Starting the study of such a science as statistics, you should understand that it contains (like any science) a lot of terms that you need to know and understand. Today we will analyze such a concept as the average value, and find out what types it is divided into, how to calculate them. Well, before we start, let's talk a little about history, and how and why such a science as statistics arose.
History
The very word "statistics" comes from the Latin language. It is derived from the word "status", and means "state of affairs" or "situation". This is a short definition and reflects, in fact, the whole meaning and purpose of statistics. It collects data on the state of affairs and allows you to analyze any situation. The work with statistical data was done in ancient Rome. There was carried out accounting of free citizens, their possessions and property. In general, initially statistics was used to obtain data on the population and their benefits. So, in England in 1061, the world's first census was conducted. The khans who reigned in Russia in the 13th century also conducted censuses to take tribute from the occupied lands.
Everyone used statistics for their own purposes, and in most cases it brought the expected result. When people realized that this is not just mathematics, but a separate science that needs to be studied thoroughly, the first scientists began to appear interested in its development. The people who first became interested in this field and began to actively comprehend it were adherents of two main schools: the English scientific school of political arithmetic and the German descriptive school. The first arose in the middle of the 17th century and aimed to represent social phenomena using numerical indicators. They sought to identify patterns in social phenomena based on the study of statistical data. Supporters of the descriptive school also described social processes, but using only words. They couldn't imagine the dynamics of events to understand it better.
In the first half of the 19th century, another, third direction of this science arose: statistical and mathematical. A well-known scientist, statistician from Belgium, Adolf Quetelet, made a huge contribution to the development of this area. It was he who singled out the types of averages in statistics, and on his initiative, international congresses dedicated to this science began to be held. WithAt the beginning of the 20th century, more complex mathematical methods began to be applied in statistics, for example, the theory of probability.
Today, statistical science is developing thanks to computerization. With the help of various programs, anyone can build a graph based on the proposed data. There are also a lot of resources on the Internet that provide any statistical data about the population and not only.
In the next section, we will look at what concepts such as statistics, types of averages, and probabilities mean. Next, we will touch on the question of how and where we can use the knowledge gained.
What are statistics?
This is a science, the main purpose of which is the processing of information to study the patterns of processes occurring in society. Thus, we can conclude that statistics studies society and the phenomena that take place in it.
There are several disciplines of statistical science:
1) General theory of statistics. Develops methods for collecting statistical data and is the basis of all other areas.
2) Socio-economic statistics. It studies macroeconomic phenomena from the point of view of the previous discipline and quantifies social processes.
3) Mathematical statistics. Not everything in this world can be explored. Something has to be predicted. Mathematical statistics studies random variables and probability distribution laws in statistics.
4) Industry and international statistics. These are narrow areas that study the quantitative side of the phenomena occurring incertain countries or sectors of society.
And now we will look at the types of averages in statistics, briefly talk about their application in other, not so trivial areas like statistics.
Types of averages in statistics
So we come to the most important thing, in fact, to the topic of the article. Of course, in order to master the material and assimilate such concepts as the essence and types of averages in statistics, certain knowledge of mathematics is necessary. First, let's remember what the arithmetic mean, harmonic mean, geometric mean, and quadratic mean are.
We took the arithmetic mean at school. It is calculated very simply: we take several numbers, the average between which must be found. Add these numbers and divide the sum by their number. Mathematically, this can be represented as follows. We have a series of numbers, as an example, the simplest series: 1, 2, 3, 4. We have 4 numbers in total. We find their arithmetic mean in this way: (1 + 2 + 3 + 4) / 4 \u003d 2.5. Everything is simple. We start with this because it makes it easier to understand the kinds of averages in statistics.
Let's also briefly talk about the geometric mean. Let's take the same series of numbers as in the previous example. But now, in order to calculate the geometric mean, we need to take the root of the degree, which is equal to the number of these numbers, from their product. Thus, for the previous example, we get: (1234)1/4~2, 21.
Let's repeat the concept of harmonic mean. As you can remember from the school mathematics course,To calculate this kind of mean, we first need to find the reciprocals of the numbers in the series. That is, we divide one by this number. So we get the reverse numbers. The ratio of their number to the sum will be the harmonic mean. Let's take the same row as an example: 1, 2, 3, 4. The reverse row will look like this: 1, 1/2, 1/3, 1/4. Then the harmonic mean can be calculated as follows: 4/(1+1/2+1/3+1/4) ~ 1, 92.
All of these types of averages in statistics, examples of which we have seen, are part of a group called power-law. There are also structural averages, which we will discuss later. Now let's focus on the first view.
Power mean values
We have already covered arithmetic, geometric and harmonic. There is also a more complex form called root mean square. Although it is not passed at school, it is quite simple to calculate it. It is only necessary to add the squares of the numbers in the series, divide the sum by their number, and take the square root of all this. For our favorite row, it would look like this: ((12+22+32+ 42)/4)1/2=(30/4)1/2 ~ 2, 74.
Actually, these are only special cases of the mean power law. In general terms, this can be described as follows: the power of the nth order is equal to the root of the degree n of the sum of numbers to the nth power, divided by the number of these numbers. So far, things are not as difficult as they seem.
However, even the power mean is a special case of one type - the Kolmogorov mean. Byin fact, all the ways in which we found different averages before can be represented in the form of one formula: y-1((y(x1)+y(x2)+y(x3)+…+y(x))/n). Here, all variables x are the numbers of the series, and y(x) is a certain function by which we calculate the average value. In the case, say, with the mean square, this is the function y=x2, and with the arithmetic mean y=x. These are the surprises sometimes given to us by statistics. We have not yet fully analyzed the types of average values. In addition to averages, there are also structural ones. Let's talk about them.
Structural averages of statistics. Fashion
This is a bit more complicated. Understanding these kinds of averages in statistics and how they are calculated requires a lot of thought. There are two main structural averages: mode and median. Let's deal with the first one.
Fashion is the most common. It is used most often to determine the demand for a particular thing. To find its value, you must first find the modal interval. What it is? Modal interval is the area of values where any indicator has the highest frequency. Visualization is needed to better represent the fashion and types of averages in statistics. The table that we will look at below is part of the problem, the condition of which is:
Determine the fashion according to the daily output of the shop workers.
| Daily output, units | 32-36 | 36-40 | 40-44 | 44-48 |
| Number of workers, people | 8 | 20 | 24 | 19 |
In our case, the modal interval is the segment of the daily output indicator with the largest number of people, that is, 40-44. Its lower limit is 44.
And now let's discuss how to calculate this very fashion. The formula is not very complicated and you can write it like this: M=x1+ n(fM-fM -1)/((fM-fM-1)+(fM-fM+1)). Here fM is the frequency of the modal interval, fM-1 is the frequency of the interval before the modal (in our case it is 36-40), f M+1 - the frequency of the interval after the modal (for us - 44-48), n - the value of the interval (that is, the difference between the lower and upper limits)? x1 - value of the lower limit (in the example it is 40). Knowing all these data, we can safely calculate the fashion for the amount of daily output: M=40 +4(24-20)/((24-20)+(24-19))=40 + 16/9=41, (7).
Structural averages statistics. Median
Let's take another look at such a type of structural values as the median. We will not dwell on it in detail, we will only talk about the differences with the previous type. In geometry, the median bisects the angle. It is not for nothing that this type of average value is so called in statistics. If you rank a series (for example, by the population of one or another weight in ascending order), then the median will be a value that divides this series into two parts equal in size.
Other types of averages in statistics
Structural types, coupled with power types, do not give everything that is requiredfor calculations in various areas. There are other types of this data. Thus, there are weighted averages. This type is used when the numbers in the series have different "real weights". This can be explained with a simple example. Let's take a car. It moves at different speeds for different periods of time. At the same time, both the values of these time intervals and the values of velocities differ from each other. So, these intervals will be real weights. Any kind of power mean can be made weighted.
In heat engineering, one more type of average values is also used - the average logarithmic. It is expressed by a rather complex formula, which we will not give.
Where does it apply?
Statistics is a science not tied to any one area. Although it was created as part of the socio-economic sphere, today its methods and laws are applied in physics, chemistry, and biology. With knowledge in this area, we can easily determine the trends of society and prevent threats in time. Often we hear the phrase "threatening statistics", and these are not empty words. This science tells us about ourselves, and with proper study it can warn of what can happen.
How are types of averages related in statistics?
Relations between them do not always exist, for example, structural types are not connected by any formulas. But with power everything is muchmore interesting. For example, there is such a property: the arithmetic mean of two numbers is always greater than or equal to their geometric mean. Mathematically it can be written like this: (a+b)/2 >=(ab)1/2. The inequality is proved by moving the right side to the left and further grouping. As a result, we get the difference of the roots, squared. And since any number squared is positive, accordingly, the inequality becomes true.
Besides this, there is a more general ratio of magnitudes. It turns out that the harmonic mean is always less than the geometric mean, which is less than the arithmetic mean. And the latter turns out to be, in turn, less than the root mean square. You can independently check the correctness of these ratios at least on the example of two numbers - 10 and 6.
What's so special about this?
It's interesting that the kinds of averages in statistics that seem to show just some kind of average, in fact, can tell a knowledgeable person much more. When we watch the news, no one thinks about the meaning of these numbers and how to find them at all.
What else can I read?
For further development of the topic, we recommend reading (or listening to) a course of lectures on statistics and higher mathematics. After all, in this article we talked about only a grain of what this science contains, and in itself it is more interesting than it seems at first glance.
HowWill this knowledge help me?
Perhaps they will be useful to you in life. But if you are interested in the essence of social phenomena, their mechanism and influence on your life, then statistics will help you understand these issues more deeply. In general, it can describe almost any aspect of our life, if it has the appropriate data at its disposal. Well, where and how information is obtained for analysis is the topic of a separate article.
Conclusion
Now we know that there are different types of averages in statistics: power and structural. We figured out how to calculate them and where and how it can be applied.
