Standard Deviation Calculator


Population
Sample

Both give the same result, but the computational formula is simpler to calculate step-by-step.
Standard
Computational

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, ${x^2}$.

$x$

$x^2$

11813924
16326569
25062500
22349729
18333489
25766049
29989401
18233124
19939601
24359049
28782369
27776729

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

$$ \sum{x} = 2681 $$

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

$$ \frac{({\sum}{x})^2}{n} = \frac{7187761}{12} = 598980.08333333 $$

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

$$ {\sum}{x^2} = 632533 $$

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

$$ {\sum}{x^2} - \frac{({\sum}{x})^2}{n} = 632533 - 598980.08333333 = 33552.916666667 $$

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. $$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1} = \frac{ 33552.916666667 }{11} = 3050.2651515151$$

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.

$$ {s} = \sqrt{3050.2651515151} = 55.2292$$

The standard-deviation calculator below calculates the standard deviation for a sample or a population.