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, ${x^2}$.

$x$	$x^2$
254	64516
116	13456
205	42025
148	21904
208	43264
174	30276
207	42849
171	29241
161	25921
202	40804
157	24649
131	17161

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

$$ \sum{x} = 2134 $$

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

$$ \frac{({\sum}{x})^2}{n} = \frac{4553956}{12} = 379496.33333333 $$

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

$$ {\sum}{x^2} = 396066 $$

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

$$ {\sum}{x^2} - \frac{({\sum}{x})^2}{n} = 396066 - 379496.33333333 = 16569.666666667 $$

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. $$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1} = \frac{ 16569.666666667 }{11} = 1506.3333333333$$

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.

$$ {s} = \sqrt{1506.3333333333} = 38.8115$$