Find the Sample Standard Deviation 212 , 249 , 212 , 248 , 239 , 212 , 216 , 234 , 248

$212$ , $249$ , $212$ , $248$ , $239$ , $212$ , $216$ , $234$ , $248$

Step 1

Find the mean.

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

$\overline{x} = \frac{212 + 249 + 212 + 248 + 239 + 212 + 216 + 234 + 248}{9}$

Simplify the numerator.

Add $212$ and $249$ .

$\overline{x} = \frac{461 + 212 + 248 + 239 + 212 + 216 + 234 + 248}{9}$

Add $461$ and $212$ .

$\overline{x} = \frac{673 + 248 + 239 + 212 + 216 + 234 + 248}{9}$

Add $673$ and $248$ .

$\overline{x} = \frac{921 + 239 + 212 + 216 + 234 + 248}{9}$

Add $921$ and $239$ .

$\overline{x} = \frac{1160 + 212 + 216 + 234 + 248}{9}$

Add $1160$ and $212$ .

$\overline{x} = \frac{1372 + 216 + 234 + 248}{9}$

Add $1372$ and $216$ .

$\overline{x} = \frac{1588 + 234 + 248}{9}$

Add $1588$ and $234$ .

$\overline{x} = \frac{1822 + 248}{9}$

Add $1822$ and $248$ .

$\overline{x} = \frac{2070}{9}$

Divide $2070$ by $9$ .

$\overline{x} = 230$

Step 2

Simplify each value in the list.

Convert $212$ to a decimal value.

$212$

Convert $249$ to a decimal value.

$249$

Convert $212$ to a decimal value.

$212$

Convert $248$ to a decimal value.

$248$

Convert $239$ to a decimal value.

$239$

Convert $212$ to a decimal value.

$212$

Convert $216$ to a decimal value.

$216$

Convert $234$ to a decimal value.

$234$

Convert $248$ to a decimal value.

$248$

The simplified values are $212, 249, 212, 248, 239, 212, 216, 234, 248$ .

$212, 249, 212, 248, 239, 212, 216, 234, 248$

Step 3

Set up the formula for sample standard deviation. The standard deviation of a set of values is a measure of the spread of its values.

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

Step 4

Set up the formula for standard deviation for this set of numbers.

$s = \sqrt{\frac{{(212 - 230)}^{2} + {(249 - 230)}^{2} + {(212 - 230)}^{2} + {(248 - 230)}^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Step 5

Simplify the result.

Simplify the expression.

Subtract $230$ from $212$ .

$s = \sqrt{\frac{{(- 18)}^{2} + {(249 - 230)}^{2} + {(212 - 230)}^{2} + {(248 - 230)}^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + {(249 - 230)}^{2} + {(212 - 230)}^{2} + {(248 - 230)}^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $249$ .

$s = \sqrt{\frac{324 + 19^{2} + {(212 - 230)}^{2} + {(248 - 230)}^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + 361 + {(212 - 230)}^{2} + {(248 - 230)}^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $212$ .

$s = \sqrt{\frac{324 + 361 + {(- 18)}^{2} + {(248 - 230)}^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + 361 + 324 + {(248 - 230)}^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $248$ .

$s = \sqrt{\frac{324 + 361 + 324 + 18^{2} + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + 361 + 324 + 324 + {(239 - 230)}^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $239$ .

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 9^{2} + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + {(212 - 230)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $212$ .

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + {(- 18)}^{2} + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + 324 + {(216 - 230)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $216$ .

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + 324 + {(- 14)}^{2} + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Raise $- 14$ to the power of $2$ .

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + 324 + 196 + {(234 - 230)}^{2} + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $234$ .

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + 324 + 196 + 4^{2} + {(248 - 230)}^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + 324 + 196 + 16 + {(248 - 230)}^{2}}{9 - 1}}$

Subtract $230$ from $248$ .

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + 324 + 196 + 16 + 18^{2}}{9 - 1}}$

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

$s = \sqrt{\frac{324 + 361 + 324 + 324 + 81 + 324 + 196 + 16 + 324}{9 - 1}}$

Add $324$ and $361$ .

$s = \sqrt{\frac{685 + 324 + 324 + 81 + 324 + 196 + 16 + 324}{9 - 1}}$

Add $685$ and $324$ .

$s = \sqrt{\frac{1009 + 324 + 81 + 324 + 196 + 16 + 324}{9 - 1}}$

Add $1009$ and $324$ .

$s = \sqrt{\frac{1333 + 81 + 324 + 196 + 16 + 324}{9 - 1}}$

Add $1333$ and $81$ .

$s = \sqrt{\frac{1414 + 324 + 196 + 16 + 324}{9 - 1}}$

Add $1414$ and $324$ .

$s = \sqrt{\frac{1738 + 196 + 16 + 324}{9 - 1}}$

Add $1738$ and $196$ .

$s = \sqrt{\frac{1934 + 16 + 324}{9 - 1}}$

Add $1934$ and $16$ .

$s = \sqrt{\frac{1950 + 324}{9 - 1}}$

Add $1950$ and $324$ .

$s = \sqrt{\frac{2274}{9 - 1}}$

Subtract $1$ from $9$ .

$s = \sqrt{\frac{2274}{8}}$

Cancel the common factor of $2274$ and $8$ .

Factor $2$ out of $2274$ .

$s = \sqrt{\frac{2 (1137)}{8}}$

Cancel the common factors.

Factor $2$ out of $8$ .

$s = \sqrt{\frac{2 \cdot 1137}{2 \cdot 4}}$

Cancel the common factor.

$s = \sqrt{\frac{2 \cdot 1137}{2 \cdot 4}}$

Rewrite the expression.

$s = \sqrt{\frac{1137}{4}}$

Rewrite $\sqrt{\frac{1137}{4}}$ as $\frac{\sqrt{1137}}{\sqrt{4}}$ .

$s = \frac{\sqrt{1137}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$s = \frac{\sqrt{1137}}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$s = \frac{\sqrt{1137}}{2}$

Step 6

The standard deviation should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$16.9$

Find the Sample Standard Deviation 212 , 249 , 212 , 248 , 239 , 212 , 216 , 234 , 248

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