Observed points, fitted line, residual stems, and extrapolation boundaryUpdates with every input
Math & Statistics
Regression Solver Calculator
Solve a simple linear least-squares regression from six entered x-y observations and report slope, intercept, fitted equation, R-squared, residual standard error, prediction, and a fully synchronized scatterplot.
Fitted equation -
Slope -
Intercept -
R-squared -
Residual standard error -
Predicted y -
Residual SSE -
Fitted direction -
LEAST-SQUARES FIT
Observed points, fitted line, residual stems, and extrapolation boundary
Residuals remain visible instead of disappearing behind R-squared; the prediction point is distinguished when it lies beyond the observed x range.
OBSERVATION LEDGER
Every fitted value and signed residual
The ledger reconciles each observation with the line and preserves the residual sign needed for diagnostics.
| Point | x | Observed y | Fitted y | Residual | Squared residual |
|---|
DATA ENTRY
Enter observations as actual x-y pairs
- Keep each y beside the x value measured in the same observation.
- Use one consistent unit for x and one for y.
- Do not sort one column without sorting the other.
- Choose prediction x within the observed range unless extrapolation is intentional.
- Inspect residuals before relying on the fitted equation.
MODEL READING
A good equation is not automatically a good data-generating story
Least squares summarizes linear association. It does not establish causation, and a high R-squared can coexist with curvature, influential points, or nonconstant variance.
The line minimizes vertical squared error, so reversing x and y generally gives a different equation.
OLS LINE
Choose slope and intercept that minimize squared vertical residuals
Centering x and y produces the slope from cross-deviation divided by x-deviation. The intercept then forces the fitted line through the sample centroid.
Detailed calculation process and general formulas
b1 = sum[(x-xbar)(y-ybar)] / sum[(x-xbar)^2]b0 = ybar - b1 xbaryhat_i = b0 + b1 x_iSSE = sum[(y_i-yhat_i)^2]R^2 = 1 - SSE/SSTSymbols, meanings, and units
- b1
- least-squares slopey units per x unit
- b0
- least-squares intercepty units
- yhat_i
- fitted value for observation iy units
- e_i
- signed vertical residualy units
- R^2
- fraction of sample y variation explained by the lineproportion
FIT DIAGNOSTICS
Four checks before using the prediction
The line, residual pattern, range, and subject-matter design must agree.
Line strength
-R-squared summarizes in-sample linear fit.
Typical residual scale
-Residual standard error keeps model error on the y scale.
Prediction range
The chart marks when prediction x extends beyond observed support.
Residual shape
Systematic bends or fans suggest the linear model is incomplete.
Decision takeaway: Use the equation only after the residual stems look compatible with a straight-line summary.
Applied decisions
Least-squares line applications
Calibration range
Six standards relate concentration to instrument response.
What the result clarifies: The fitted line is useful only within a validated response range.
Operational trend
Six workload levels are paired with cycle time.
What the result clarifies: Residual structure can reveal a capacity bend hidden by the overall slope.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
This calculator performs unweighted simple linear regression on six observations. It does not correct for measurement error in x, repeated observations, time dependence, clustering, heteroscedasticity, influential outliers, nonlinear response, or omitted variables.
Regression Solver Calculator | Six-Point Least Squares Line FAQ
Why are residuals vertical?
Ordinary least squares minimizes squared deviations in y for entered x values.
Does high R-squared prove causation?
No. It describes sample fit, not causal identification.
Can I predict beyond the largest x?
The calculator can compute it, but the chart marks extrapolation because model behavior outside observed support is unverified.