Essence and types of mean values in statistics and ways to calculate them. Types of average values in statistics are brief: examples, table

Starting the study of such a science as statistics, it should be understood that it contains (like any science) many terms that need to be known and understood. Today, we will understand such a notion as the average value, and find out which species it shares, how to calculate them. Well, before starting, let's talk a little about history, and about how and why there was such a science as statistics.

History

The very word "statistics" is derived from the Latin language. It is derived from the word "status", and means "state of things" or "situation." This short definition and reflects, in fact, the entire meaning and purpose of statistics. She collects data on the state of affairs and allows you to analyze any situation. The work with statistical data was carried out even in ancient Rome. There the account was taken of free citizens, their possessions and property. In general, statistics were originally used to obtain data on the number of people and their benefits. Thus, in England in 1061 the world's first population census was conducted. Khans, who reigned in Russia in the 13th century, also conducted censuses in order to take tribute from the occupied lands.

Everyone used the statistics for their own purposes, and in most cases this brought the expected result. When people realized that it is not just math, but a separate science that needs to be studied thoroughly, the first scientists who are interested in its development began to appear. People who first became interested in this area and began actively to 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 present 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 socio-social processes, but using only words. They could not imagine the dynamics of events in order to better understand it.

In the first half of the 19th century, another, third direction of this science arose: statistical and mathematical. A great contribution to the development of this direction was made by the famous scientist, statistician from Belgium Adolf Quetelet. It was he who distinguished the types of average values in statistics and, on his initiative, international congresses devoted to this science began to be held. Since the beginning of the 20th century, more complex mathematical methods have begun to be applied in statistics, for example, probability theory.

Today, statistical science is developing through computerization. With the help of different programs, everyone can build a graph based on the proposed data. On the Internet, there are also a lot of resources that provide any statistical data about the population and not only.

In the next section, we will analyze what mean such concepts as statistics, types of mean values and probabilities. Next, we touch on the question of how and where we can use the knowledge gained.

What is statistics?

This is a science whose main purpose is to process information for studying the laws of processes occurring in society. Thus, we can formulate the conclusion that statistics study society and those phenomena that occur in it.

There are several disciplines of statistical science:

1) General theory of statistics. Develops methods of collecting statistical data and is the basis of all other areas.

2) Socio-economic statistics. She studies macroeconomic phenomena from the point of view of the previous discipline and quantitatively characterizes social processes.

3) Mathematical statistics. Not everything in this world can be explored. Something has to be foreseen. Mathematical statistics studies random variables and the laws of probability distribution in statistics.

4) Industry and international statistics. These are narrow areas that study the quantitative aspect of phenomena occurring in certain countries or sectors of society.

And now we will look at the types of mean values in statistics, briefly describe their application in other, not so trivial areas as statistics.

Types of average values in statistics

So we came to the most important, actually, to the topic of the article. Of course, for the mastery of the material and the assimilation of such concepts as the essence and types of mean values in statistics, certain knowledge of mathematics is needed. To begin with, remember that the average is arithmetic, harmonic, geometric and quadratic.

We took the arithmetic average at school. It is very easy to calculate: we take a number of numbers, the middle between which you need to find. Add these numbers and divide the sum by their number. Mathematically, this can be represented as follows. We have a number of numbers, as an example, the simplest series: 1,2,3,4. In total we have 4 numbers. Their average arithmetic is found thus: (1 + 2 + 3 + 4) / 4 = 2.5. It's simple. We begin with this, because it is easier to understand the types of mean values in statistics.

Let us also briefly describe the geometric mean. Take the same series of numbers as in the previous example. But now, in order to calculate the geometric mean, we need to extract the root of the degree, which is equal to the number of these numbers, from their product. Thus, for the previous example, we get: (1 * 2 * 3 * 4) ^1/4 ~ 2.21.

Let us repeat the notion of the mean harmonic. As you can recall from the school course of mathematics, in order to calculate this kind of average, we need to first find the numbers inverse to the numbers of the series. That is, we divide the unit by this number. So we get the inverse numbers. The ratio of their number to the sum and will be the average harmonic. Take for example the same series: 1, 2, 3, 4. The reverse series will look like this: 1, 1/2, 1/3, 1/4. Then the average harmonic can be calculated as follows: 4 / (1 + 1/2 + 1/3 + 1/4) ~ 1,92.

All these kinds of average values in statistics, the examples of which we have considered, are part of the group called power law. There are also structural averages, which we will discuss later. Now we will stop on the first form.

Power averages

We have already analyzed the arithmetic, geometric and harmonic. There is also a more complex view, called the mean square. Although it does not pass at school, it is quite easy to calculate it. It is only necessary to add the squares of the numbers of the series, divide the sum by their number, and extract the square root of all this . For our favorite series it will look like this: ((1 ² +2 ² +3 ² +4 ² ) / 4) ^1/2 = (30/4) ^1/2 ~ 2.74.

In fact, these are all only particular cases of the average power. In a general form, this can be described as follows: a nth power power is equal to a root of degree n from the sum of numbers in the nth power divided by the number of these numbers. While everything is not as difficult as it seems.

However, even the power mean is a particular case of one type - the Kolmogorov average. In fact, all the ways in which we found different averaged values before this can be represented as a single formula: y ^-1 * ((y (x ₁ ) + y (x ₂ ) + y (x ₃ ) + ... + Y (x _n )) / n). Here all the variables x are the numbers of the series, and y (x) is a function by which we take the average value. In the case, say, with the mean square, this is the function y = x ² , and with the arithmetic mean y = x. These are some surprises that statistics sometimes give us. We sorted out the types of average values yet to the end. In addition to the medium there are also structural ones. Let's talk about them.

Structural mean values of statistics. Fashion

Here everything is a little more complicated. To disassemble these types of averages in statistics and how to calculate them, you need to think carefully. There are two main structural averages: fashion and median. We'll deal with the first one.

Fashion is 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? The modal interval is the range of values where any index has the largest frequency. Need for clarity, in order to better represent the mode and types of mean values in statistics. The table, which we consider below, is part of the task whose condition is:

Determine the fashion according to the data of the workshop on day-to-day production.

In our case, the modal interval is a segment of the daily output with the largest number of people, that is, 40-44. Its lower limit is 44.

And now we will discuss how to calculate this very fashion. The formula is not very complicated and can be written as follows: M = x ₁ + n * (f _M -f _{M -1} ) / ((f _M -f _{M -1} ) + (f _M -f _{M + 1} )). Here f _M is the frequency of the modal interval, f _M-1 is the frequency of the interval before the modal (in our case this is 36-40), f _{M + 1} is the frequency of the interval after the modal (for us - 44-48), n is the interval That is, the difference between the lower and upper bounds)? X ₁ is the value of the lower bound (in the example this is 40). Knowing all these data, we can safely calculate the mode for the amount of daily output: M = 40 + 4 * (24-20) / ((24-20) + (24-19)) = 40 + 16/9 = 41, ( 7).

Structural mean values are statistics. Median

We will analyze still such kind of structural sizes, as a median. We will not go into detail on it, we will only talk about the differences with the previous type. In geometry, the median divides the angle in half. It is not in vain in statistics that this kind of medium is so called. If we rank the series (for example, by the size of the population of one or another weight in order of increasing numbers), then the median will be such a value that divides this series into two parts, equal in number.

Other types of averages in statistics

Structural types, along with power grades, do not give everything that is required for calculations in various fields. Allocate and other types of this data. Thus, there are average weights. This type is used when the numbers in the series have a different "real weight". This can be explained by a simple example. Let's take the car. It moves at different speeds at different times. In this case, the values of these time intervals and the values of the velocities differ from each other. So, these intervals will be real weights. Any form of power averages can be made weighted.

In heating engineering, another type of average value is also used: the average logarithmic mean. It is expressed by a rather complex formula, which we will not quote.

Where does this apply?

Statistics - a science that is not tied to any one sphere. Although it was created as part of the social and economic sphere, today its methods and laws are applied in physics, chemistry, and biology. Possessing knowledge in this field, we can easily determine the trends of society and in time to prevent threats. Often we hear phrases "threatening statistics", and these are not empty words. This science tells us about ourselves, and with proper study it can warn about what can happen.

How are the types of means in statistics related?

Relations between them do not always exist, for example, structural types are not related by any formulas to each other. But with power everything is much more interesting. For example, there is a property: the arithmetic mean of two numbers is always greater than or equal to their geometric mean. Mathematically, it can be written as: (a + b) / 2> = (a * b) ^1/2 . The inequality is proved by carrying the right-hand side to the left and further grouping. As a result, we get the root difference, squared. And since any number in the square is positive, respectively, the inequality becomes true.

In addition, there is a more general relationship of magnitudes. It turns out that the average harmonic is always less than the geometric mean, which is less than the arithmetic mean. And the latter turns out, in turn, to be less than the standard mean. You can independently verify the correctness of these relations, at least on the example of two numbers - 10 and 6.

What is interesting about this?

It is interesting that the types of average values in statistics that seem to show just some middle level, in fact can tell a knowing person much more. When we watch the news, no one thinks about the meaning of these figures and how to find them at all.

What else can I read?

To further develop the topic, we recommend reading (or listening) a course of lectures on statistics and higher mathematics. After all, in this article we talked only about a grain of what this science contains, and in itself it is more interesting than it seems at first glance.

How will 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, the statistics will help you to better understand these issues. In general, she can describe almost any side of our life, if there are relevant data at her disposal. Well, then, where and how information is extracted for analysis - the topic of a separate article.

Conclusion

Now we know that there are different kinds of average values in statistics: power and structural. We have figured out how to calculate them and in where and how it can be applied.