Curve fitting models

Each built-in model in curve.fit’s Model menu 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.

• 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.

• 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.

• 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 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.

• Hill Sigmoid (4 Parameter)

  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.

• 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 (5 Parameter)

  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.

• Lorentzian

  The amplitude of a driven, damped oscillator against driving frequency: a resonance peak on a constant baseline. Use it for resonance curves of mechanical oscillators and AC circuits.

• 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.

• 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.

• 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. Fitting a custom equation explains how to write one and choose starting values.