Normal Probability Density Calculator

Normal Probability Density Function:

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

Probability Density (PDF):

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1. What is Normal Probability Density?

The normal probability density function describes the relative likelihood of a continuous random variable taking on a given value in a normal distribution. It is characterized by its bell-shaped curve and is fundamental in statistics and probability theory.

2. How Does the Calculator Work?

The calculator uses the normal probability density formula:

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

Where:

\( \mu \) — Mean of the distribution
\( \sigma \) — Standard deviation of the distribution
\( x \) — Value at which to evaluate the density
\( \pi \) — Mathematical constant pi (approximately 3.14159)
\( e \) — Euler's number (approximately 2.71828)

Explanation: The formula calculates the height of the probability density curve at point x for a normal distribution with given mean and standard deviation.

3. Importance of Probability Density Calculation

Details: Probability density calculations are essential for statistical analysis, hypothesis testing, quality control, risk assessment, and many scientific applications where normal distribution assumptions apply.

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. All values must be valid numerical inputs.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between probability density and probability?
A: Probability density gives the relative likelihood at a specific point, while probability gives the area under the curve between two points. For continuous distributions, probability at a single point is zero.

Q2: What does the probability density value represent?
A: It represents the height of the probability density function at that point, indicating how "dense" the probability is around that value.

Q3: Can probability density be greater than 1?
A: Yes, probability density can be greater than 1. The important property is that the total area under the curve equals 1, not that individual density values are less than 1.

Q4: When is the normal distribution assumption appropriate?
A: The normal distribution is often appropriate for natural phenomena, measurement errors, and many biological characteristics where values cluster around a mean.

Q5: What if my standard deviation is zero?
A: Standard deviation must be greater than zero. A standard deviation of zero would represent a degenerate distribution with all probability mass at the mean.