Fashion, average and median

Average, fashion and averageare measurements obtained from sets of data that can be used to represent the entire set. The tendency of these measures is to result in a valuecentral. For this reason, they are called centrality measures.

Fashion

The most frequent data in a set is called fashion. See an example:

In a music school, classes are made up of only 8 students. In the “A” class, Mateus, Mateus, Rodrigo, Carolina, Ana, Ana, Ana and Teresa are enrolled.

Note that there are two boys named Matthew and three girls named Hannah. The name that is repeated the most is Ana and, therefore, it is the fashion for this dataset.

Now an example with numbers: in a music school, the eight students in class “A” have the following ages: 12 years old, 13 years old, 13 years old, 12 years old, 11 years old, 10 years old, 14 years old and 11 years old.

Note that ages 11, 12 and 13 repeat the same number of times and no age appears more than these three. In this case, the set has three modes (11, 12 and 13) and is called trimodal.

There may also be sets bimodal, that is, with two fashions; amodal, with no fashion etc.

Mind Map: Central Trend Measures

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median

If the set of information is numeric and is arranged in ascending or descending order, its average will be the number that occupies the center position in the list. Consider that the aforementioned music school has nine teachers and that their ages are:

32 years, 33 years, 24 years, 31 years, 44 years, 65 years, 32 years, 21 years and 32 years

To find the average of the ages of the teachers, we must organize the list of ages in ascending order:

21, 24, 31, 32, 32, 32, 33, 44 and 65

Note that number 32 is the fifth. On your right there are another 4 ages as well as on the left. Therefore, 32 is the median of the list of teachers' ages.

21, 24, 31, 32, 32, 32, 33, 44, 65

If the list has a number pair of information, to find the average (M_The), we must find the two core values ​​(a₁ and the₂) from the list, add them up and divide the result by 2.

M_The = The₁ + the₂
2

If the teachers' ages were 19 years, 19 years, 18 years, 22 years, 44 years, 45 years, 46 years, 46 years, 47 years and 48 years, the growing list with both measurescentrals would be:

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18, 19, 19, 22, 44, 45, 46, 46, 47, 48

Note that the amount of information to the right and left of these two numbers is exactly the same. THE average of this dataset is therefore:

M_The = The₁ + the₂
2

M_The = 44 + 45
2

M_The = 89
2

M_The = 44.5 years

Average

Average (M), more precisely called simple arithmetic mean, it is the result of summing all the information in a dataset divided by the number of pieces of information that has been summed. THE simple arithmetic mean between 14, 15 and 25, for example, is as follows:

M = 14 + 15 + 25
3

Since there are three dice in the list, we divide the sum of these dice by the number 3. The result is:

M = 54
3

M = 18

THE average and the measureincentrality most used because it more evenly blends the lowest and highest values ​​in a list. In the previous set, for example, the average is equal to 44.5, even with so many ages close to 20 years. Note the average simple arithmetic of that same set:

M = 18 + 19 + 19 + 22 + 44 + 45 + 46 + 46 + 47 + 48
10

M = 35.4 years

weighted average

THE weighted average (M_P) is an extension of the simple mean and considers weights for the information in the dataset. It is done by summing the product of an information by its respective weight and then dividing this result by the sum of all weights used.

Consider the data in the following table as an example, which lists the ages of the sixth graders in school A. Let's calculate the average of ages.

It is possible to calculate the simple average by adding 10 years four times, 11 years fifteen times, etc. However, through a averageweighted, we can consider the number of students aged 11 as the weight of that age in this classroom; the number of students who are 10 years old as the weight of that age, and so on until all ages have been added. Thus, the calculation of the weighted average would be:

M_P = 4·10 + 15·11 + 10·12 + 1·13
4 + 15 + 10 + 1

M_P = 40 + 165 + 120 + 13
30

M_P = 338
30

M_P = 11.26 years.

By Luiz Paulo Moreira
Graduated in Mathematics