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Use scipy.stats.fit when you have sample values and want to estimate parameters for a probability distribution. Use scipy.optimize.curve_fit when you have paired x-y observations and a function to fit. They solve different problems; neither API’s parameter estimates alone prove that the model is appropriate.
The examples and API details below follow the SciPy v1.18.0 documentation. Check the reference pages for your installed SciPy version before relying on version-sensitive signatures or defaults.
Choose the SciPy fit API that matches your data
| Your task | API | What it does |
|---|---|---|
| Estimate a named probability distribution from sample values | scipy.stats.fit or a continuous distribution object’s fit method |
Estimates distribution parameters, with the top-level API accepting parameter bounds. |
| Fit a specified function to paired x-y measurements | scipy.optimize.curve_fit |
Estimates the function’s parameters and returns an approximate covariance matrix. |
| Minimize a residual vector with robust loss or additional control | scipy.optimize.least_squares |
Lets you define residuals directly and configure bounds, loss, Jacobian, and optimizer controls. |
| Test whether a sample is compatible with a distribution family | scipy.stats.goodness_of_fit |
Uses Monte Carlo calibration, refitting unknown parameters in simulated samples. |
These APIs are not interchangeable. A distribution fit estimates parameters under a family you choose; a curve fit estimates parameters in a function you specify; a goodness-of-fit test addresses compatibility with a distribution family.
Fit a probability distribution to sample values
For one-dimensional sample data from an assumed discrete or continuous family, the top-level scipy.stats.fit function estimates parameters subject to the bounds you provide. Its FitResult includes named parameters and optimizer status.
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Example: fit a negative-binomial distribution
import numpy as np
from scipy import stats
# Replace this demonstration data with your one-dimensional observations.
rng = np.random.default_rng(7)
data = stats.nbinom.rvs(5, 0.4, size=500, random_state=rng)
result = stats.fit(
stats.nbinom,
data,
bounds={"n": (1, 20), "p": (0.05, 0.95)},
)
print(result.params)
print(result.success)
print(result.message)
print(result.nllf())
result.plot()The data-generation line is for a runnable illustration, not a claim about fixed fit results. SciPy’s v1.18.0 documentation likewise notes that its negative-binomial example is stochastic and that numerical results can vary with the default optimizer. Inspect the returned result rather than expecting particular parameter values. See the SciPy stats.fit reference.
Set bounds that reflect plausible parameters
Bounds define the optimizer’s search space; they should reflect the distribution’s parameter domain and what is plausible for the data. SciPy notes that convergence is more likely when bounds are tight while still containing the maximum-likelihood estimate. A parameter can be fixed by giving it equal lower and upper bounds.
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Bounds may be a mapping from parameter names to intervals or a sequence. With a sequence, provide intervals for all shape parameters; location and scale bounds may follow. If you omit location and scale bounds, the top-level function fixes them at 0 and 1, respectively. Non-finite input values raise ValueError.
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Read the result and check the fit
- Check
result.successandresult.messageto see whether the optimizer reports success. - Inspect
result.paramsfor the fitted named parameters andresult.nllf()for the negative log likelihood. - Use
result.plot()to compare the fitted probability mass function or density with a normalized histogram. - Check whether the chosen family and parameter constraints make sense for the data; a successful optimization is not proof of an adequate model.
How distribution-instance fitting differs
Continuous univariate distribution objects also provide a fit method. The SciPy v1.18.0 reference describes these methods as maximum-likelihood estimation and documents support for regular data and censored observations represented by CensoredData. This is distinct from the top-level scipy.stats.fit, which supports discrete as well as continuous distributions and accepts explicit bounds.
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Starting values matter: defaults do not work well for every distribution. Review the SciPy probability-distributions tutorial and the documentation for the specific distribution and API you use.
Fit a curve to paired x-y observations
For measurements modeled as ydata = f(xdata, *params) + eps, define a callable with the independent variable first and the parameters after it. curve_fit returns popt, the estimated parameters, and pcov, an approximate covariance matrix.
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Example: fit an exponential decay
import numpy as np
from scipy.optimize import curve_fit
def decay(x, amplitude, rate, offset):
return amplitude * np.exp(-rate * x) + offset
rng = np.random.default_rng(7)
xdata = np.linspace(0, 4, 60)
ydata = decay(xdata, 3.0, 1.2, 0.4) + 0.15 * rng.normal(size=xdata.size)
popt, pcov = curve_fit(
decay,
xdata,
ydata,
bounds=([0, 0, -1], [10, 5, 2]),
)
print(popt)
print(pcov)The bounds in this example constrain amplitude, rate, and offset to the intervals shown. The noisy synthetic observations mean the exact estimates can vary. For a further example and the full call signature, see SciPy’s curve_fit reference.
Use measurement uncertainty when it matters
The sigma argument can be scalar or one-dimensional standard deviations, or a two-dimensional covariance matrix. By default, absolute_sigma=False, so SciPy scales the parameter covariance estimate using residual variance. Set absolute_sigma=True when the supplied uncertainty values should be treated as absolute rather than relatively scaled.
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Check numerical reliability
- Use
float64inputs and ensure the model returns compatible values. - Constrain parameters with bounds when their valid ranges are known.
- If parameters have very different scales, take care with scaling and interpretability.
- Treat
pcovas a local approximation, not a guarantee. A poor Jacobian rank or a large covariance condition number can signal unreliable estimates.
Use least squares for direct residual control
Choose scipy.optimize.least_squares when it is clearer to construct the residual vector yourself or when you need a robust loss function. For example, define residuals as the model’s predictions minus the observed values, then pass that residual function to the optimizer. SciPy’s official example contrasts ordinary least squares with soft-L1 and Cauchy losses on data containing outliers. Robust losses reduce the influence of large residuals; they do not validate the model or guarantee correct estimates.
See the least_squares reference for bounds, loss choices, Jacobian options, and optimizer controls, and the SciPy optimization tutorial for an outlier example.
Test distribution compatibility separately
Parameter estimation answers which parameters best fit a chosen family under the fitting method; it does not test whether that family is a plausible description of the sample. For a formal check, scipy.stats.goodness_of_fit simulates samples under the null model and refits unknown parameters in each simulated sample. It supports Anderson-Darling, Kolmogorov-Smirnov, Cramér-von Mises, and Filliben statistics.
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Quick Recap
Practical workflow
- Decide whether your input is a sample for a probability distribution or paired x-y measurements for a function.
- Choose the matching API, then inspect its documentation for your installed SciPy version.
- Set plausible parameter bounds or constraints; avoid unnecessarily broad ranges.
- Inspect optimizer status and estimates, then compare the fitted model with the observations.
- For a distribution, use a goodness-of-fit procedure when you need a formal compatibility test; do not infer model adequacy from fitted parameters alone.
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