Normal Probability Distribution Calculator

Normal Probability Distribution Function:

\[ P_{Normal} = \frac{1}{\sigma_{Normal} \cdot \sqrt{2\pi}} \cdot e^{-\frac{1}{2} \left( \frac{x - \mu_{Normal}}{\sigma_{Normal}} \right)^2} \]

Normal Probability Distribution:

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1. What is the Normal Probability Distribution?

The Normal Probability Distribution, also known as the Gaussian distribution, is a mathematical function that describes a symmetrical bell-shaped curve. It is one of the most important probability distributions in statistics due to its natural occurrence in many natural phenomena.

2. How Does the Calculator Work?

The calculator uses the Normal Probability Distribution formula:

\[ P_{Normal} = \frac{1}{\sigma_{Normal} \cdot \sqrt{2\pi}} \cdot e^{-\frac{1}{2} \left( \frac{x - \mu_{Normal}}{\sigma_{Normal}} \right)^2} \]

Where:

\( \mu_{Normal} \) — Mean of the distribution
\( \sigma_{Normal} \) — Standard deviation of the distribution
\( x \) — Number of successes (value at which to evaluate)
\( \pi \) — Archimedes' constant (≈3.14159)
\( e \) — Napier's constant (≈2.71828)

Explanation: The formula calculates the probability density at a specific point x in a normal distribution with given mean and standard deviation.

3. Importance of Normal Distribution

Details: The normal distribution is fundamental in statistics and is used in various fields including natural sciences, social sciences, and engineering. Many statistical tests and methods assume normal distribution of data.

4. Using the Calculator

Tips: Enter the mean value, standard deviation (must be positive), and the value at which you want to evaluate the probability density function.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between probability and probability density?
A: For continuous distributions like the normal distribution, we calculate probability density rather than discrete probability. The area under the curve between two points gives the probability.

Q2: What are the properties of a normal distribution?
A: It is symmetric about the mean, bell-shaped, and defined by two parameters: mean and standard deviation. About 68% of values fall within one standard deviation from the mean.

Q3: When is the normal distribution appropriate to use?
A: When data clusters around a central value with no bias left or right, and when extreme values become increasingly rare.

Q4: What is the standard normal distribution?
A: A special case where mean = 0 and standard deviation = 1. Any normal distribution can be converted to standard normal using z-scores.

Q5: How does standard deviation affect the shape of the curve?
A: Larger standard deviation makes the curve wider and flatter, while smaller standard deviation makes it narrower and taller.