Bias-Variance Curve Calculator

Result summary

Simulation mode

Recommended complexity

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Minimum validation error

Minimum expected error

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Bias² + variance + noise

Generalisation gap

—

Validation minus training error

Selected model condition

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Choose a complexity point

Interactive bias-variance graph

Data source and model setup

Data mode

Complexity measure

Custom axis title

Complexity direction

Optimise by

Selected complexity

Simulation controls

Minimum complexity

Maximum complexity

Complexity points

Initial bias

Bias decay rate

Initial variance

Variance growth rate

Irreducible noise

Training-data noise

Training sample size

Random seed

Curve smoothness

Higher values reduce simulated jitter.

Underfitting strength

Overfitting strength

Base standard deviation

Manual data entry

Use columns: complexity, bias, variance, training, validation, test, standard deviation. Headers are optional.

Complexity	Bias	Variance	Training	Validation	Test	Std.	Remove

Upload CSV

Load example

Validation and experiment settings

Training split (%)

Validation split (%)

Cross-validation folds

Repeated runs

Confidence level

Interval method

Feature-noise level

Outliers (%)

Regularisation strength

Detailed results table

The optimal row is highlighted.

Complexity	Bias	Bias²	Variance	Noise	Expected	Training	Validation	Test	Gap	Condition

Model comparison

Add current models to compare their validation curves.

Model	Optimal complexity	Minimum validation	Minimum expected	Generalisation gap	Ranking score
No comparison models added.

Formula used

Expected Prediction Error = Bias² + Variance + Irreducible Noise

Bias² measures systematic error from restrictive assumptions.
Variance measures sensitivity to different training samples.
Irreducible noise represents randomness no model can remove.
Generalisation gap equals validation error minus training error.

How to use

Choose simulation mode or enter measured experiment results.
Select the model complexity measure and optimisation target.
Adjust noise, sample size, validation, and confidence settings.
Generate the curves and inspect the recommended complexity.
Select any point to review its fitting condition.
Add models for comparison, then export the analysis.

Example data

Complexity	Bias	Variance	Training	Validation	Test	Std.
1	1.12	0.08	0.62	1.43	1.47	0.08
4	0.47	0.22	0.27	0.61	0.64	0.06
6	0.25	0.45	0.18	0.59	0.62	0.08
9	0.09	1.18	0.10	1.28	1.34	0.16

Interpretation guide

Underfitting region

Bias is high and predictions remain too simple. Training and validation errors can both stay elevated.

Balanced region

Bias and variance remain reasonably controlled. Validation error usually reaches its lowest practical level.

Overfitting region

Variance and the generalisation gap rise. Training improves while unseen-data performance becomes worse.

Frequently asked questions

What is the bias-variance trade-off?

It describes tension between simple stable models and flexible sensitive models. Good complexity controls both error sources.

Why is validation error often U-shaped?

Bias dominates at low complexity. Variance dominates after complexity becomes excessive.

Does bias always decrease with complexity?

It usually decreases in teaching examples. Real measurements can contain irregular movements.

How does sample size affect variance?

Larger datasets usually stabilise fitted parameters. This can reduce variance and confidence-band width.

What does irreducible noise mean?

It is unexplained randomness in outcomes. Better model complexity cannot completely remove it.

Which error should determine the optimum?

Validation error is normally preferred during tuning. Test error should support final evaluation.

Can I use cross-validation results?

Yes. Enter mean errors and their standard deviations for every complexity point.

How does regularisation change the curves?

Stronger regularisation can increase bias. It often reduces variance and limits overfitting.

What is the bias-variance crossover?

It is the point where bias squared and variance are closest. It is informative but not always optimal.

Can multiple models be compared?

Yes. Add each current curve to the comparison panel and review their rankings.