It might be hard to grasp what that means. Often, it's more intuitive if the variation statistic is in the same units as the original measurements. It's easy to turn variance into that kind of statistic. All you have to do is take the square root of the variance.
Like the variance, this square root is so important that it is has a special name: standard deviation.
Population standard deviation
The standard deviation of a population is the square root of the population variance. The symbol for the population standard deviation is Σ (sigma). Its formula is
For this 5-score population of measurements (in inches):
50, 47, 52, 46, and 45
the population variance is 6.8 square inches, and the population standard deviation is 2.61 inches (rounded off).
Sample standard deviation
The standard deviation of a sample — an estimate of the standard deviation of a population — is the square root of the sample variance. Its symbol is s and its formula is
For this sample of measurements (in inches):
50, 47, 52, 46, and 45
the estimated population variance is 8.4 square inches, and the estimated population standard deviation is 2.92 inches (rounded off).
Using R to compute standard deviation
As is the case with variance, using R to compute the standard deviation is easy: You use thesd() function. And like its variance counterpart, sd() calculates s, not Σ:> sd(heights)
[1] 2.915476
For Σ — treating the five numbers as a self-contained population, in other words — you have to multiply the sd() result by the square root of (N-1)/N:
> sd(heights)*(sqrt((length(heights)-1)/length(heights)))
[1] 2.607681
Again, if you're going to use this one frequently, defining a function is a good idea:
sd.p=function(x){sd(x)*sqrt((length(x)-1)/length(x))}
And here's how you use this function:
> sd.p(heights)
[1] 2.607681
