Sum of Squares: Definition & Application
In many situations, it is important to know how much variation there is in a set of measurements. One way to quantify this is to calculate the sum of squares. In this lesson, we’ll learn how to calculate sum of squares and how to interpret the results.
What is Sum of Squares?
Rachel is a nurse at City Hospital and is closely monitoring two patients. Their doctors have asked Rachel to check the patients’ blood oxygen concentration levels every hour to make sure they don’t vary too much from hour to hour. How will Rachel be able to tell if the oxygen concentration of these patients changes too much? One way that this could be done is by calculating the sum of squares of the data.
In statistics, the sum of squares measures how far individual measurements are from the mean. To calculate the sum of squares, subtract each measurement from the mean, square the difference, and then add up (sum) all the resulting measurements. We’ll look at this in a little more detail later.
definition of sum of squares
The sum of squares is also sometimes known as variation, because it measures the amount of variability in the data. If Rachel measured the oxygen concentration of both patients every hour, she could tell if the oxygen concentration was varying too much by looking at the sum of squares. A high sum of squares would indicate a lot of variability in the data, while a low sum of squares (most of the measurements close to the mean) would indicate a low amount of variability.
Calculating the Sum of Squares
To find the sum of squares for a set of data, first find the mean by adding all the measurements and then dividing by the total number of measurements in the data set. Data on the oxygen concentration of the two patients during a 12 hour period is shown below, including the average or mean for each patient.
oxygen concentration data table
Did you notice that the two patients had the same mean blood oxygen concentration? Does this mean that they had the same amount of variability in oxygen concentration during this time period? Definitely not! To determine the variation in the data for each patient, you’ll have to calculate the sum of squares.
To do this, subtract each measurement from the mean to find the difference from the mean. Then, square all these differences and add them up to find the total sum of squares.
Let’s look at how to calculate sum of squares using the data gathered from the two patients. Remember that x represents each measurement, while xbar represents the mean. So, (x-xbar) is the difference between the measurement and the mean, which is 93 for both patients.
sum of squares data table
Wow! There is a big difference between the patients. The sum of squares showed that the variation for Patient #1 was 238, while the variation for Patient #2 was only 28. This is a big difference, even though both had the same mean. Calculating the sum of squares indicates that Rachel should monitor Patient #1 a little more closely, because his oxygen concentration is varying a lot from hour to hour. Patient #2 is much more stable, with a variation of only 28.
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