Standard Deviation Calculator

Solution:

1. Start by writing the computational formula for the standard deviation of a sample: $$ {s}= \sqrt{\frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1}}$$

2. Create a table of 2 columns and 13 rows. There will be a header row and a row for each data value. The header row should be labeled with ${x}$ and $ x^2$. Enter the data values in the ${x}$ column, with each data value in its own row. In the second column, put the square of each of the data values, ${x^2}$.

$x$	$x^2$
204	41616
221	48841
298	88804
205	42025
253	64009
124	15376
174	30276
257	66049
127	16129
255	65025
297	88209
166	27556

3. Find the sum of all the values in the first column, ${\sum}{x}$.

$$ \sum{x} = 2581 $$

4. Square the answer from step 3, then divide that number by the size of the sample.

$$ \frac{({\sum}{x})^2}{n} = \frac{6661561}{12} = 555130.08333333 $$

5. Find the sum of all the values in the second column, ${\sum}{x^2}$.

$$ {\sum}{x^2} = 593915 $$

6. Subtract the answer in step 4 from the answer in step 5.

$$ {\sum}{x^2} - \frac{({\sum}{x})^2}{n} = 593915 - 555130.08333333 = 38784.916666667 $$

7. Divide the answer in step 6 by n - 1, one less than the size of the sample. This answer is the variance of the sample. $$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1} = \frac{ 38784.916666667 }{11} = 3525.9015151515$$

8. Take the square root of the answer found in step 7 above. This number is the standard deviation of the sample. It is symbolized by ${s}$ . Here, we round the standard deviation to at most 4 decimal places.

$$ {s} = \sqrt{3525.9015151515} = 59.3793$$