Probability Calculator Normal Distribution

Normal Distribution Formula:

\[ P = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \]

Probability (P):

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1. What is Normal Distribution?

The normal distribution, also known as 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 normal distribution formula:

\[ P = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \]

Where:

\( P \) — Probability (0-1)
\( \mu \) — Mean of the distribution
\( \sigma \) — Standard deviation
\( x \) — Value to evaluate

Explanation: The formula calculates either the probability density (PDF) or cumulative probability (CDF) at a given point in a normal distribution.

3. Importance of Normal Distribution

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

4. Using the Calculator

Tips: Enter the mean (μ), standard deviation (σ), and the value (x) you want to evaluate. Select whether you want the probability density (PDF) or cumulative probability (CDF).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between PDF and CDF?
A: PDF gives the probability density at a specific point, while CDF gives the probability that a random variable is less than or equal to a certain value.

Q2: What are typical values for mean and standard deviation?
A: The standard normal distribution has μ=0 and σ=1, but these parameters can be any real number (with σ > 0).

Q3: When is the normal distribution not appropriate?
A: When data is highly skewed or has heavy tails, other distributions like log-normal or Student's t might be more appropriate.

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: How accurate is the CDF approximation?
A: The error function approximation used here is accurate to about 7 decimal places.