The Gaussian distribution, also known as the normal distribution, is one of the most important probability distributions in statistics. It forms a characteristic bell-shaped curve that is symmetric about its mean. The distribution is completely defined by two parameters: the mean mu, which determines the center of the distribution, and the standard deviation sigma, which controls how spread out the curve is.
The mathematical formula for the Gaussian distribution is given by the probability density function shown here. This formula contains the exponential function e raised to a negative quadratic term, which creates the characteristic bell shape. The parameters mu and sigma control the position and width of the curve respectively. Let me demonstrate how changing these parameters affects the shape of the distribution.
The Empirical Rule, also known as the sixty eight ninety five ninety nine point seven rule, is a fundamental property of the normal distribution. It tells us that approximately sixty eight percent of all data falls within one standard deviation of the mean, ninety five percent falls within two standard deviations, and ninety nine point seven percent falls within three standard deviations. This rule is extremely useful for understanding data spread and making statistical predictions.
The Standard Normal Distribution is a special case of the Gaussian distribution where the mean is zero and the standard deviation is one. This standardized form is extremely important because any normal distribution can be converted to the standard normal distribution using the Z-score transformation. The Z-score tells us how many standard deviations a value is from the mean, allowing us to compare values from different normal distributions on the same scale.
To summarize what we have learned about the Gaussian distribution: It is a fundamental bell-shaped probability distribution characterized by its mean and standard deviation. The empirical rule tells us how data is distributed within standard deviations. The standard normal distribution provides a universal reference frame. This distribution is essential in statistics and appears throughout natural and social sciences for modeling random phenomena.