Mode, mean and median: what they are and how to calculate them

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 valuesin 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 observationscalled 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 bimodalthat 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 valuesso 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. \(7\).