Explaining Gaussian Distribution
The Gaussian distribution, also known as the normal distribution, is one of the most fundamental concepts in statistics and probability theory (mentioned it in my article on Gauss and knew an article was needed). Named after the renowned mathematician Carl Friedrich Gauss, this probability distribution model underpins a wide range of natural phenomena and is instrumental in various fields, from science and engineering to finance and social sciences (yeah, believe it or not).
Understanding the Gaussian Distribution
At its core, the Gaussian distribution is characterised by its bell-shaped curve, symmetrically centered around its mean value. This bell curve illustrates the probability distribution of a continuous random variable, where the data tends to cluster around the mean, becoming less likely as it moves further away from the mean in either direction. Sounds very strange but is incredibly useful!
The probability density function (PDF), not Portable Document Format of the Gaussian distribution is defined by the equation:
f(x) = (1 / (σ√(2π))) * e^(-(x-μ)² / (2σ²))
where: μ is the mean of the distribution. σ is the standard deviation, which represents the spread or dispersion of the data. e is the base of the natural logarithm, approximately equal to 2.71828. e, Is obviously Euler’s number.
Properties of the Gaussian Distribution
Symmetry: The curve of the Gaussian distribution is symmetrical about its mean, μ. This means that the probability of observing a value x above the mean is the same as the probability of observing a value below the mean at the same distance from it.
Unimodality: The Gaussian distribution has a single peak at its mean, and it is the most likely value to occur.
Empirical Rule: The Gaussian distribution follows the empirical rule, also known as the 68–95–99.7 rule. Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Funny, the concept of empiricism is also mentioned quite a bit in philosophy.
Applications of the Gaussian Distribution
Usually, these tend to be short, concise lists. But not this time. There are A LOT of applications, so what you’ve learned today is actually very useful!
Natural Phenomena: Many physical and biological phenomena naturally follow the Gaussian distribution. Examples include the height of individuals in a population, errors in measurements, and the distribution of exam scores in a large student population (thank Gauss for your A’s I guess…)
Statistical Inference: The Gaussian distribution is widely used in statistical inference, including hypothesis testing and confidence intervals. Its well-defined properties and central limit theorem make it a useful tool for making predictions and drawing conclusions from sample data.You might not think so, but this is actually very important — complicated-looking words sometimes make you forget that!
Machine Learning: Gaussian distributions are commonly used in machine learning algorithms, such as Gaussian Naive Bayes and Gaussian Mixture Models, for classification and clustering tasks. Impressive if you know what any of those are! Let me know!
Finance and Economics: In finance, the Gaussian distribution is employed to model stock returns, asset prices, and risk management processes. So yeah, thank gauss for your money… sort of…
Quality Control: Gaussian distributions play a vital role in quality control processes to monitor variations in manufacturing processes and ensure product consistency. It’s always when you don’t expect it, that’s when Gauss appears.
Done!
