Calculating Variance by the Raw Score Method

When we actually calculate the variance for a set of scores, we generally use the raw score method. This is also the method used by computers and calculators to find the variance. The formula for the variance by the raw score method is mathematically equivalent to the deviation score method. Although we will not do it now, it is possible to derive the raw score formula from the deviation score formula.

The raw score formula for the variance is:

We can see from this formula that to find the population variance we sum up the squared individual scores and subtract from them the sum of the scores quantity squared divided by the number of scores . The variance then is this quantity divided by the number of scores. We can see that the only data we need to make this calculation are the sum of the scores, the sum of the squared scores, and the number of scores. The worksheet below shows these quantities.

Worksheet for Calculating the Variance for 7 scores

5	25
3	9
4	16
4	16
3	9
4	16
5	25
28	116

The quantity recorded at the bottom of the first column is the sum of the scores and the quantity at the bottom of the second column is the sum of the squared scores . The number of scores can be obtained by counting the number of scores in the first column. We can put these quantities into the formula for population variance to obtain 0.57, the same answer as we got with the deviation score method.

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 raw score method. That formula is

There are three differences between the formula for the sample variance and the formula for the population variance

1. The symbol for the sample variance is s squared ()
2. The symbol for the number of scores is n rather than N
3. The denominator of the equation uses the sample size -1 () rather than N.

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).

It is good to understand how we calculate variance, but in actual practice we will use a computer running the Excel spreadsheet program to calculate the variance. Before we consider how this is done let us consider another measure of variability, the standard deviation.