Even More About Sum of Squares (SS)

Like all measures of dispersion, Sum of Squares (SS) gets larger when the distribution of scores is more dissimilar (heterogeneous). The more homogeneous (similar) the scores, the smaller the distribution.

Deviation Method

Conceptually, Sum of Squares is an extension of mean variance. Instead of taking the absolute values of the deviations, we square the critters. For example:

            X         mean         d          d²
            7            5            2           4
            5            5            0           0         
            5            5            0           0         
            5            5            0           0
            3            5          -2           4 

The sum of the squared deviations is called Sum of Squares. In this example, the sum of the squared deviations is 8. The Sum of Squares (SS for short) is the sum of the squared deviations. Like all measures of dispersion, the larger the number, the more dispersed the distribution of raw scores. The smaller the SS, the less dispersed the scores are.

This “deviation method” of calculating Sum of Squares is illustrate that it is a measure of dispersion from the mean. After using this method several times, it should be clear that the Sum of Squares is the sum of the squared deviations. It is a measure of squared deviations from the mean.

Once this concept is clear, you’ll be ready to know the secret: there is an easier way to calculate Sum of Squares.

Raw Score Method

The problem with the deviation method is clearest when the mean is not an integer. When the mean is 5, it’s not hard to subtract it from every score. When the mean 5.387, it is difficult to know how many places to carry out each of the sub-answers. It’s not impossible to do; it’s just a pain.
It seems like some mathematician with nothing better to do must have come up with a easier way to calculate Sum of Squares. And, in truth, there is an easier way.

The raw score method only uses the raw scores; there are no deviations to calculate. Here is the formula we use:

Here’s how the process works. Assume this is the distribution at issue:

              X
            11
              7
              3
              4
              5
              8

First, each number is squared:

             X           X²
            11        121
              7          49
              3            9
              4          16
              5          25
              8          64

Now, we sum each column. The sum of the first column is 38 (it’s called the Sum of X). And the sum of the second column is 284; this number is called the Sum of the X-squareds.

Next, we square the sum of X (38 times itself = 1444) and divide it by N. Since N = 6, we divide 1444 by 6 (which equals 240.67).

Fourth, we recall the sum of the X² and subtract 240.67 from it. So 284 minus 240.67 = 43.33. The Sums of Squares is 43.33.

Both Ways At Once

Let’s compare the two methods. They will produce the same results but the raw score method is much easier to calculate.

             X           X²         d          d²
            11        121      4.67    21.78
             7           49        .67        .44
             3            9    -3.33     11.11
             4          16    -2.33       6.44
             5          25    -1.33       1.78

Sum    38        284      0          43.33
N = 6
Mean = 6.66
Sum of Squares    44.33         43.33

NOW YOU CHOOSE:
    Day 3: Dispersion
    A Bit More About Dispersion
    Even More About Dispersion
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    How To Calculate
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    Formulas For Dispersion
    Practice Problems
    More Practice Problems
    Basic Facts About Dispersion
    Vocabulary
    Quiz 3
    Summary