The Gaussian distribution, also known as the normal distribution or bell curve, is a type of continuous probability distribution for a real-valued random variable. It is one of the most important and widely used distributions in statistics and in the natural and social sciences.
The graph of the Gaussian distribution is a bell-shaped curve, symmetric around the mean (average), with its spread determined by its standard deviation. Here are its main properties:
1. **Symmetry:** The distribution is symmetric around the mean. The left and right halves of the distribution are mirror images.
2. **Mean, Median, Mode:** In a Gaussian distribution, the mean (average), median (middle value), and mode (most common value) are all equal and located at the center of the distribution.
3. **Standard Deviation:** The standard deviation measures the dispersion of the distribution. About 68% of the values fall within one standard deviation from the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations. This is known as the 68-95-99.7 rule, or the empirical rule.
4. **Unimodal:** The distribution has one peak. This is where the mode occurs, which also corresponds to the mean and median.
5. **Infinite Range:** Theoretically, the normal distribution extends to positive and negative infinity. Practically, values very far from the mean are extremely unlikely.
The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. Transforming a normal distribution to the standard normal distribution is called standardizing or normalizing.
The equation for the Gaussian distribution is:
f(x) = (1 / √(2πσ^2)) * e^(- (x – μ)^2 / 2σ^2)
where:
– μ is the mean
– σ is the standard deviation
– π is approximately 3.14159
– e is Euler’s number, approximately 2.71828
– x is the point up to which the probability is sought.
The Gaussian distribution is commonly used in statistics and in the natural and social sciences as a simple model for complex random variables. It’s used in a wide range of applications, from finance to physics, due to the central limit theorem, which states that the sum of many independent and identically distributed (i.i.d.) random variables approximately follows a normal distribution, regardless of the shape of the original distribution.