At the beginning of this lesson we said that the standard deviation is the square root of the average squared deviation of the scores from the mean.
If we look at the formula for the standard deviation for a population using the deviation score method we will see that this is true.
The formula tells us to take a score and subtract the mean from it and square the difference. We then sum up all of these squared differences and divide by the number of scores. Finally we take the square root of the product and this is the standard deviation.
Let's do a problem in finding the standard deviation using the same data we used earlier to find the variance.
|
|
|
|---|---|---|
| 5 | 1 | 1 |
| 3 | -1 | 1 |
| 4 | 0 | 0 |
| 4 | 0 | 0 |
| 3 | -1 | 1 |
| 4 | 0 | 0 |
| 5 | 1 | 1 |
| 28 | 4 |
The sum of squared deviation scores is found at the bottom of the third column and this is the quantity we need (as well as N) to find the standard deviation. So using the formula we find the standard deviation to be 0.755
If we had already calculated the variance, we could find the standard deviation by taking the square root of the variance, since the standard deviation is the square root of the variance.
If our data is considered a sample instead of a population we must use the formula for the standard deviation of a sample using the deviation score method. This differs from the population formula by substituting n-1 for N in the formula. So for our example the sample standard deviation by the deviation score method is 0.816
We can also find the sample standard deviation directly from the variance if we have already calculated the sample variance.
I hope this discussion of the deviation score method for finding the standard deviation for a population and for a sample gives you a better understanding of what the standard deviation really is. In actuality however, the standard deviation is usually calculated using the raw score method, so let's turn to that topic now.