1. What Is The Error Function?

The error function (erf) is a special function that occurs in probability, statistics, and partial differential equations. It describes the probability that a random variable falls within a certain range in a normal distribution.

2. How Does The Calculator Work?

The calculator uses the mathematical definition:

Where:

• \( x \) — Input argument (real number)
• \( \pi \) — Mathematical constant pi ≈ 3.14159
• \( e \) — Euler's number ≈ 2.71828
• The integral represents the area under the Gaussian curve

Explanation: The calculator implements an approximation algorithm (Abramowitz and Stegun) since the integral cannot be expressed in elementary functions.

3. Importance Of Error Function

Details: The error function is fundamental in statistics for calculating normal distribution probabilities, in physics for heat conduction problems, and in engineering for signal processing applications.

4. Using The Calculator

Tips: Enter any real number as the argument x. The calculator will return the corresponding error function value between -1 and 1. For large |x|, erf(x) approaches ±1.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of the error function?
A: The error function returns values between -1 and 1. erf(-∞) = -1, erf(0) = 0, erf(∞) = 1.

Q2: How accurate is this approximation?
A: The Abramowitz and Stegun approximation used here has maximum error less than 1.5×10⁻⁷, which is sufficient for most practical applications.

Q3: What is the relationship to normal distribution?
A: For a standard normal variable Z, P(-x ≤ Z ≤ x) = erf(x/√2). This connects directly to probability calculations.

Q4: Are there built-in functions for erf?
A: Many programming languages and mathematical software have built-in erf() functions, but this calculator provides a reliable approximation for general use.

Q5: What about the complementary error function?
A: erfc(x) = 1 - erf(x). This is often used for tail probabilities in statistics.