Gaussian curve and distribution curve

Gaussian curve and distribution curve

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Gaussian curve, a mathematical concept but not at all disconnected from reality because it is able to represent many everyday situations by simplifying their interpretation. So let's see what it is and what it tells about what is happening around us. We find it drawn in a plane of Cartesian coordinates but also as a distribution or as a surface.

Gaussian curve

The Gaussian, or Gaussian curve, was "invented" by the German mathematician Karl Friedrich Gauss. Its formulas and all that is mathematical behind it, is known to few, but its general meaning and its usefulness is known to many.

Gaussian curve and distribution

When we draw a Gaussian curve we try to represent a certain event representing graphically the distribution of its possible values. Let's take the outcome of a coin toss or something more complex such as the people who support a certain team divided by age groups.

To obtain the Gaussian distribution of a value we are measuring, it is necessary to carry out many measurements of the same quantity with an instrument, collecting the various results. The same number will not always be obtained due to the accuracy errors of our instrument and also for those related to our work, called accidental errors. The more they are many of our measures, plus their representation on a graph will be a Gaussian curve.

Gaussian curve and table

In accompaniment to a Gaussian curve we can also find a table with the values ​​that correspond to the various points in the plane represented and joined. Watching the Gaussian curve and the table, we see that there is a maximum point of the “bell”Which then goes down more or less evidently.

It depends on the dispersion of the values ​​around the mean which is measured with the standard deviation. Table in hand, we can say that for the Gaussian curve o 68% of the measurements differ from the mean by less than the standard deviation and that 95% put two standard deviations. If the standard deviation then has a high value, we will have a bell, therefore one Gaussian which descends more softly before and after its maximum and the maximum, at this point, corresponds to a value yes more probable but not always representative.

Since we are talking about probability of obtaining a certain value, also for the Gaussian curve, the area underlying it is worth 1. The sum of the probabilities of all values ​​must give 1.

Gaussian curve and surface

The Gaussian surface is a concept related to Gauss's law. Specifically, given an electric field, to identify the Gaussian surface it is necessary to find in a three-dimensional space the surface normal to the electric field at each point. Closed in three dimensional space, crossed by an electric field flux, this surface can also be simply an infinite sphere or cylinder. This happens respectively when the field we consider is produced by a point charge and a conductor wire of infinite length.

Gaussian curve and function

Forwarding more and more in the field of physics and mathematics, we also find the Gaussian functions in which integral is the function of errors. Some examples of Gaussian function, indeed, one for all: the wave function of the ground state ofquantum harmonic oscillator. This is why we hear about Gaussian functions in the quantum field theory.

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Video: Integrating Normal Density Function (June 2022).


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