Least Squares

Another Fit

Same Machinery, New Data

A second dataset runs through the same exact machinery. The negative slope and zero-sum residual table are recomputed from the displayed values, then compared with a coherent trial line.

highlighted = computed this step

Another dataset

The second dataset has sum x=6, sum y=14, sum x times x=14, and sum x times y=15.

[461461415]\begin{bmatrix}4&6&14\\6&14&15\end{bmatrix}
Second dataset normal equationsExact sums from the second dataset assemble this system.Second dataset system461461415

Same solve machinery

Solving gives a=53/10 and b=-6/5. The negative slope comes from the displayed data.

a=53/10,b=−6/5a=53/10,\quad b=-6/5
Second dataset exact solveThe displayed system recomputes the negative-slope solution.Second dataset solve461461415solution: 53/10, -6/5

Fit and residuals

The fit residuals are -3/10, -1/10, 11/10, and -7/10. Their sum is 0 and SSE is 9/5.

∑r=0,SSE=9/5\sum r=0,\quad SSE=9/5
Second dataset fitThe negative-slope fit and residual table share the same exact data.fitABCDVisible residual certificatexyfitresid0553/10-3/101441/10-1/102429/1011/103117/10-7/10sum residuals = 0

Zero-sum is not enough

The trial line y=5 minus 1x has residuals 0, 0, 1, and -1. The sum is 0, but SSE is 2, which is greater than 9/5.

∑r=0,2>9/5\sum r=0,\quad 2>9/5
Zero-sum trial is not enoughA generic trial table can be coherent without being the least-squares fit.Trial residualsxyfitresid055014402431312-1sum residuals = 0

Summary

The same procedure re-applies to new shown data. This is a machinery check, not a capability claim about the model.

same procedure, new shown data\text{same procedure, new shown data}
Second dataset fitThe negative-slope fit and residual table share the same exact data.fitABCDVisible residual certificatexyfitresid0553/10-3/101441/10-1/102429/1011/103117/10-7/10sum residuals = 0