Normal Distribution Point Calculator

Normal Distribution Formula:

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]

Probability Density:

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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 normal distribution probability density function:

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]

Where:

\( \mu \) — Mean of the distribution
\( \sigma \) — Standard deviation of the distribution
\( x \) — The point at which to evaluate the function
\( \pi \) — Mathematical constant Pi (~3.14159)
\( e \) — Euler's number (~2.71828)

Explanation: The function gives the relative likelihood of a random variable taking on a given value.

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 the mean (μ), standard deviation (σ > 0), and the value (x) at which you want to evaluate the probability density function.

5. Frequently Asked Questions (FAQ)

Q1: What does the probability density value mean?
A: It represents the relative likelihood of the random variable taking on that exact value. For continuous distributions, the probability at any single point is technically zero.

Q2: What's the difference between PDF and CDF?
A: PDF (Probability Density Function) gives the density at a point, while CDF (Cumulative Distribution Function) gives the probability up to a point.

Q3: What are typical values for μ and σ?
A: μ can be any real number, σ must be positive. In standard normal distribution, μ=0 and σ=1.

Q4: Why is the normal distribution so common?
A: Due to the Central Limit Theorem, many random variables are normally distributed when aggregated.

Q5: How is this different from a probability?
A: For continuous distributions, we calculate probabilities over intervals, not single points. The PDF helps calculate these probabilities.