DEGREE-TWO CURVE FIT

Quadratic Regression Calculator

Fit a parabola to paired observations using least squares. Inspect the fitted curve, vertex, R-squared, RMSE and residuals. Use a quadratic model when a single bend is meaningful for your data.

Ordinary least squaresBrowser-basedNo sign-in

YOUR DATA

Enter paired observations

Paste or upload numeric X and Y columns. Use a comma, tab or semicolon between columns and a period as the decimal mark. The first non-empty line is a header by default. Up to 5 MB and 5,000 observations are accepted.

Upload a CSV or TSV file

The file is read locally in your browser.

Or paste data

At least three complete pairs with distinct X values are needed.

X variable

Y variable

First row contains column namesTurn this off when the first line is data.

YOUR RESULTS

Quadratic fit

Fitted equation in centered and scaled X——

R-squared (R²)—Observed variation relative to the fitted values

RMSE—Typical residual size in Y units

Vertex X—Turning point location when the curve has a vertex

Vertex Y—Fitted value at the turning point

Curvature—Quadratic coefficient for centered/scaled X

Observed values and fitted curve

Points are observations; the curve shows the degree-two least-squares fit.

Observed Fitted

Residuals by observation

Residual = observed Y − predicted Y.

Row	Observed Y	Predicted Y	Residual

Formula and calculation method

The calculator fits ŷ = α + βz + γz² by least squares, where z = (x − μ) / s, μ is the mean X value, and s is the root mean square distance of X from μ. Centering and scaling X helps the numerical calculation. The displayed equation uses these normalized coordinates; the exported JSON includes α, β, γ, μ and s.

R² = 1 − SSE/SST and RMSE = √(SSE/n). The vertex is reported when γ is nonzero and the values are within the numeric range.

When a quadratic fit is useful

A degree-two curve can describe one broad bend, such as a U-shaped or inverted-U-shaped pattern. It should be chosen because the shape is plausible and supported by the data, not simply because it raises R². The curve can turn sharply outside the observed X range, so avoid extrapolating without domain knowledge.

Assumptions and limitations

At least three valid observations and enough distinct X values to identify a degree-two curve are required.
R² measures fit on these input observations. It does not test significance, causation, or performance on new data.
A quadratic curve has one turning point at most and may not represent more complex patterns.
RMSE is sensitive to outliers and uses the same units as Y.
The current limit is 5,000 observations and 5 MB per upload; larger inputs are rejected instead of partly calculated.

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