The term as "Sβ" appered in the blog QE8 last week. Since ST is the sum of squares of the data itself, and Se is the sum of squares of the errors, it is easy to have an image. Then, do you think what is Sβ? Can you imagine what is the sum of squares of the change in slope? Then, I tried various trial and error using Excel. Please see the following materials.
Click here for material → Sβ
When the signals M1, M2 and M3 are 1,2 and 3, I calculate Sβ, Se, Sβ + Se, β and β2 changing y1, y2 and y3 as shown in rows 4 to 9 of columns A to C in the table. A plot of y against the signal M is shown in the lower left. The slope of this straight line is β. The middle figure is a plot of β2 vs. Sβ. I thought it would be a straight line, but the part where β is small is a curve. The formula of ST = Sβ + Se is works. I plot with ST on the horizontal axis and Sβ + Se on the vertical axis. It's almost linear. Although the image of Sβ has not been grasped yet, if β is large to some extent, it seems to correlate with β2 and ST = Sβ + Se holds. I think that the error fluctuation affect to ST in the part where β is small.
Anyway, if you don't know what it means in Quality engineering and other study, I recomend you try to put in a value,to calculate and to plot graph.