A model's expected error on new data splits cleanly into two parts that pull in opposite directions:

• Bias — error from the model being too rigid to capture the truth. Simple models have high bias.
• Variance — error from the model being so sensitive that it changes wildly with each new training set. Flexible models have high variance.

As you crank up complexity, bias falls but variance rises. Total test error is their sum (plus unavoidable noise), so it traces a U-shape — and the best model sits at the bottom of the U.

The U-shaped curve

Watch the three curves as complexity grows. Bias² slides down, variance climbs up, and their sum — the total error — dips to a minimum and then rises again. Move the marker to the bottom of the total curve: that's the complexity you want.

Steering the tradeoff

Every knob in machine learning is, secretly, a bias–variance knob: a tree's depth, a polynomial's degree, the strength of regularization, the size of a network. More data is the rare free lunch — it lowers variance without raising bias, shifting the whole sweet spot toward more powerful models. Knowing which way a change pushes you is half of practical ML.