Normal CDF Calculator
Calculate the Cumulative Distribution Function (CDF) for a normal distribution.
What is the Normal CDF?
The Normal Cumulative Distribution Function (CDF), often denoted as \( \Phi(x) \) for the standard normal distribution (mean=0, std dev=1) or \( F(x; \mu, \sigma) \) for a general normal distribution with mean \( \mu \) and standard deviation \( \sigma \), represents the probability that a random variable following a normal distribution will take on a value less than or equal to a specified value \( x \).
In simpler terms, it answers the question: "What is the probability that our observed value is less than or equal to \( x \)?".
The Math Behind It
The probability density function (PDF) of a normal distribution is given by:
$$ f(x; \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $$
The CDF is the integral of the PDF from negative infinity up to \( x \):
$$ F(x; \mu, \sigma) = \int_{-\infty}^{x} \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^2} dt $$
This integral does not have a closed-form solution in terms of elementary functions. It is typically approximated using numerical methods or special functions, such as the error function (erf).
A common approach is to standardize the variable \( x \) into a z-score:
$$ z = \frac{x – \mu}{\sigma} $$
Then, the CDF of the general normal distribution can be related to the CDF of the standard normal distribution:
$$ F(x; \mu, \sigma) = \Phi(z) = \Phi\left(\frac{x – \mu}{\sigma}\right) $$
The standard normal CDF, \( \Phi(z) \), is often calculated using approximations or lookup tables. The relationship with the error function is:
$$ \Phi(z) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{z}{\sqrt{2}}\right)\right] $$
Where erf(x) is the error function.
Use Cases
- Statistics: Determining p-values in hypothesis testing.
- Finance: Modeling asset prices and calculating risk measures like Value at Risk (VaR).
- Quality Control: Assessing the probability of a product meeting certain specifications.
- Engineering: Analyzing tolerances and performance variations.
- Biology and Medicine: Studying distributions of biological measurements (e.g., height, blood pressure).