Normal Distribution Calculator

Understanding Normal Distribution

The Normal Distribution, also known as the Gaussian Distribution or simply the Bell Curve, is one of the most important probability distributions in statistics. It is a continuous probability distribution that is perfectly symmetrical around its center, meaning that data near the mean are more frequent in occurrence than data far from the mean.

A normal distribution is fully defined by two parameters:

• Mean (μ): The location of the peak of the bell curve. Since the distribution is symmetrical, the mean is also the median and the mode.
• Standard Deviation (σ): The measure of spread or variability. A smaller standard deviation results in a taller, narrower curve, while a larger standard deviation spreads the curve wider and flatter.

The Empirical Rule (68-95-99.7 Rule)

For any normal distribution, the probability of a data point falling within a certain distance from the mean is predictable:

• 68.27% of the data falls within 1 standard deviation (μ ± 1σ).
• 95.45% of the data falls within 2 standard deviations (μ ± 2σ).
• 99.73% of the data falls within 3 standard deviations (μ ± 3σ).

Standard Normal Distribution & Z-Scores

The Standard Normal Distribution is a special case of normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. Any normal distribution can be standardized by converting its raw values (x) into standard scores (Z-scores) using the formula:

Z = (x - μ) / σ

The Z-score represents how many standard deviations a value is above or below the mean. If Z is positive, the value is above average; if negative, it is below average. Once a Z-score is calculated, we can look up standard normal tables (or use this calculator) to find the cumulative probability.

Probability Density Function (PDF) vs. Cumulative Distribution Function (CDF)

This calculator handles two core statistical concepts:

• PDF (Probability Density Function): Represents the height of the curve at a given x value. It does not directly represent probability, but represents the density of probability at that point.
• CDF (Cumulative Distribution Function): Calculates the area under the curve to the left of a given x value. The total area under the bell curve is always exactly 1.0 (representing 100% probability).

For related statistical tools, try the Standard Deviation Calculator, Variance Calculator, and Average Calculator.