Mode, mean and median are characteristics of any set of statistical data and allow to synthetically express the properties of a statistical survey. They are classified as statistical position indices.

BRIEF

### FASHION, MEDIA AND MEDIA: WHAT ARE THEY FOR?

**Mode, mean and median** I am **statistical position indices** which allow you to **evaluate the order of magnitude of a set of statistical data** and locate the distribution. They are used to summarize information from a set of data, collected, for example, through a statistical survey.

### Medium

**The average is the most used position index** perform an analysis of a data set; there are different types of media that can be used in the analysis of a phenomenon. The general definition of the mean was proposed by the Italian mathematician **Oscar Chesini**: given a sample of \(n\) elements, \((x_1, x_2, \cdots, x_n)\), and a function \(f\) of \(n\) variables, this value is defined as mean \(M\) such that replacing it with all variables causes the value of the function to remain unchanged, i.e. \(f(x_1, x_2, \cdots, x_n) = f(M, M, \cdots,M)\). All the most common averages are obtained as special cases of this definition, using a particular \(f\).

### The arithmetic mean

**The arithmetic mean is the most commonly used type of mean** and is calculated by adding all the values obtained then dividing by the number of values themselves:

$$ M_a=\frac{1}{n}\cdot\sum_{i=1}^{n}x_i; $$

The weighted arithmetic mean (or weighted average) is used if the available values have a “weight”, or rather a different importance; is calculated by adding all the available values multiplied by their weights then dividing by the sum of the weights:

$$ M_{a, pond}=\frac {\sum_{i=1}^{n}x_i f_i}{\sum_{i=1}^{n}f_i}, $$

where \ (f_i \) is the weight of the term \ (i-th \).

The arithmetic mean has several properties. Among the most important are:

- \(\sum_{i=0}^{n}(x_i-M_a)=0\);
- The arithmetic mean is a linear operator: \ (M[ax+b]= aM_x + b \), where \ (M_x \) is the arithmetic mean of character \ (x \).

For example, given the numbers \ (x_1 = 5, x_2 = 8, x_3 = 10, x_4 = 3 \), the arithmetic mean is calculated as follows:

$$ M_a = \frac {5 + 8 + 10 + 3} {4} = 6.5. $$

### The geometric mean

**The geometric mean** of \(n\) elements is calculated by taking the root \(nth\) of the product of \(n\) elements:

$$M_g=\sqrt[n]{\prod_{i=1}^{n}x_i}. $$

If you want to assign a weight to the items, you can use the weighted geometric mean calculation:

$$M_{g, pond}=\sqrt[\sum_{i=1}^{n} f_i]{\prod_{i=1}^{n}x_i^{f_i}}, $$

where \ (f_i \) is the weight of the term \ (i-th \).

The geometric mean** it is used when positive values are available and when the considered values are multiplied rather than added**, for example when calculating the interest rate or the growth rate. A characteristic of this type of media is that **small values are much more influential than large values**in the total calculation.

For example, given the numbers \ (x_1 = 5, x_2 = 8, x_3 = 10, x_4 = 3 \), the geometric mean is calculated as follows:

$$M_g=\sqrt[4]{5\cdot 8\cdot 10\cdot 3}=\sqrt[4]{1200} = 5.9. $$

### The harmonic mean

**The harmonic mean** of \(n\) elements is defined as the inverse of the arithmetic mean of the inverses of the individual values:

$$M_h=\frac{n}{\sum_{i=1}^{n}\frac{1}{x_i)}. $$

If you want to assign a weight to the elements, you can use the weighted harmonic mean calculation:

$$ M_h=\frac{\sum_{i=1}^{n}f_i}{\sum_{i=1}^{n}\frac{f_i}{x_i)}, $$

where \ (f_i \) is the weight of the term \ (i-th \).

The harmonic mean **it is noticeably affected by minor module elements** **and is less affected by terms that differ greatly from other observations**called outliers.

For example, given the numbers \ (x¬_1 = 5, x_2 = 8, x_3 = 10, x_4 = 3 \), the harmonic mean is calculated as follows:

$$ M_h = \frac{4}{\frac{1}{5}+\frac{1}{8}+\frac{1}{10}+\frac{1}{3}}=3.03 . $$

### The average power

**The average power** (or generalized) is a **generalization of Pythagorean means** and it is calculated by taking the root \(k-th \) of the arithmetic mean of the exponent powers \(k\) of the \(n\) observed values:

$$ M_p=\biggl(\frac{1}{n}\cdot\sum_{i=1}^{n}x_i^k\biggr)^{\frac{1}{k)}. $$**The other types of media are special cases of the average power**: for \ (k = 1 \) we obtain the arithmetic mean; for \ (k = -1 \) we obtain the harmonic mean; for \ (k \ rightarrow 0 \) we get the geometric mean; for \ (k = 2 \) we obtain the quadratic mean; \ (\ cdots \).

If you want to assign a weight to the elements, you can use the weighted average power calculation:

$$ M_p=\biggl(\frac{1}{\sum_{i=1}^{n}f_i}\cdot\sum_{i=1}^{n}f_i\cdot x_i^k\biggr)^{ \frac {1} {k}}, $$

where \ (f_i \) is the weight of the term \ (i-th \).

### Fashion

**Fashion** of a frequency distribution X **is the value, if any, that occurs most frequently** and is often denoted by the symbol \ (v_0 \). It may happen that a distribution is *bimodal*that is, there are two values that appear with the same frequency (or *trimodal* if there are three, and so on); this characteristic indicates that the distribution may not be homogeneous. **In the case of a Gaussian distribution, the value of the mode coincides with that of the median and the mean.** For example, considering a set of data: \ (x_1 = 5, x_2 = 8, x_3 = 10, x_4 = 3, x_5 = 10, x_6 = 7 \), the mode is \ (10 \) .

### The median

in statistics **the median is the value that is in the middle position of the data set ranked in ascending or descending order.** In case you have a **odd number** **values**, **the median is simply the central value**, i.e. the one at position \(\frac{n+1}{2}\); in case you have a **even number** **values**so **the median will be given by the arithmetic mean of the two central data**, i.e. between the data at position \(\frac{n}{2}\) and \(\frac{n+1}{2}\). Also, the median is the value for which the cumulative relative frequency is \(0.5\). For example, considering a set of data: \ (x_1 = 5, x_2 = 8, x_3 = 10, x_4 = 3, x_5 = 10, x_6 = 7 \), the median, since there is an even number of terms, is given by the arithmetic mean between \(7\) and \(8\), or \(\frac{7+8}{2}=7.5\). Considering an odd data set: \ (x_1 = 5, x_2 = 8, x_3 = 10, x_4 = 3, x_5 = 7 \), the median will be the term that is in the middle position after arranging the data in ascending order or descending order, i.e. \(7\).

Source

*STATISTICAL INDEXES MEDIA, MODE, MEDIAN, VARIANCE***Unimi***Synthesizing data: descriptive statistics***SIN-RIDT**