?embed=1 for iframe mode These points show floor area versus price for Western Sydney homes. Instead of a clean straight-line trend, the cloud bends upward and spreads out, hinting that a simple linear model will struggle.
What pattern do you see in the scatter?
A straight regression line is laid over the data. It cuts through the middle but misses the curvature, so its predictions are biased in the low and high ends.
How does the linear fit summarise the curved pattern?
A quadratic curve bends upward to follow the middle of the cloud. It captures the lift in mid-range floor areas that a straight line ignores.
How does the quadratic compare to the linear fit?
A cubic curve adds another bend. Here it only tweaks the shape slightly, which might not justify the extra complexity.
See the linear, quadratic, and cubic fits together. The quadratic hugs the data best; the linear underfits; the cubic adds tiny wiggles.
Which model explains the curved pattern best here?
To predict price from floor area, trace along the quadratic curve. Predictions come from the curve, not just an average.
Where do these predictions come from?
Residuals from the linear model show a clear curve: the line underestimates mid-range prices and overestimates at the edges.
What pattern do you see in the linear residuals?
Residuals from the quadratic model look more random, signalling that the curve has removed most of the systematic bend.
How do the quadratic residuals look compared to linear?
Compare mean squared error for linear, quadratic, and cubic fits. Lower MSE indicates tighter predictions.
Which model has the lowest MSE here?
R² increases when curvature explains more variance. Compare how much variance each model explains.
How does adding curvature affect R²?
The cubic curve starts to chase noise at the edges. Small wiggles signal overfitting when the data do not support the extra bend.
This dataset climbs and then flattens toward a ceiling. A linear fit would miss the plateau and overshoot.
What pattern does this dataset suggest?
A polynomial fit on this saturation data oscillates near the ends, showing instability when extrapolating.
How does the polynomial behave on the saturation data?
A logistic-shaped curve rises and then levels off. This shape better matches the ceiling behaviour than a high-degree polynomial.
Comparing the polynomial to the logistic concept shows why polynomials cannot robustly model saturation without instability.
Which model is conceptually better for saturation?
Polynomial regression can capture curvature and improve fit, but overfitting and instability are risks. For saturation patterns, a logistic model is often more appropriate.