statistics

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The symbol (–) placed above an expression signifies that its mean value applies.

definitions

sum

formula: statistics - Sum

arithmetic mean

formula: statistics - Arithmetic mean

weighted arithmetic mean

formula: statistics - Weighted arithmetic mean

mediane

The median for a series of numbers arranged by increasing value is the middle value, or the arithmetic mean of the two central values (when there is an even number of values).

mode

The mode of a series of numbers is the number most frequently encountered, i.e. the one that has the greatest frequency. The mode may not exist or it may not be unique.

Empirical relation between the mean, the median and the mode (figure 1) :

Mode Secured image
Figure 1. Mode

The following empirical relation applies to moderately asymmetrical unimodal density curves :

formula: statistics - Empirical relation applies to moderately asymmetrical unimodal density curves

geometric mean G :

formula: statistics - Geometric mean G

quadatic mean QM :

formula: statistics - Quadratic mean QM

average deviation AD :

formula: statistics - Average deviation AD

standard deviation SD

formula: statistics - Standard deviation SD

variance V

formula: statistics - Variance V

property of the type deviation

In the case of a normal distribution or of a Laplace-Gauss distribution (figure 2), it would seem that :

Distribution- Laplace-Gauss distributionSecured image
Figure 2. Distribution or of a Laplace-Gauss distribution
formula: statistics - Property of the standard deviation

graphic method for adjusting a curve

Using a scatter chart (figure 3), it is often possible to represent a continuous curve that is close to the data. This type of curve is called an adjustment curve.

Dispersion diagramSecured image
Figure 3. Dispersion diagram

The most classic method is the least squares method.

definition

Among all the curves that are close to a series of data, the one that provides the best match is the one that confirms the following property :

formula: statistics - definition

D1, D2, DN being the distances between the curve and the experimental points (figure 4).

curve experimental pointsSecured image
Figure 4. The curve and the experimental points

This is called the curve of the least squares.

regression line of the least squares

formula: statistics -  Straitght line of regression line of the least squares