The standard-deviation calculator below calculates the standard deviation for a sample or a population.
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 10 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, ${x^2}$.
$x$ | $x^2$ |
132 | 17424 |
218 | 47524 |
217 | 47089 |
129 | 16641 |
162 | 26244 |
138 | 19044 |
144 | 20736 |
171 | 29241 |
108 | 11664 |
3. Find the sum of all the values in the first column, ${\sum}{x}$.
$$ \sum{x} = 1419 $$4. Square the answer from step 3, then divide that number by the size of the sample.
$$ \frac{({\sum}{x})^2}{n} = \frac{2013561}{9} = 223729 $$5. Find the sum of all the values in the second column, ${\sum}{x^2}$.
$$ {\sum}{x^2} = 235607 $$6. Subtract the answer in step 4 from the answer in step 5.
$$ {\sum}{x^2} - \frac{({\sum}{x})^2}{n} = 235607 - 223729 = 11878 $$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. $$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1} = \frac{ 11878 }{8} = 1484.75$$
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.
$$ {s} = \sqrt{1484.75} = 38.5325$$