Gaussian Distribution Calculator
Statistical Results
Understanding the Gaussian Distribution
The Gaussian distribution, often called the Normal Distribution or the "Bell Curve," is a continuous probability distribution that is symmetrical around the mean. It is defined by two primary parameters: the Mean (μ), which represents the center or peak of the curve, and the Standard Deviation (σ), which dictates the spread or width of the distribution.
Key Components Explained
- Mean (μ): This is the average value. In a perfectly normal distribution, the mean, median, and mode are all equal.
- Standard Deviation (σ): This measures how much the data deviates from the mean. A small σ creates a tall, narrow curve; a large σ creates a short, wide curve.
- Z-Score: This identifies how many standard deviations a value (x) is away from the mean. A Z-score of 0 is exactly at the mean.
- Probability Density Function (PDF): This indicates the relative likelihood of the random variable taking on the value x.
- Cumulative Distribution Function (CDF): This represents the probability that the variable will take a value less than or equal to x.
Realistic Examples
Example 1: Human Height
If the mean height of men in a city is 175 cm (μ) with a standard deviation of 7 cm (σ), and you want to find the percentile of someone who is 182 cm (x), your Z-score would be 1.0. The CDF would tell you that roughly 84.13% of the population is shorter than this person.
Example 2: Manufacturing Quality Control
A factory produces bolts with a mean diameter of 10mm. The tolerance allows for a standard deviation of 0.05mm. If a bolt measures 10.1mm, it is 2 standard deviations away from the mean, falling into the upper 2.27% of production.
The Math Behind the Curve
The probability density is calculated using the formula:
f(x) = [1 / (σ√(2π))] * e^(-0.5 * ((x-μ)/σ)²)