Where you would use it
R²=0.70 means 70% of target variance relative to the mean baseline is explained on that evaluated sample.
Use when
Use as a relative goodness-of-fit summary alongside absolute error.
9 · Evaluation, Metrics & Diagnostics · Regression Metrics & Residuals
R² and adjusted R²
R² compares residual variance with a mean baseline. A good R² does not guarantee unbiased or well-calibrated predictions; adjusted R² is mainly a classical linear-model diagnostic. The important practical question is not only how the technique is defined, but what assumptions it introduces, which data are allowed to influence it, and how its effect should be validated on unseen evidence.
What this concept means in practice
R² compares residual variance with a mean baseline. A good R² does not guarantee unbiased or well-calibrated predictions; adjusted R² is mainly a classical linear-model diagnostic. The important practical question is not only how the technique is defined, but what assumptions it introduces, which data are allowed to influence it, and how its effect should be validated on unseen evidence.
The practical value of R² and adjusted R² comes from understanding both the transformation and the boundary around it: what information is allowed to enter, what assumption is being made, and how you know the result is still valid after the transformation.
A beginner-friendly way to reason about it is to start with a tiny case where the correct result can be checked independently. Once the mechanism is clear, scale the exact same reasoning to larger tables, pipelines or models.
PurposeUse as a relative goodness-of-fit summary alongside absolute error.
MechanismCompare residual sum of squares with the variance of a mean-only baseline. Adjusted R² adds a penalty for predictor count in classical linear models.
EvidenceInspect intermediate and final output; compare with an independent expectation.
Main cautionNegative test R² is possible; high R² does not prove calibration, causality or low bias.
Trace the operation from input to decision
Compare residual sum of squares with the variance of a mean-only baseline. Adjusted R² adds a penalty for predictor count in classical linear models.
1Input→
2Apply rule→
3Inspect state→
4Validate→
5Use result
Key rule
R² = 1 - SSE/SST
Visual explanation
Make the structure visible
The interactive view uses a concept-specific plot when the topic maps naturally to one; otherwise it uses a workflow view instead of leaving a broken placeholder.
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What can make the result misleading
Watch out
Negative test R² is possible; high R² does not prove calibration, causality or low bias.
A useful diagnostic question is: Could the same code still run successfully if the analytical assumption were wrong? If yes, add an explicit validation check rather than relying on execution success.
Implementation
PythonMiniature Python example
Keep the example small enough that you can inspect each stage manually.
# Purpose: demonstrate R² and adjusted R² with a small, inspectable example.
# Follow the comments and printed stages to connect each operation with its result.
# Import the library or helper used in this example.
# Step 1 — Import the module so its functions/classes are available to the rest of this example.
import numpy as np
# Import the library or helper used in this example.
# Step 2 — Import only the named objects needed by the following steps, keeping dependencies explicit.
from sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score
# Create the numerical values used in the calculation.
# Step 3 — Construct `y` as an array so vectorised numerical operations can be applied consistently.
y = np.array([20,24,27,31,35,39,42,47,51,56,60,65],dtype=float)
# Step 4 — Construct `p` as an array so vectorised numerical operations can be applied consistently.
p = np.array([21,23,29,30,34,41,40,48,54,55,59,67],dtype=float)
# Store this intermediate value with a descriptive name for the next step.
# Step 5 — Compute the right-hand expression and store its result in `residual` for the next step.
residual = y-p
# Print this intermediate result so you can verify the workflow step by step.
# Step 6 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 1 · Mean residual:", round(residual.mean(),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 7 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · MAE:", round(mean_absolute_error(y,p),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 8 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · RMSE:", round(mean_squared_error(y,p)**0.5,3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 9 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · R2:", round(r2_score(y,p),3))
# Print this intermediate result so you can verify the workflow step by step.
# Step 10 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · Largest absolute error:", round(np.abs(residual).max(),3))Expected / illustrative output
STEP 1 · Mean residual: -0.333 STEP 2 · MAE: 1.5 STEP 2 · RMSE: 1.633 STEP 2 · R2: 0.986 STEP 3 · Largest absolute error: 3.0
Before you move on
- Can you state what data or object enters the operation?
- Can you explain what changes and what must remain invariant?
- Have you checked the result on a tiny case you can verify independently?
- Have you considered the main failure mode described above?
- Can the operation be reproduced from code/formulas and documented assumptions?