Calc Sum of Squares
Sum of Squares is easy to calculate. Plus it’s one of the most useful measures of dispersion. Here’s a video and then another example:
Hi, this is Ken Tangen with a note about Sum of Squares.
Sum of Squares (SS) is used to find out how similar or dispersed a groups of scores is.
You may have seen people use the deviation method for calculating the Sum of Squares. They subtract the mean from each score, square the deviations and add them up. But no one actually calculates SS that way.
The deviation method is for teaching the concept of dispersion. After subtracting the mean from each score in a long list, you get the idea that this score deviates from the mean, this score deviates from the mean…until you’re really tired of the process.
Dispersion is how much scores deviate from the center (the mean). It is a measure of group heterogeneity. The more different the scores are within a group, the larger they disperse from the mean. So dispersion is how many scores deviate from the mean and by how much.
But once you have the idea that SS is a measure of dispersion, move on to the Raw Score Method. It’s soooooo much easier to calculate.
Let me give you the formula and then walk you through it. There are only 4 things to calculate and only 5 steps to the whole thing.
Let’s get the numbers first and then plug them into the formula.
First, count how many raw scores you have.
Second, add up your raw scores. That is, take the sum of the X’s.
Third, square the number you just calculated.
Fourth, make a new column and fill it by squaring each number in the first column.
Fifth, add up this new column.
Now we have all the numbers. Let’s plug them into the formula.
The Sum of Square equals the sum of X-squared (column 2) minus the sum of column one squared divided by the number of people in our study. It sounds worse than it is.
In math rules, we square before we divide, and we divide before we subtract. The result is the sum of squares.
This is much easier than the deviation method. You don’t have to calculate the mean or do all of the subtraction. And if you have 100 or 1000 scores, that’s a lot of time. Relax. Use the raw score method.
For more examples and practice problems, go to StatNut.com.
Let me give you the formula and then talk you through the process. Here’s the formula:
Like range, variance and standard deviation, Sum of Squares (SS for short) is a measure of dispersion. The more inconsistent the scores are (less homogeneous) the larger the dispersion. The more homogenous the scores (alike), the smaller the dispersion.
Using the formula above, let’s go through it, step by step Assume this is the distribution at issue:
First, each number is squared, and put into another column:
Second, we sum each column. The sum of the first column is 45. This is called the sum of X.
The sum of the second column is the sum of X-squared. Remember, we squred the scores and then added them up. The sum of the squared-X’s is 355.
Third, we square the sum of X (45 times itself = 2025) and divide it by N (number of scores).
Since N = 7, we divide 2025 by 7 (which equals 289.29).
Fourth, we recall the sum of the X2 and subtract 240.67 from it. So 355 minus 289.29 = 65.71. The Sum of Squares is 65.71.