Starting to study a science such as statistics, it should be understood that it contains (like any science) many 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 you start, let's talk a little bit about history, and how and why such a science as statistics arose.
Story
The word "statistics" itself originates from the Latin language. This is a derivative of the word “status”, and means “state of affairs” or “situation”. This is a short definition and reflects, in fact, the whole point and purpose of statistics. It collects data on the state of things and allows you to analyze any situation. Work with statistics was carried out in ancient Rome. There, free citizens, their possessions and property were recorded. In general, initially statistics were used to obtain data on the number of people and their benefits. So, in England in 1061, the world's first census was conducted. The khans that reigned in Russia in the 13th century also conducted censuses to take tribute from the occupied lands.
Each used statistics for their own purposes, and in most cases this brought the expected result. When people realized that this was not just mathematics, but a separate science that needed to be studied thoroughly, the first scientists interested in its development began to appear. People who first became interested in this area and began to comprehend it actively were adherents of two main schools: the English scientific school of political arithmetic and the German descriptive school. The first arose in the mid-17th century and set the goal of representing social phenomena using numerical indicators. They sought to identify patterns in social phenomena through the study of statistical data. Supporters of the descriptive school also described social and 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 area of this science arose: statistical and mathematical. A huge contribution to the development of this direction was made by a famous scientist, statistician from Belgium Adolf Ketle. It was he who identified the types of averages in statistics, and on his initiative international congresses dedicated to this science began to be held. From the beginning of the 20th century, more complex mathematical methods, for example, probability theory, began to be used in statistics.
Today, statistical science is developing through computerization. With the help of various programs, everyone can build a graph based on the proposed data. There are also many resources on the Internet that provide any statistics on the population and not only.
In the next section, we will examine what such concepts as statistics, types of averages, and probabilities mean. Next, we will address the question of how and where we can use the knowledge gained.
What is statistics?
This is a science whose main purpose is the processing of information to study the laws of processes occurring in society. Thus, we can formulate the conclusion that statistics study society and the phenomena that occur in it.
There are several disciplines of statistical science:
1) The general theory of statistics. Develops methods for 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 study random variables and probability distribution laws in statistics.
4) Industry and international statistics. These are narrow areas that study the quantitative side of phenomena occurring in certain countries or sectors of society.
And now we will look at the types of average values in statistics, briefly talk about their application in other, not so trivial areas, as statistics.
Types of averages in statistics
So we come to the most important, in fact, to the topic of the article. Of course, for the development of the material and the assimilation of such concepts as the essence and types of average values in statistics, certain knowledge of mathematics is necessary. To begin with, recall what the arithmetic mean, harmonic, geometric and quadratic mean.
Arithmetic mean we passed in school. It is calculated very simply: we take several numbers, the average between which must be found. Add up these numbers and divide the amount 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. In total we have 4 numbers. We find their arithmetic mean in this way: (1 + 2 + 3 + 4) / 4 = 2.5. Everything is simple. We start with this, because it is easier to understand the types of averages in statistics.
We also briefly describe the geometric mean. Take the same series of numbers as in the previous example. But now, 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 concept of harmonic mean. As you can recall from the school course in mathematics, in order to calculate this kind of average, we must first find the numbers that are inverse to the numbers in the series. That is, we divide the unit by this number. So we get the inverse numbers. The ratio of their quantity to the total will be the harmonic mean. Take for example the same row: 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 these types of averages in statistics, the examples of which we have examined, are part of a group called power. There are also structural averages, which we will analyze later. Now dwell on the first form.
Power Average Values
We have already disassembled arithmetic, geometric and harmonic. There is also a more complex view called the root mean square. Although they don’t go to school, calculating it is quite simple. 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 from all this . For our favorite series, it will look like this: ((1 2 +2 2 +3 2 2 +4 2 ) / 4) 1/2 = (30/4) 1/2 ~ 2.74.
In fact, these are just special cases of the middle power. In general terms, this can be described as follows: a power of the nth order is equal to a root of degree n from the sum of numbers in the nth power divided by the number of these numbers. So far, everything is not as complicated as it seems.
However, even a power-law average is a special case of one species — the Kolmogorov mean. In fact, all the ways in which we found different averaged values before that can be represented as one 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 certain function by which we calculate the average value. In the case of, say, the quadratic mean, this is the function y = x 2 , and with the arithmetic mean y = x. Here are some surprises we are sometimes presented with statistics. Types of average values, we have not yet figured out. In addition to the average, there are also structural ones. Let's talk about them.
Structural averages of statistics. Fashion
Everything is a bit more complicated here. To analyze these types of averages in statistics and how to calculate them, you need to think carefully. There are two main structural averages: mode and median. We will deal with the first.
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 indicator has the highest frequency. Visibility is needed to better represent the mode and types of averages in statistics. The table that we will consider below is part of the task, the condition of which is this:
Determine the fashion according to the daily production workshop.
| Daily output, pcs. | 32-36 | 36-40 | 40-44 | 44-48 |
| The number of workers, people | 8 | twenty | 24 | 19 |
In our case, the modal interval is a segment of the daily output indicator with the largest number of people, i.e. 40-44. Its lower bound is 44.
And now we will discuss how to calculate this very mode. The formula is not very complicated and you can write it like this: 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 it is 36-40), f M + 1 is the frequency of the interval after the modal (for us, 44-48), n is the value of the interval ( that is, the difference between the lower and upper bounds)? x 1 is the value of the lower boundary (in the example it 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 averages are statistics. Median
We will also analyze such a kind of structural quantities 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 divides the angle in half. It is not for nothing that in statistics this type of average size was so called. If you rank a series (for example, by the population of a given weight in ascending order of number), then the median will be a value that divides this series into two parts, equal in number.
Other types of averages in statistics
Structural types, coupled with power ones, do not give everything that is required for calculations in various fields. Other types of this data are also distinguished. Thus, there are weighted averages. This type is used when the numbers in the series have different "real weight". This can be explained with a simple example. Take a car. It moves at different speeds at different intervals. In this case, the values of these time periods and the values of the speeds differ from each other. So, these gaps will be material weights. Any kind of power means can be weighted.
In heat engineering, another type of average value is also used - the average logarithmic. It is expressed by a rather complicated formula, which we will not give.
Where does this apply?
Statistics is a science that is 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 identify social trends 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 is able to warn about what might happen.
How are the 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 much more 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, you can write this: (a + b) / 2> = (a * b) 1/2 . The inequality is proved by moving the right side to the left and further grouping. As a result, we get the root difference squared. And since any number squared is positive, accordingly, the inequality becomes true.
In addition, there is a more general ratio of values. It turns out that the harmonic mean is always less than the geometric mean, which is less than the arithmetic mean. And the latter is, in turn, less than the mean square. You can independently verify the correctness of these relations, at least for the example of two numbers - 10 and 6.
What is interesting about this?
It is interesting that the types of average values in statistics, which, it would seem, show just some average level, actually 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 you read?
For further development of the topic, we recommend reading (or listening to) a course of lectures on statistics and higher mathematics. Indeed, 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 impact on your life, then statistics will help you better understand these issues. In general, she can describe almost any side of our life if she has the relevant data. Well, where and how information is extracted for analysis is the topic of a separate article.
Conclusion
Now we know that there are different types of average values in statistics: power and structural. We figured out how to calculate them and how and where to apply it.