Une distribution gaussienne, communément appelée distribution normale, is a fundamental concept in statistics and théorie des probabilités. It is characterized by its symmetric, bell-shaped curve, where most observations cluster around the central peak, and probabilities for values further away from the mean taper off equally in both directions. The Gaussian distribution is defined by two parameters: the mean (μ), which indicates the center of the distribution, and the standard deviation (σ), which measures the spread or dispersion of the data.
Mathématiquement, la fonction de densité de probabilité (FDP) d'une distribution gaussienne s'exprime comme suit :
f(x) = (1 / (σ√(2π))) * e^(-(x – μ)² / (2σ²))
In this equation, ‘e’ is the base of the natural logarithm, and ‘π’ is the constant pi. The shape of the distribution is determined by the mean and standard deviation: a larger standard deviation results in a wider and flatter curve, while a smaller standard deviation yields a steeper and narrower peak.
The Gaussian distribution is crucial in various fields, including statistics, natural and sciences sociales, and machine learning, due to the Théorème Central Limite, which states that, given a sufficiently large sample size, the distribution of sample means will be approximately normally distributed, regardless of the original distribution of the data. This property makes Gaussian distributions a key component in statistical modeling, hypothesis testing, and data analysis.