Understanding Gaussian Error Functions
The error function connects integration, probability, and measurement error. It appears when a bell curve must be accumulated from its center. The symbol erf(x) measures the signed area under exp(-t²), scaled by 2 divided by square root pi. That scaling makes the value approach one as x grows.
Why It Matters
Normal distributions use the same exponential shape. A standard normal curve has exp(-z²/2). With the substitution t = z / √2, its cumulative probability becomes one half times 1 plus erf(z / √2). This link is useful in statistics, quality control, physics, and signal analysis. It turns a difficult Gaussian integral into a reusable function.
Proof Idea
The proof starts with the normal cumulative function. Write Φ(z) as the integral from negative infinity to z of the standard normal density. Split the integral at zero. The left half contributes one half by symmetry. For the part from zero to z, set u = s / √2. Then ds equals √2 du. After simplification, the remaining integral matches the definition of erf. The result is Φ(z) = 0.5[1 + erf(z/√2)].
Numerical Calculation
A computer cannot usually express erf with elementary functions. It must approximate the integral. This calculator offers a fast approximation, Simpson integration, and a power series. The fast method works well for common values. Simpson integration is transparent because it sums slices under the curve. The series is useful near zero, but it may converge slowly for large inputs.
Using Results
For a raw Gaussian value, the calculator first standardizes it with z = (x - μ) / σ. It then finds lower tail probability, upper tail probability, density, and complement values. For intervals, it subtracts two cumulative values. This gives the probability between two observations. Always use a positive standard deviation. Increase precision when comparing close probabilities. Use more Simpson slices for a stronger proof style result.
Limit Checks
Good results should pass simple checks. erf(0) equals zero. Φ(0) equals 0.5. Very large positive z values approach one. Very large negative z values approach zero. The two tails should sum to one. These checks help detect bad inputs, rounding issues, or a wrong standard deviation before using the answer in reports.