1. What is the Normal Distribution?

The normal distribution, also known as the Gaussian distribution, is a continuous 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 distribution formula:

Where:

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

Explanation: The formula converts your value to a z-score and then calculates the probability using the standard normal CDF.

3. Importance of Normal Distribution

Details: The normal distribution is important in statistics and is often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.

4. Using the Calculator

Tips: Enter your value (x), the mean (μ), and standard deviation (σ). Standard deviation must be greater than 0.

5. Frequently Asked Questions (FAQ)

Q1: What does P(X < x) mean?
A: It represents the probability that a random variable X takes a value less than x.

Q2: What is the range of possible probabilities?
A: Probabilities range from 0 to 1, where 0 means impossible and 1 means certain.

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

Q4: What is the 68-95-99.7 rule?
A: In a normal distribution, about 68% of values fall within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.

Q5: When is the normal distribution not appropriate?
A: When data is skewed or has heavy tails, other distributions like log-normal or Student's t might be more appropriate.