How to Calculate Standard Deviation by Hand
- 1). Look at your data to figure out what exactly you are calculating. For example, if you are looking at the standard deviation of how student in a class scored on a test, the individual test scores are what you are looking at. They are the Xi, or individual values of the variable in question.
- 2
Create a table with 4 columns and label each variable in an individual row in the first column. For the example illustrated above, in the first cell of each row, list one of the student scores, as shown in the attached photo. - 3
Find the mean, or average, of your variables. This is done by adding up all of the individual values and dividing by number of observations you have. For example, the 11 values in the example add to 855. That would leave you with a mean of 77.73. Place this value in each cell in the second column, as indicated in the picture below. - 4
Subtract each observation from the mean. For example, the first student scored an 82. If you take that score and subtract the average of 77.73, you are left with 4.27. This is how much the individual observation, in this case the student, varied or deviated from the mean. - 5
Take each individual deviant and square it. For example, for the first observation your would take 4.27^2 or 4.27 * 4.27, which gives you 18.26. For observations that are far away from the mean, then you will have a very high result. Similarly, by squaring the results, all of your figures will become positive. - 6). Add up all of the figures in the final column. This is what the large ¦² means: add up the difference between each observation and the mean, squared. For the example below, the sum should equal 2566.18
- 7). Divide that number by one minus the total number of observations. For example, in this class there are 11 students. Thus, 11 students - 1 would give us 10. Then 2566.18 / 10 would equal 256.618. This gives you the variance-- an important statistical measure.
- 8). Find the square root of the variance, that you found above. This should give you a deviation of 16.019 for the example we used.
- 9). Interpret the results. The majority of the results are 1 standard deviation above or below the mean. So for this example, the mean is 77.73 and the majority of results will fall between 77.73-16.02 and 77.73+16.02 or between 61.71 and 93.75. Look at the data to see if it makes sense. It does-- there is only one observation that does not fall within one standard deviation.
