Calculator Inputs
Plotly Graph
The chart displays the Gaussian curve over the selected range and highlights the evaluated x value.
Example Data Table
This example uses A = 1, μ = 0, σ = 1, and C = 0.
| x | f(x) | Interpretation |
|---|---|---|
| -3 | 0.011109 | Far into the left tail |
| -2 | 0.135335 | Low density away from the mean |
| -1 | 0.606531 | One standard deviation below the mean |
| 0 | 1.000000 | Peak of the Gaussian curve |
| 1 | 0.606531 | One standard deviation above the mean |
| 2 | 0.135335 | Low density on the right side |
| 3 | 0.011109 | Far into the right tail |
Formula Used
General Gaussian function:
y = A × exp( -((x - μ)2) / (2σ2) ) + C
Normalized Gaussian density:
y = 1 / (σ√(2π)) × exp( -((x - μ)2) / (2σ2) )
Full width at half maximum:
FWHM = 2√(2 ln 2) × σ
Core area between two bounds: Aσ√(π/2) × [ erf((b-μ)/(σ√2)) - erf((a-μ)/(σ√2)) ]
The calculator evaluates the function at one x value, computes width and slope metrics, and estimates interval area using the error function.
How to Use This Calculator
Choose either a general Gaussian or a normalized density.
Enter the x value you want to evaluate.
Set amplitude, mean, standard deviation, and optional offset.
Provide lower and upper bounds to estimate area over an interval.
Define the chart range and step for plotting the curve.
Click the calculate button to generate results above the form.
Use the CSV or PDF buttons to export the current summary.
Frequently Asked Questions
1. What does this Gaussian function calculator compute?
It evaluates a Gaussian curve at a selected x value, plots the shape, and reports peak, width, slope, standardized CDF, and interval area metrics.
2. What is the difference between general and normalized modes?
General mode uses your chosen amplitude and optional offset. Normalized mode converts the curve into a probability density, so the core Gaussian area becomes 1.
3. Why must standard deviation be greater than zero?
Standard deviation controls spread. A zero or negative value breaks the Gaussian formula, because division by σ and σ² would become invalid.
4. What does the FWHM result mean?
FWHM means full width at half maximum. It measures how wide the curve is at half of its peak height, which is useful in signal analysis.
5. What are the inflection points?
For a Gaussian curve, the inflection points occur at μ - σ and μ + σ. They mark where curvature changes from bending downward to upward.
6. What does the interval area represent?
It measures the accumulated curve value between the lower and upper bounds. In normalized mode, that area is directly comparable to probability mass.
7. Why might the chart step be adjusted automatically?
Very tiny steps can create too many points for smooth browser rendering. The calculator may widen the plotting step to keep the chart responsive.
8. Where is this calculator useful?
It helps in statistics, machine learning, physics, imaging, spectroscopy, finance, and quality control whenever bell-shaped curves describe variation or noise.