(sigma squared). The formula says to take a score
(
)
and subtract the mean (
), then square this difference
(
) and sum up all of these squared differences
(
), and divide the sum by the number of scores
(
).Let us first consider the deviation score method for calculating the variance for a population of scores. Although in practice the deviation score is rarely used, it does help us understand how variance is calculated. It follows our previous definition of variance and could thus be referred to as the definitional method of calculating variance.
The formula for the variance for a population using the deviation score method is as follows:
We can see from the formula that the symbol for the population variance
is (sigma squared). The formula says to take a score
()
and subtract the mean (), then square this difference
() and sum up all of these squared differences
(), and divide the sum by the number of scores
().
We can see that the formula truly does say what the definition of variance says - variance is the average squared deviation of the scores from the mean.
We will use this formula to find the variance for the seven scores found in the following worksheet. In the first column we have the seven scores. At the bottom of the column is the sum of the scores and this can be used to find the mean. The mean = the sum of the scores divided by the number of scores or 28/7 = 4. In the second column is recorded the value for the score minus the mean and in the third column is recorded this value squared, that is the square of the score minus the mean. At the bottom of the third column is the sum of the squared deviations of the scores from the mean and this is the quantity we need to calculate 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 |
For this problem the population variance is 0.57
If we were to consider the scores above as a sample rather than a population, we would use the formula for the variance for a sample using the deviation score method. That formula is
The are three differences between the formula for the sample variance and the formula for the population variance
)
)
) rather than N. This has to do with
a statistical concept regarding degrees of freedom. A sample has one less
degree of freedom than a population so for a population we divide by N, while
for a sample we divide by n-1.
Thus we can see that for the sample variance for our problem above we simply divide by n-1 rather than N. For our problem then, the sample variance is 0.667 (slightly larger than the population variance).
I hope that this discussion of variance, using the deviation score method, gives you a good understanding of variance even if you will not be using this formula in the actual computation of variance. Actual computations generally use the raw score formula and we will turn to that next.