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 7 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, ${x^2}$.

$x$	$x^2$
178	31684
231	53361
103	10609
167	27889
161	25921
118	13924

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

$$ \sum{x} = 958 $$

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

$$ \frac{({\sum}{x})^2}{n} = \frac{917764}{6} = 152960.66666667 $$

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

$$ {\sum}{x^2} = 163388 $$

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

$$ {\sum}{x^2} - \frac{({\sum}{x})^2}{n} = 163388 - 152960.66666667 = 10427.333333333 $$

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. $$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1} = \frac{ 10427.333333333 }{5} = 2085.4666666667$$

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.

$$ {s} = \sqrt{2085.4666666667} = 45.6669$$