Normal CDF Calculator

Normal Cumulative Distribution Function:

\[ P = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-\frac{t^2}{2}} dt \]

Cumulative Probability (P):

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1. What is the Normal CDF?

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 the integral of the normal probability density function from negative infinity to x.

2. How Does the Calculator Work?

The calculator computes the integral:

\[ P = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(t-\mu)^2}{2\sigma^2}} dt \]

Where:

\( x \) — The value at which to evaluate the CDF
\( \mu \) — Mean of the distribution
\( \sigma \) — Standard deviation of the distribution
\( P \) — Cumulative probability (0 to 1)

Explanation: The function calculates the area under the normal curve to the left of the given x value.

3. Importance of Normal CDF

Details: The normal CDF is fundamental in statistics for hypothesis testing, confidence intervals, and probability calculations across many fields including science, engineering, and finance.

4. Using the Calculator

Tips: Enter the x value, mean (default 0), and standard deviation (default 1). For standard normal distribution, use mean=0 and stddev=1.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between PDF and CDF?
A: PDF gives the probability density at a point, while CDF gives the cumulative probability up to that point.

Q2: What are typical uses of the normal CDF?
A: Calculating p-values, determining percentiles, and finding probabilities for ranges of values in normally distributed data.

Q3: What does a CDF value of 0.5 mean?
A: It means 50% of the distribution lies below that point (the median, which equals the mean for normal distributions).

Q4: Can I calculate CDF for non-normal distributions?
A: This calculator is specifically for normal distributions. Other distributions require different formulas.

Q5: How accurate is this calculator?
A: Very accurate (6 decimal places) for standard normal distribution. Accuracy depends on the approximation method used.