*square*inches. This is because you square the deviations before you average them. So the variance in the five-score population in the preceding example is 6.8 square inches.

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 the`sd() `

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`