Curve fitting models

Each model in curve.fit’s Model menu, including Custom Function, has a page with its equation, what each parameter means and tips for fitting it. From a model’s page you can open the workspace with that model already selected.

Fit Statistics report AIC and BIC where they are defined, and Previous ranks fits of the same data by AIC, so you can compare models. Parameters can be fixed or kept within limits, and Evaluate reads the fitted model in either direction: y at a chosen x, or x for a measured y.

• Arrhenius

  Exponential dependence on 1/x. Use it for reaction rate constants, vapor pressures and other thermally activated quantities measured against absolute temperature.

• Cauchy

  Cauchy’s dispersion formula: refractive index against wavelength as a constant plus terms in 1/x² and 1/x⁴. Use it for the refractive index of a transparent material measured at several wavelengths, for example with a prism or refractometer.

• Cubic

  A third-degree polynomial. Use it as an empirical curve for smooth data with up to two turns, such as a calibration curve that a quadratic cannot follow.

• Custom Function

  Your own equation in x with up to five parameters, A to E. Use it when the theory of your experiment gives a form that no built-in model has, and fit that form directly instead of linearizing it.

• Damped Oscillator

  A sine wave whose amplitude decays exponentially, centered on a constant level. Use it for damped pendulums, mass–spring systems and ringing electrical circuits.

• Double Exponential

  The sum of two exponential decays on a constant background. Use it when a decay has a fast and a slow component, such as a mixture of two radioactive isotopes or a two-stage relaxation.

• Driven Oscillator Amplitude

  The displacement amplitude of a driven, damped oscillator against driving frequency: a resonance peak on a constant baseline. Use it for a mechanical oscillator’s amplitude or the capacitor voltage of a driven series RLC circuit.

• Exponential Decay (Half-life)

  Exponential decay toward a constant level, written with the half-life t½. Use it for radioactive decay above a background and other first-order decays where the half-life is the result you want.

• Exponential Decay (Time Constant)

  Exponential decay toward a constant level, written with the time constant τ. Use it for RC and RL circuits, Newton’s law of cooling and other first-order decays where the time constant is the result you want.

• Exponential with Offset

  Exponential growth or decay toward a constant level. Use it for cooling curves, capacitor charging and discharging, and decays measured on a background.

• Gaussian

  A bell-shaped peak on a constant baseline. Use it for spectral lines, detector peaks and measured distributions.

• Gaussian on Linear Background

  A Gaussian peak on a sloped, straight-line baseline. Use it for spectral lines and detector peaks whose background changes across the measured range.

• Inverse

  A constant plus a term in 1/x. Use it when y varies inversely with x, such as gas pressure against volume at constant temperature.

• Linear

  A straight line. Use it for proportional relationships, calibration lines and any data you expect to change at a constant rate.

• Logistic

  An S-shaped rise from 0 to a plateau A. Use it for population growth, autocatalytic reactions and other growth that starts slowly, speeds up and then saturates.

• Logistic 4PL (Hill)

  The four-parameter logistic, or Hill, curve: an S-shaped step between two plateaus that is symmetric on a logarithmic x axis. Use it for dose–response, ligand-binding and immunoassay standard curves.

• Logistic 5PL

  The five-parameter logistic curve: an S-shaped step between two plateaus that can be asymmetric. With S = 1 it is the four-parameter Hill curve. Use it for dose–response and immunoassay standard curves whose two halves differ.

• Logistic with Baseline

  An S-shaped step between two plateaus, symmetric in x. Use it for melting and denaturation curves, activation curves and other transitions that start and end at nonzero levels.

• Lorentzian Peak

  The Lorentzian, or Cauchy, line shape on a constant baseline, with its center and full width at half maximum. Use it for spectral lines, magnetic resonance and other peaks whose wings fall off more slowly than a Gaussian’s.

• Malus’s Law

  Malus’s law with a background: the intensity of light transmitted through two polarizers against the angle between them. Use it for polarization experiments with a rotating analyzer.

• Natural Log

  A logarithmic response: y changes by the same amount each time x is multiplied by the same factor. Use it for sound levels in decibels and other responses proportional to the logarithm of a stimulus.

• Oscillator

  A sine wave centered on a constant level. Use it for periodic data without noticeable damping, such as an AC voltage or a pendulum’s angle over a few swings.

• Poisson

  The Poisson distribution scaled by a total count. Use it for histograms of counts, such as the number of radioactive decays recorded in many equal time windows.

• Power Law

  y proportional to a power of x. Use it for scaling laws, such as a pendulum’s period against its length or radiated power against temperature.

• Power Law with Offsets

  A power law shifted along both axes. Use it when a power-law dependence starts at a threshold xₒ and sits on a constant background.

• Power Law with Vertical Offset

  A power law plus a constant offset. Use it when y follows a power of x but does not approach zero.

• Quadratic

  A parabola. Use it for height against time in free fall or projectile motion, and for calibration curves with slight, steady curvature.

• Quartic

  A fourth-degree polynomial. Use it as an empirical curve when lower-degree polynomials leave a pattern in the residuals.

• RLC Resonance

  The current in a driven series RLC circuit against frequency, or equally the voltage across its resistor: a peak at the resonant frequency f₀ whose sharpness is set by the quality factor Q. Use it for AC resonance labs and other resonances measured as a current or a velocity.

• Saturation (Michaelis–Menten)

  A rise that levels off at V. Use it for enzyme kinetics, with reaction rate against substrate concentration, and for binding and other saturating responses.

• Shifted Reciprocal

  A hyperbola with a vertical asymptote at x = xₒ and a horizontal one at y = C. Use it for inverse relationships with offsets, such as a thin lens’s image distance against object distance.

• Single Slit Diffraction

  The single-slit diffraction pattern: a central maximum with much weaker side maxima, on a constant background. Use it for light intensity measured across the diffraction pattern of one slit.

• Stokes Law

  Stokes’ law for the terminal speed of a small ball falling through a viscous fluid, with a correction for the wall of the tube. Use it for falling-ball viscometer experiments, with ball diameter as x and terminal speed as y.

• Two Slit Interference

  The double-slit intensity pattern: interference fringes inside a single-slit diffraction envelope, on a constant background. Use it for light intensity measured across a two-slit pattern.

• Weighted Mean

  Combines independent measurements of the same quantity, giving more weight to the more precise ones. Use it to average repeated measurements that have different uncertainties.

• X Log X

  A constant plus a term in x ln x. Use it when theory predicts this form, for example the running time of an algorithm that grows as n log n.

Your own equation

For any other equation, choose Custom Function in the Model menu and type the equation with x and up to five parameters, A to E. Saved fits and reports show it as typeset mathematics. Fitting a custom equation explains how to write one and choose starting values.