1. What is Normal Distribution?

The normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

2. How Does the Calculator Work?

The calculator uses the standard normal cumulative distribution function:

Where:

• \( x \) — The value you want to evaluate
• \( \mu \) — The mean of the distribution
• \( \sigma \) — The standard deviation of the distribution
• \( \Phi \) — The standard normal CDF

Explanation: The formula converts your value to a z-score and then calculates the probability that a random variable from the distribution would be less than your value.

3. Importance of Normal Distribution

Details: Normal distribution is fundamental in statistics and appears in many natural phenomena. It's used in hypothesis testing, quality control, and risk assessment.

4. Using the Calculator

Tips: Enter your value, the mean of the distribution, and the standard deviation. The calculator will return the probability that a random variable from this distribution would be less than your value.

5. Frequently Asked Questions (FAQ)

Q1: What does the probability result mean?
A: The result represents the probability that a randomly selected value from this normal distribution would be less than your input value.

Q2: What if I want P(X > x) instead?
A: Subtract the result from 1 (P(X > x) = 1 - P(X < x)).

Q3: What is the standard normal distribution?
A: A special case where μ = 0 and σ = 1. Any normal distribution can be converted to standard normal using z-scores.

Q4: When is normal distribution not appropriate?
A: When data is skewed or has heavy tails. Other distributions like log-normal or Student's t may be better.

Q5: How accurate is this calculator?
A: It uses a good approximation of the normal CDF, accurate to about ±0.0002 for most values.