What makes a peak Gaussian, and what happens when its center becomes sharper or flatter than a normal bell curve? This interactive tutorial compares the standard Gaussian function with a generalized Gaussian (exponential-power) function. Adjust four parameters, compare shapes, and learn when each model may be useful in biology, biochemistry, and data analysis.
Gaussian Function: The Familiar Bell Curve
A Gaussian-shaped peak can be written as f(x) = A exp[−(x − μ)2 / (2σ2)]. The parameter A is the peak height, μ locates its center, and σ > 0 controls its width. For a peak with A = 100, the maximum is 100 at x = μ, but 100 is not a required mathematical limit.
This peak-height form is not a normalized probability density unless A is chosen to make its integral equal to 1. In statistics, the familiar normal-distribution density uses A = 1/(σ√(2π)).
Generalized Gaussian Function: One Extra Shape Parameter
Here we use the following four-parameter generalized Gaussian:
g(x) = A exp[−½ |(x − μ)/σ|p]
Its parameters are A > 0 (peak height), μ (center), σ > 0 (horizontal scale), and p > 0 (shape). The standard Gaussian is the special case p = 2. This page uses a specific scale convention: some textbooks instead write exp[−(|x−μ|/α)p]. Their α is not numerically identical to our σ.
For p < 2, the peak is more pointed and the far tails decay more slowly than a Gaussian with the same A, μ, and σ. For p > 2, the top becomes flatter and the far tails decay faster. Every curve in this family is symmetric about μ. Thus, the generalized Gaussian changes peakedness and tail shape, but it does not create left–right asymmetry.
Interactive Gaussian vs. Generalized Gaussian Graph
Use the presets to compare a Laplace-like pointed peak, a Gaussian peak, and flatter super-Gaussian peaks. Both displayed curves use the same A, μ, and σ; the generalized curve additionally uses p. All examples are synthetic mathematical illustrations, not experimental observations.
Horizontal axis: x (arbitrary units). Vertical axis: peak response (arbitrary units). Initial plot window: x = −10 to 10; y = 0 to 120. The graph automatically expands vertically when needed.
Shared peak parameters
These three parameters affect both curves so the shape comparison remains fair.
Generalized Gaussian shape
p = 2 reproduces the Gaussian; p = 1 gives an exponential-power cusp; p > 2 creates a flatter top. For p ≤ 1, the curve has a pointed, nondifferentiable peak at its center (for p = 1) or a sharper cusp (for p < 1).
Mathematical constraints: A > 0, σ > 0, p > 0, μ and x can be any real number. Slider limits are demonstration choices, not theoretical restrictions.
Current equations
Gaussian: —
Generalized Gaussian: —
FWHM means full width at half maximum. Gaussian: 2σ√(2 ln 2). Generalized Gaussian: 2σ(2 ln 2)1/p. Here σ is a scale parameter; for p ≠ 2, it is not generally the statistical standard deviation.
Move a slider to update the curves.
Gaussian vs. Generalized Gaussian: Key Differences
| Property | Gaussian peak | Generalized Gaussian peak |
|---|---|---|
| Equation | A exp[−½((x−μ)/σ)²] | A exp[−½|(x−μ)/σ|p] |
| Parameters | A, μ, σ (3) | A, μ, σ, p (4) |
| Symmetric about μ? | Yes | Yes |
| Peak height | A | A |
| Tail | Gaussian | Heavier if p < 2; lighter if p > 2 |
| Special case | — | p = 2 is Gaussian |
Why Does the Shape Exponent Change the Curve?
The exponent p acts on the absolute standardized distance |(x−μ)/σ|. Near the center, where that distance is less than 1, increasing p makes its power smaller, leaving the response closer to A and flattening the top. Beyond a standardized distance of 1, increasing p makes the power larger, so the tails decrease more rapidly. This explains the flat-top / light-tail trade-off without changing the curve's symmetry.
Try this: Set A = 100, μ = 0, σ = 2, and move p from 1 to 6. Notice how the peak changes while the curve stays centered. Then vary σ: both peaks broaden together, but only the generalized curve changes its shape when p moves.
Applications in Biology and Biochemistry
1. Fluorescence Microscopy: Intensity Across a Spot
A cross-section through a fluorescent spot is often modeled with a Gaussian when the optical point-spread function is approximately Gaussian. Here, x can represent distance across an image and the curve height represents background-subtracted fluorescence intensity. A generalized Gaussian may be explored as a phenomenological alternative when the measured cross-section is more pointed or flat-topped; it does not, by itself, establish an optical mechanism.
2. Spatial Fluorescence Profiles in Cells or Tissues
Intensity along a line through a cell or tissue image can form a peak. Comparing p = 2 with an adjustable p can help students describe whether an observed profile is Gaussian-like or has a different peak shape. Real profiles may be asymmetric, have multiple peaks, or include a nonzero background; those situations require other model terms or a different model.
3. Biological Data Analysis: Choosing a Distribution
The ordinary Gaussian probability density is ubiquitous in statistical modeling of approximately symmetric measurements. The generalized Gaussian (also called the exponential-power family) allows different central sharpness and tail behavior. To use either as a probability density, normalize the total area to 1; the amplitude A in our interactive peak function is not a free parameter in a normalized density.
For strictly positive, skewed concentrations or count data, neither symmetric model should be chosen automatically. Start from the data's support, possible measurement noise, residual patterns, and underlying mechanism, rather than choosing the curve that merely looks attractive.
Understanding the Limits: Peak Height, Domain, and Normalization
The display starts at a peak height of 100 for convenience, but neither function is inherently capped at 100. Both satisfy 0 < f(x) ≤ A when A > 0, with the maximum attained at x = μ. The mathematical x-domain is all real numbers; the finite plotted interval is a visualization choice.
For a Gaussian probability density, A must equal 1/(σ√(2π)); for the generalized Gaussian under this page's scale convention, A must equal p / [2σ·21/pΓ(1/p)]. Here Γ denotes the gamma function. The simulator intentionally displays unnormalized peaks so users can independently change the visible height A.
When Is a Generalized Gaussian Better?
A generalized Gaussian can fit some symmetric peaks more flexibly because p changes peakedness and tails. Yet the extra parameter can overfit noise, and an improved training fit does not guarantee better prediction. For asymmetry, multimodal signals, or baseline offsets, neither of the current formulas is sufficient on its own. Compare residuals and predictive performance before choosing a model.
Frequently Asked Questions
Is a generalized Gaussian the same as a Gaussian?
No. The standard Gaussian is the p = 2 special case of this generalized Gaussian family.
Can the generalized Gaussian represent asymmetry?
Not in the symmetric four-parameter formulation used here. Changing p affects peak shape and tails on both sides equally.
Is σ always the standard deviation?
No. In the Gaussian p = 2 form shown here, σ equals the standard deviation after density normalization. For p ≠ 2, σ is a scale parameter, and the variance depends on p as well as σ.
Why do both curves have a maximum of 100 at first?
The default slider sets A = 100 for easy comparison. Unlike percentage-based normalized response functions, these peaks can be assigned other positive amplitudes.
What happens when p = 1?
The generalized peak becomes A exp[−|x−μ|/(2σ)], a Laplace-shaped cusp under the scale convention used here.
Can I use this graph in PowerPoint?
You can open the page during a presentation and operate its sliders. Interactive embedding inside a PowerPoint slide depends on the available web-view add-in.
References and Further Reading
NIST/SEMATECH. Normal distribution. SciPy documentation. Generalized normal distribution (version 1). Both resources discuss density forms; the interactive graph uses a peak-height parameterization.