I have a simple set of 10 data points
My ten Data Points
2
3
3
4
5
8
9
11
11
13
(mean = 6.9)
My prediction nodel predicts the following values for the ten data points (listed in same order)
2
3
4
5
6
7
9
10
11
12
I calculate
SST = SIGMA (VALUE - MEAN)^2 = 142.9
SSM = SIGMA (MODEL - MEAN)^2 = 108.9
SSE = SIGMA (VALUE - MODEL)^2 = 6
I thought SSM + SSE = SST, but
SSM + SSE = 114.9 which doesn't equal 142.9
Where am I going wrong?
40
To understand where you might be going wrong, let's break down the calculations for the sum of squares.
The sum of squares total (SST) measures the total variability in the data. It is calculated by taking the sum of the squared differences between each data point and the mean.
In your case, SST is calculated as:
SST = (2 - 6.9)^2 + (3 - 6.9)^2 + (3 - 6.9)^2 + (4 - 6.9)^2 + (5 - 6.9)^2 +
(8 - 6.9)^2 + (9 - 6.9)^2 + (11 - 6.9)^2 + (11 - 6.9)^2 + (13 - 6.9)^2
= 55.59 + 15.21 + 15.21 + 9.61 + 3.61 + 1.21 + 5.29 + 18.89 + 18.89 + 39.69
= 142.9
The sum of squares model (SSM) measures the variability explained by the prediction model. It is calculated by taking the sum of the squared differences between each predicted value and the mean.
In your case, SSM is calculated as:
SSM = (2 - 6.9)^2 + (3 - 6.9)^2 + (4 - 6.9)^2 + (5 - 6.9)^2 + (6 - 6.9)^2 +
(7 - 6.9)^2 + (9 - 6.9)^2 + (10 - 6.9)^2 + (11 - 6.9)^2 + (12 - 6.9)^2
= 21.16 + 15.21 + 9.61 + 3.61 + 0.81 + 0.01 + 5.29 + 5.29 + 18.89 + 18.89
= 108.9
The sum of squares error (SSE) measures the remaining unexplained variability. It is calculated by taking the sum of the squared differences between each data point and its corresponding predicted value.
In your case, SSE is calculated as:
SSE = (2 - 2)^2 + (3 - 3)^2 + (3 - 4)^2 + (4 - 5)^2 + (5 - 6)^2 +
(8 - 7)^2 + (9 - 9)^2 + (11 - 10)^2 + (11 - 11)^2 + (13 - 12)^2
= 0 + 0 + 1 + 1 + 1 + 1 + 0 + 1 + 0 + 1
= 6
So, based on your calculations, SST = 142.9, SSM = 108.9, and SSE = 6. If we add up SSM and SSE, we get 108.9 + 6 = 114.9, which is indeed not equal to SST.
The reason SSM + SSE does not equal SST is likely due to rounding errors in the calculations. The actual values might be slightly different from what you've provided. I would recommend re-checking your calculations and make sure there are no rounding errors or mistakes in your input values.