Claimed and OLS lines compete against the same residual evidenceUpdates with every input
Math & Statistics
Regression Verification Calculator
Audit a claimed regression slope and intercept against five entered observations by checking residual balance, x-weighted orthogonality, centroid passage, SSE difference from the true OLS line, and coefficient tolerances.
Claim verdict -
Rebuilt OLS equation -
Slope residual -
Intercept residual -
Claimed residual sum -
Claimed x-residual sum -
Claimed SSE penalty -
Centroid miss -
CLAIMED-LINE AUDIT
Claimed and OLS lines compete against the same residual evidence
Two lines share one scatterplot while a normal-equation compass shows residual balance and x-weighted orthogonality.
VERIFICATION LEDGER
Point-level residuals under claimed and optimal lines
The same five observations expose whether the claimed coefficients merely round the OLS solution or define a materially worse line.
| Point | x | Observed y | Claimed fit | Claimed residual | OLS fit | OLS residual |
|---|
CLAIM AUDIT
Do not refit the claim before recording it
- Enter the five source observations exactly.
- Transcribe the claimed slope and intercept as published.
- Set coefficient tolerance from reported rounding precision.
- Compare the rebuilt OLS coefficients independently.
- Use normal-equation residuals and SSE penalty to diagnose any mismatch.
OPTIMALITY EVIDENCE
Coefficient closeness and least-squares optimality are related but distinct
A rounded coefficient pair may miss the exact normal equations slightly while remaining within an explicit tolerance. A copied or transposed coefficient can produce a large SSE penalty.
Residual balance identities require an intercept. They are arithmetic properties of OLS, not proof that a straight line is scientifically appropriate.
NORMAL-EQUATION AUDIT
A valid intercept OLS line satisfies two residual identities
The independent OLS line is rebuilt from data. A claimed line is compared by coefficient residuals, SSE, centroid passage, residual sum, and x-weighted residual sum.
Detailed calculation process and general formulas
e_i(claim) = y_i - (b0_claim + b1_claim x_i)sum e_i(OLS) = 0sum x_i e_i(OLS) = 0b1_OLS = Sxy/SxxSSE penalty = SSE_claim - SSE_OLSSymbols, meanings, and units
- b1_claim
- claimed slope under audity per x
- b0_claim
- claimed intercept under audity units
- e_i
- signed vertical residualy units
- SSE penalty
- extra squared error versus OLSy units squared
- tau
- entered coefficient tolerancecoefficient units
AUDIT SIGNATURES
Four ways a claimed line can fail
The pattern of failures often identifies the source.
Slope mismatch
-Residuals rotate systematically across x.
Intercept mismatch
-Residuals shift together and the line misses the centroid.
SSE penalty
-Extra squared error measures the practical arithmetic cost.
Normal-equation failure
Residual sums expose whether the claimed line is an OLS solution.
Decision takeaway: A verified claim should match coefficients within declared precision and preserve the OLS residual identities.
Applied decisions
Regression audit cases
Rounded report coefficients
A report displays slope and intercept to two decimals.
What the result clarifies: Small normal-equation residuals may be explained by declared rounding.
Swapped spreadsheet cells
Slope and intercept are accidentally transposed.
What the result clarifies: The coefficient residuals, centroid miss, and SSE penalty all expose the defect.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
Passing this audit verifies arithmetic consistency with ordinary least squares for the entered observations. It does not validate model assumptions, linearity, data provenance, causal claims, representativeness, or predictive performance.
Regression Verification Calculator | Claimed-Line Normal Equation Audit FAQ
Why can a rounded line fail the exact normal equations?
Rounding perturbs the coefficients, so residual identities may be small rather than exactly zero.
Why compare SSE?
OLS is defined as the line with minimum squared vertical residuals for these data.
Does a passing audit mean the model is useful?
No. It means the coefficients are arithmetically consistent with OLS under the selected tolerance.