GIFSQ Tools · Module 16 — Statistics, Experimental Design & Research · Tool 311 of 500
Fit a straight line y = a + bx to paired data, check R², and test whether the slope differs from zero — the calibration-curve and dose-response workhorse.
Try sample values — fills the form with the worked example below.
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Slope (b)
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Intercept (a)
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R²
Equation: b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)²; a = ȳ − b·x̄. R² = 1 − Σ(y − ŷ)² ÷ Σ(y − ȳ)². Slope test: t = b ÷ SE(b), SE(b) = √(MSE ÷ Σ(x − x̄)²), df = n − 2.
Method: Ordinary least-squares linear regression per ISO 16269-1. The slope p-value comes from the t distribution itself (incomplete beta function).
Assumptions: the relationship is linear, the errors are independent with constant variance and approximately normal; x is measured without important error.
Limitations: extrapolation beyond the x range is unreliable; a high R² does not prove the straight line is the right model — always look at the residuals. Outliers and leverage points can dominate the fit.
Rounding: slope and intercept to 4 decimal places; R² to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Method: Ordinary least-squares linear regression per ISO 16269-1. The slope p-value comes from the t distribution itself (incomplete beta function).
Assumptions: the relationship is linear, the errors are independent with constant variance and approximately normal; x is measured without important error.
Limitations: extrapolation beyond the x range is unreliable; a high R² does not prove the straight line is the right model — always look at the residuals. Outliers and leverage points can dominate the fit.
Rounding: slope and intercept to 4 decimal places; R² to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Worked example: X 1–5 with Y 2, 4, 5, 4, 5: slope 0.6000, intercept 2.2000, R² = 0.6000, slope p = 0.1240 — not significant at α = 0.05.
Enter values above to see results.
A calibration curve is the same fit: concentration on x, response on y, then read unknowns from the line. Tool 311 of 500 — GIFSQ
