Edited by John Healy
You use your calculator or spreadsheet, or even a statistics software package to calculate standard deviation, which is an estimate of the population standard deviation. Yet, understanding how one could calculate standard deviation without such advanced tools may prove useful. The knowledge of basic sum of squares methods provides a foundation for ANOVA and DOE analysis techniques.
If nothing else, this little bit of historical knowledge may enhance the reputation of those that did these calculations by hand or with mechanical adders and slide rules. Statisticians in the past had to be resourceful individuals, just to accomplish the calculations we take for granted today.
Recall that the formula for the sample standard deviation is
$$ \large\displaystyle s=\sqrt{\frac{\sum\nolimits_{i-1}^{n}{{{\left( {{x}_{i}}-\bar{X} \right)}^{2}}}}{n-1}}$$
Where xi is the data, XÌ„ is the data average, and n is the number of data points.
Let’s say we have 12 points of data and with to calculate the standard deviation.
| Sample | x | X-bar | (X-XÌ„) | (X-XÌ„)^2 |
| 1 | 244 | 324 | -80 | 6400 |
| 2 | 322 | 324 | -2 | 4 |
| 3 | 391 | 324 | 67 | 4489 |
| 4 | 313 | 324 | -11 | 121 |
| 5 | 337 | 324 | 13 | 169 |
| 6 | 321 | 324 | -3 | 9 |
| 7 | 276 | 324 | -48 | 2304 |
| 8 | 299 | 324 | -25 | 625 |
| 9 | 343 | 324 | 19 | 361 |
| 10 | 333 | 324 | 9 | 81 |
| 11 | 383 | 324 | 59 | 3481 |
| 12 | 327 | 324 | 3 | 9 |
1. Calculate the average (XÌ„ or mean)
$$ \large\displaystyle \bar{X}=\frac{\sum\nolimits_{i=1}^{n}{{{x}_{i}}}}{n}=\frac{2190}{12}=324$$
2. Compute the deviation between xi and X̄
$$ \large\displaystyle ({{x}_{i}}-\bar{X})$$
3. Square each deviation
$$ \large\displaystyle {{({{x}_{i}}-\bar{X})}^{2}}$$
4. Sum the squares of the deviations
$$ \large\displaystyle \sum\nolimits_{i=1}^{n}{{{({{x}_{i}}-\bar{X})}^{2}}}$$
5. Calculate the sample standard deviation
$$ \large\displaystyle s=\sqrt{\frac{\sum\nolimits_{i-1}^{n}{{{\left( {{x}_{i}}-\bar{X} \right)}^{2}}}}{n-1}}=\sqrt{\frac{18053}{11}}=40.5$$
Of course if you have the full dataset of the population, you can calculate the population standard deviation using the same method, just do not subtract 1 from n in the denominator of step 5.
In summary, use your calculator and use n-1 in the denominator when calculating the sample standard deviation.
Related:
Central Limit Theorem (article)
Point and Interval Estimates (article)
8 Steps to creating an X-bar and s control chart (article)
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