 51 + 18 + 32}{8}$

Add $123$ and $16$ .

$\overline{x} = \frac{139 + 24 + 51 + 18 + 32}{8}$

Add $139$ and $24$ .

$\overline{x} = \frac{163 + 51 + 18 + 32}{8}$

Add $163$ and $51$ .

$\overline{x} = \frac{214 + 18 + 32}{8}$

Add $214$ and $18$ .

$\overline{x} = \frac{232 + 32}{8}$

Add $232$ and $32$ .

$\overline{x} = \frac{264}{8}$

Divide $264$ by $8$ .

$\overline{x} = 33$

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

$s^{2} = \sum_{i = 1}^{n} \frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}$

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

$s = \frac{{(35 - 33)}^{2} + {(67 - 33)}^{2} + {(21 - 33)}^{2} + {(16 - 33)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Simplify the result.

Simplify the numerator.

Subtract $33$ from $35$ .

$s = \frac{2^{2} + {(67 - 33)}^{2} + {(21 - 33)}^{2} + {(16 - 33)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

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

$s = \frac{4 + {(67 - 33)}^{2} + {(21 - 33)}^{2} + {(16 - 33)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Subtract $33$ from $67$ .

$s = \frac{4 + 34^{2} + {(21 - 33)}^{2} + {(16 - 33)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Raise $34$ to the power of $2$ .

$s = \frac{4 + 1156 + {(21 - 33)}^{2} + {(16 - 33)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Subtract $33$ from $21$ .

$s = \frac{4 + 1156 + {(- 12)}^{2} + {(16 - 33)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Raise $- 12$ to the power of $2$ .

$s = \frac{4 + 1156 + 144 + {(16 - 33)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Subtract $33$ from $16$ .

$s = \frac{4 + 1156 + 144 + {(- 17)}^{2} + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Raise $- 17$ to the power of $2$ .

$s = \frac{4 + 1156 + 144 + 289 + {(24 - 33)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Subtract $33$ from $24$ .

$s = \frac{4 + 1156 + 144 + 289 + {(- 9)}^{2} + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

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

$s = \frac{4 + 1156 + 144 + 289 + 81 + {(51 - 33)}^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Subtract $33$ from $51$ .

$s = \frac{4 + 1156 + 144 + 289 + 81 + 18^{2} + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Raise $18$ to the power of $2$ .

$s = \frac{4 + 1156 + 144 + 289 + 81 + 324 + {(18 - 33)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Subtract $33$ from $18$ .

$s = \frac{4 + 1156 + 144 + 289 + 81 + 324 + {(- 15)}^{2} + {(32 - 33)}^{2}}{8 - 1}$

Raise $- 15$ to the power of $2$ .

$s = \frac{4 + 1156 + 144 + 289 + 81 + 324 + 225 + {(32 - 33)}^{2}}{8 - 1}$

Subtract $33$ from $32$ .

$s = \frac{4 + 1156 + 144 + 289 + 81 + 324 + 225 + {(- 1)}^{2}}{8 - 1}$

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

$s = \frac{4 + 1156 + 144 + 289 + 81 + 324 + 225 + 1}{8 - 1}$

Add $4$ and $1156$ .

$s = \frac{1160 + 144 + 289 + 81 + 324 + 225 + 1}{8 - 1}$

Add $1160$ and $144$ .

$s = \frac{1304 + 289 + 81 + 324 + 225 + 1}{8 - 1}$

Add $1304$ and $289$ .

$s = \frac{1593 + 81 + 324 + 225 + 1}{8 - 1}$

Add $1593$ and $81$ .

$s = \frac{1674 + 324 + 225 + 1}{8 - 1}$

Add $1674$ and $324$ .

$s = \frac{1998 + 225 + 1}{8 - 1}$

Add $1998$ and $225$ .

$s = \frac{2223 + 1}{8 - 1}$

Add $2223$ and $1$ .

$s = \frac{2224}{8 - 1}$

Subtract $1$ from $8$ .

$s = \frac{2224}{7}$

Approximate the result.

$s^{2} \approx 317.7143$

Find the Variance 35 , 67 , 21 , 16 , 24 , 51 , 18 , 32

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