1. What is the Normal Cumulative Distribution Function?

The Normal Cumulative Distribution Function (CDF) gives the probability that a normally distributed random variable will be less than or equal to a given value. It's fundamental in statistics for probability calculations and hypothesis testing.

2. How Does the Calculator Work?

The calculator uses the Normal CDF formula:

Where:

• \( x \) — The value for which you want the probability
• \( \mu \) — Mean of the distribution
• \( \sigma \) — Standard deviation of the distribution
• \( \text{erf} \) — The Gauss error function

Explanation: The formula transforms the value to a standard normal scale and calculates the area under the curve to the left of that point.

3. Importance of CDF Calculation

Details: The CDF is essential for determining probabilities in normal distributions, calculating p-values in hypothesis testing, and creating confidence intervals in statistical analysis.

4. Using the Calculator

Tips: Enter the value (x), mean (μ), and standard deviation (σ). Standard deviation must be positive. The result will be between 0 and 1, representing the cumulative probability.

5. Frequently Asked Questions (FAQ)

Q1: What does a CDF value of 0.5 mean?
A: A CDF of 0.5 means there's a 50% probability the random variable will be less than or equal to the given value, which occurs exactly at the mean for symmetric distributions.

Q2: How is this related to the standard normal distribution?
A: The standard normal has μ=0 and σ=1. Any normal distribution can be converted to standard normal using z-scores.

Q3: What's the difference between CDF and PDF?
A: PDF gives the probability density at a point, while CDF gives the accumulated probability up to that point.

Q4: What are common uses of the normal CDF?
A: Used in quality control, risk assessment, statistical hypothesis testing, and many other statistical applications.

Q5: How accurate is this calculator?
A: It uses a numerical approximation of the error function accurate to about 7 decimal places.