Gaussian Fit Width Calculator

Example Data Table

Sample symmetric peak data (baseline 0). Paste into “Fit width from data” to test.

Formulas Used

• y = A·exp(-(x-μ)²/(2σ²)) + C (standard Gaussian with baseline).
• FWHM = 2√(2 ln2)·σ (full width at half maximum).
• W(1/e) = 2√2·σ, W(1/e²) = 4σ (standard form widths).
• Beam optics mapping: I = I0·exp(-2r²/w²) ⇒ w = 2σ (1/e² radius).

Units: keep x and widths consistent. Amplitude uses your y scale.

How to Use This Calculator

1. Select Convert width types to switch among common definitions.
2. Enter a positive width value and choose its type.
3. Press Calculate to see σ, FWHM, and other widths.
4. Select Fit width from data to estimate width from measurements.
5. Provide baseline C if the signal has an offset.
6. Paste (x,y) pairs; include points around the peak and tails.
7. Download CSV or PDF for documentation and sharing.

If your peak is asymmetric or multi-peaked, consider preprocessing first.

The notes below summarize common Gaussian width conventions used in labs.

Gaussian Fit Width Notes

1) Why width is a key metric

Gaussian width compresses a complex peak into one physically meaningful number. In spectroscopy it tracks line broadening, in timing it quantifies jitter, and in imaging it summarizes blur. Width comparisons are often more stable than peak height because amplitude can change with gain, exposure, or alignment.

2) Sigma as the natural scale

The standard model y = A·exp(-(x-μ)²/(2σ²)) + C uses σ to set the spread. A Gaussian contains about 68.27% of its area within ±1σ and 95.45% within ±2σ. Those coverage values connect “width” to probability and expected repeatability.

3) FWHM for quick reading

Full width at half maximum is convenient on plots and instrument sheets. The conversion is fixed: FWHM = 2√(2 ln2)·σ ≈ 2.35482·σ. Because the constant is universal, converting between σ and FWHM does not depend on amplitude or baseline, only on the chosen definition.

4) 1/e and 1/e² conventions

For the standard Gaussian, the full width at 1/e of the peak is W(1/e)=2√2·σ, and the full width at 1/e² is W(1/e²)=4σ. These levels are useful when a detector or algorithm naturally thresholds at exponential fractions rather than half maximum.

5) Beam optics mapping

In laser work, profiles often use I = I0·exp(-2r²/w²), where w is the 1/e² radius. If your x-axis represents radius r, the equivalent relation is w = 2σ, and the 1/e² diameter is 2w = 4σ. Reporting w avoids confusion between radius and diameter conventions.

6) Fast fitting from measured points

The fitting mode estimates σ by transforming the data with z = ln(y − C) and fitting a quadratic z = a x² + b x + c. From the coefficients, σ = √(-1/(2a)) and μ = b·σ². This is efficient for single, symmetric peaks with adequate sampling on both sides. For accuracy, include tail points down to roughly 5–10% of peak height and avoid clipping. If noise dominates, average repeats or smooth lightly before fitting, then rerun to confirm.

7) Baseline, quality, and reporting

Choose baseline C to represent background; points with y ≤ C cannot be log-transformed and are ignored. A valid peak requires negative curvature (a < 0). Use the exported CSV or PDF to record the width definition, units label, baseline, and fit diagnostics for traceable comparisons.

FAQs

1) Which width definition should I publish?

Use σ for modeling and uncertainty analysis, and FWHM for specifications and quick plot reading. If you work in optics, also state whether you mean 1/e² radius w or 1/e² diameter.

2) Why does the fit ignore points at or below baseline?

The fit uses ln(y − C). When y − C is zero or negative, the logarithm is undefined, so those points cannot contribute to the regression.

3) Can I fit a dip instead of a peak?

Yes. Convert the dip into a positive peak by fitting baseline minus signal, or invert the data after subtracting the baseline. The method requires positive values after baseline handling.

4) How many data points are recommended?

Five usable points are the minimum, but 15–50 points spanning both sides of the peak and the tails is preferred. More points improve curvature estimation and reduce sensitivity to noise.

5) My peak is asymmetric. Will results be accurate?

Asymmetry violates the Gaussian assumption, so the reported width becomes an approximation. Try fitting only the central region, narrowing the x-range, or using a different model when skewness is significant.

6) What does log-space R² represent here?

R² is computed on z = ln(y − C), the values actually regressed. It is useful for screening fit consistency, but you should also inspect how the fitted curve matches the original y data.

7) How do I interpret beam w vs σ?

For I = I0·exp(-2r²/w²) with x=r, the equivalent Gaussian has σ = w/2. The calculator reports both w (1/e² radius) and the corresponding 1/e² diameter.