Statistics related papers 1 on the application of probability in statistics
Abstract: Probability is a mathematical subject that studies random phenomena. Its theory is rigorous, widely used and developing rapidly. At present, the theory and method of probability have been widely used in statistics. This paper mainly introduces some applications of probability in statistics from two aspects: normal distribution and small probability events.
Keywords: random phenomenon; Events; Sample; Mother; Normal distribution; Small probability principle
Statistics are mainly divided into descriptive statistics and inferential statistics. Given a set of data, statistics can summarize and describe these data. This usage is called descriptive statistics. In addition, the observer establishes a mathematical model in the form of data to explain its randomness and uncertainty, thus inferring the steps and matrices in the research. This usage is called applied statistics. In addition, there is a subject called mathematical statistics, which is devoted to discussing the theoretical basis behind this subject.
When the same instrument measures the weight of the same object many times, the results are always slightly different from each other, which is caused by accidental factors such as the influence of the measuring instrument on the atmosphere and the physiological or psychological changes of the observer. Similarly, the same gun fires multiple similar shells at the same target, and the impact point is different, because various accidental factors in the manufacture of shells will also affect the quality of shells. In addition, the position error of the barrel, the slight change of weather conditions and so on all affect the impact point. For example, the life of light bulbs produced by the same process in a production line is also different.
In short, a similarity of these phenomena is that under the same basic conditions, after a series of experiments or observations, different results will be obtained. In other words, as far as individual test results or observation results are concerned, this result will appear from time to time, and that result will appear from time to time, showing a kind of contingency. This phenomenon is called random phenomenon. For random phenomena, we are usually concerned about whether a certain result appears in experiments or observations. This result is called a random event, or simply an event. For practical reasons, a subset of the research group is selected to replace every piece of data in the research matrix, and this subset is called a sample. Inference statistics is used to model the data in the data, calculate its probability and infer the mother. This inference can present (hypothesis testing) the prediction of future observation, the prediction of correlation, or the modeling of (regression) relationship with correct or wrong answers.
Random phenomenon has its contingency and inevitability. This inevitability is manifested in the stability of the frequency of random events in a large number of experiments, that is, the frequency of a random event often swings around a fixed constant. This law is called statistical regularity. The stability of frequency shows that the probability of random events is an inherent objective attribute of random events and does not change with people's will, so it can be measured. For random event A, a number p(A) is used to represent the probability of the occurrence of the event. This number p(A) is called the probability of random event A, so the probability measures the probability of the occurrence of the random event.
If the sample is representative of the matrix, the inferences and conclusions made by the sample can be extended to the whole matrix. Statistics provides many methods to estimate and correct the randomness (error) in the process of sample data. To understand the randomness of a probability, we must have basic mathematical concepts. Mathematical statistics is a branch of applied mathematics, which uses probability theory to analyze and verify the theoretical basis of statistics.
Probability plays an important role in statistics, including population, sampling research, statistical description, statistical inference, normal distribution law and so on. Normal distribution is the most important distribution in probability. On the one hand, normal distribution is the most common distribution in nature, such as measurement error; Distribution of impact points of shells; Dimensions of human physiological characteristics: body length, weight, etc. The output of crops; The dimensions of factory products: diameter, length, width and height, all obey the normal distribution approximately.
Generally speaking, if there are many random factors affecting a certain quantity index, and each factor plays a small role, then it can be proved that it obeys normal distribution by the limit theorem of probability theory. On the other hand, normal distribution has many good properties, many of which can be approximated by normal distribution, and some of which can be derived from normal distribution, so it is very important in theoretical research. For example, if we use the normal distribution law to count the distribution of school grades, we can find out whether the students in a stage have made overall progress, and then find out the reasons and get improvement methods. Analyze the economic development in one year and forecast the income in the coming year. Find out the main factors that affect development, seek ways to improve and so on.
A small probability event is an event with a small probability of occurrence (P? 0.05), which has an important application in statistics. In theory, the possibility of such an event is almost zero. For example, winning the lottery is a typical small probability event. Maybe every issue will win the grand prize (very unlikely), but for every lottery player, the possibility of winning the lottery is very small (the probability of a small probability event in an experiment). In fact, this is an important theory in the application of small probability events in statistics, based on the principle of small probability. ) that is, the possibility of small probability events in the experiment is very small. If it does happen, its authenticity can be doubted according to statistics.
If a reception station receives five people to visit alone one day, and all five people visit on Monday as a result, can it be inferred that the reception station has a prescribed reception day? Suppose there is no prescribed reception day, and a visitor visits on any of the five days, which is expressed as Am(m= 1, 2, 3, 4, 5,)? I receive m people a week, all on Mondays? Event, the probability of Am is as follows:
Event A 1 probability 0.2 event A2 probability 0.22
Event A3 probability 0.23 event A4 probability 0.24
The probability of event A5 is 0.25.
The probability that all five people will go on Monday is 0.00032, which is about three in ten thousand. Now there is a small probability event in an experiment, so I doubt the correctness of the hypothesis and infer that the receiving station has a specified receiving day.
18 14 years, Laplace recorded an interesting statistic in his new work. The ratio of boys to girls in the world is 22∶2 1, that is, among the babies born, boys account for 5 1.2% and girls account for 48.8%, but strangely, it is 1745. And the other ratio is 25∶24, with boys accounting for 565,438+0.02%, which is 0.65,438+0.8% different from the former. After investigation and study, it is found that Parisians have? Prefer girls over boys? There is a bad habit of abandoning baby boys, which distorts the birth rate. The revised birth ratio is still 22: 2 1. Statistics based on the principle of small probability have high accuracy, but there is also a low risk of error.
The principle of small probability has a very important application in statistics. For example, we judge the conclusion of hypothesis testing, which is a statistical inference method to infer the population with sample information. Because of the existence of sampling error, the sample information and overall characteristics may be different, so hypothesis testing is actually to judge whether the differences between the compared parties are caused by sampling error. The value of p in the hypothesis test reflects the probability that the difference is caused by sampling error. In hypothesis testing, the relationship between P value and test level A is compared (usually set to 0.05), so as to make statistical differences.
If the value of p is less than a certain statistical value, the probability that the difference is caused by sampling error is very low. Then according to the principle of small probability, the probability of a small probability event in one sampling is almost zero, so the difference in judgment may be caused by the essential difference between the two parties. Otherwise, it is considered that the difference is caused by sampling error. The test level here is considered to be set before the hypothesis test, which is the probability that the researcher can bear to give up the truth and falsehood in this hypothesis test, and can also be understood as the probability of a small probability event set by the researcher. P value is calculated, that is, the difference is the probability caused by sampling error when the test hypothesis is established.
Statistics plays an increasingly important role in modern management and social life. With the development of social economy and science and technology, statistics play an increasingly important role in modern state management and enterprise management. People's daily life is inseparable from statistics. The influence of statistics is so great that the probability closely related to it plays an increasingly important role.
Statistics related papers 2 talk about the teaching methods of statistics foundation and the cultivation of students' application ability
The basic knowledge of statistics is a technical subject that studies data. It is comprehensive, abstract and widely used. Through the teaching of this course, students can be trained to use statistical tools to systematically analyze and solve problems. In the teaching of secondary vocational schools, we should combine the characteristics of this subject, constantly improve teaching methods and improve students' comprehensive application of statistical knowledge.
Keywords: statistics teaching method design ability training
The basic knowledge of statistics is a technical subject that studies data. The methods of investigation, research and analysis in the subject content are not only applied to all work, but also to data collection, collation, analysis and summary in the research process of other disciplines. Therefore, statistics is comprehensive, abstract and widely used. Through the teaching of this course, students' ability to systematically analyze and solve problems by using statistical tools is cultivated. Based on the characteristics of this subject, this paper discusses its teaching methods and the cultivation of students' application ability.
First, the characteristics of basic statistics teaching
The basis of statistics is also the principle of socio-economic statistics, and its subject content has the following characteristics: first, there are many basic concepts and theoretical teaching is abstract; Second, there are many kinds of indicators, and it is difficult for beginners to draw a clear line between them when learning. Third, there are many investigation and analysis methods, so it is difficult to correctly understand and choose the appropriate investigation methods; Fourthly, the correct investigation method, the setting of method index system and the definition of statistical scope are directly related to whether the correct conclusion reflecting things can be drawn; Fifth, investigating the scientific setting of the index system of things is directly related to finding out the relevant indicators that reflect the objective and internal essence of things. Therefore, for the teaching objects of secondary vocational school students who are young and have poor analytical skills, even if they master the statistical principles conceptually, it is difficult to correctly apply statistical knowledge to solve problems in real social economy without combining actual statistical case data and adopting appropriate teaching methods, and even draw wrong conclusions about things because of the wrong use of methods.
Two, combined with the characteristics of subject knowledge, using appropriate teaching methods to strengthen the cultivation of application ability.
In teaching, first of all, through the comprehensive analysis of the content system of teaching materials and the knowledge structure of teaching objects, as well as the summary of students' interest, understanding depth and mastery of statistical knowledge, different teaching methods are appropriately implemented in different links of teaching.
1, through the introduction of subject content system and work tasks, enhance students' interest in learning.
When teaching the content of this subject, first, introduce the basic framework of the content of the basic statistics textbook to students: the meaning, research object, nature, function and basic research methods of statistics. Secondly, it introduces the subject knowledge system: the basic concepts in statistics, the ways and means of statistical data investigation and arrangement, the display and provision of statistical data, the analysis of statistical data provided by various index methods (total index method-reflecting the scale of things, average index method-reflecting the concentration trend and general law of things, relative index method-reflecting the vertical and horizontal comparison of things and the connection between things, and standard deviation method-reflecting the discrete trend between the overall unit mark values in things. Statistical index method-reflects the influence of various direct factors in things.
Time series method-reflects the development and change trend of things in a time period. Sampling survey method-the most scientific method among statistical special survey methods. Correlation regression analysis-analysis of causality in things. ) Through the simple explanation and introduction of the content system, students can have a general perceptual knowledge of the subject and get interested before learning specific theoretical knowledge. Learn with the awareness and purpose of solving practical problems by mastering statistical knowledge.
2. Let students learn from rational knowledge to perceptual knowledge and enhance their application ability.
When I introduce the basic concepts of statistics and the contents of statistical investigation methods in teaching, besides giving examples to each knowledge point, I give several typical statistical investigation schemes after some knowledge is finished, so that students can understand the statistical population, the definition of overall scope, overall units, signs, indicators and investigation methods involved in these investigation schemes. This not only enables students to transfer the abstract knowledge of statistical concepts from rational knowledge to perceptual knowledge, but also further makes students understand that the choice of investigation methods must be based on the objects to be investigated and the problems to be solved, not on the purpose of investigation, and anything can be investigated in any way. Only by correctly choosing statistical methods and means to investigate and analyze objective things can we draw correct conclusions and be able to correctly use statistical knowledge to analyze and solve problems.
3. Combine the application of comprehensive indicators with typical data to improve students' application ability. When teaching the comprehensive index method, the understanding of each index is
Give examples to help students understand the significance and function of this index. In order to make students understand and distinguish the functions of various indicators correctly, after introducing all indicators, I choose the annual statistical bulletin of national economy as a case, so that students can find out every comprehensive indicator they have learned from the statistical bulletin, such as the gross national product and population in 2007. The percentage of GDP completion this year is a relative indicator of planned completion, and the percentage of GDP growth this year is a dynamic relative indicator. Per capita GDP is a relative index of intensity.
The proportion of GDP is a relative structural indicator. The percentage of average annual growth in five years is the application of average development speed and average growth speed to be learned later. Through such a case, students not only have a correct understanding of the application of various comprehensive index methods, but also transform their understanding of various indicators into their application ability, and at the same time lay a foundation for learning dynamic sequence knowledge in the future. It plays a potential role in consolidating and understanding knowledge and previewing the content of the next teaching link in teaching. It also has a comprehensive role in mastering knowledge. Through such a case, students further make it clear that when studying a common problem, they can use various indicators to analyze different aspects of the problem and find out the objective relationship between things, all of which must be based on certain statistical data. Therefore, it further emphasizes the necessity for students to learn statistical knowledge, and also makes them realize the scientific and practical nature of statistical knowledge.
4. Comprehensive use of old and new knowledge in real cases to improve students' application ability.
When teaching the content of statistical index, the principle of the basic method of compiling statistical index is taught to students. The compilation of commodity price, commodity quantity and employee wage index illustrated in the textbook is only an introduction to the basic calculation method. In order to cultivate students' application ability, we must also explain the case of compiling actual statistical index, so that students can apply theoretical knowledge and its calculation method to practical work. Therefore, after I have finished the theoretical knowledge and calculation method, I will especially introduce the compilation of retail price index in practical work. This economic index is also a common concern of the public, which is closely related to people's living standards.
Tell the students that the compilation of price index uses the knowledge of sampling survey, and it is impossible to investigate the price of every commodity in practical work, but to collect the prices of shopping malls and bazaars in different categories. For example, the market price of vegetables is collected at least three times a week, and the transaction price is collected three times each time. The price entered in the compilation of retail commodity price index is actually a simple average price for many times, while the three prices of a commodity should be simply averaged every day, and the average price three times a week should be simply averaged again. If the commodity prices in the shopping mall are relatively stable, they can be averaged at the beginning and end of the period. Through such a case, it not only taught students new knowledge, but also reviewed and consolidated the specific application of the average index calculation method, which is not only used in daily life, but also widely used in economic research. Further tell students the application of weighted average method and harmonic average method in compiling price index and other socio-economic phenomenon index.
5. Typical investigation case teaching method to cultivate students' comprehensive application of statistical knowledge and ability to analyze and solve problems.
In teaching, I put the cultivation of students' ability to apply statistical knowledge and analyze problems in the teaching content of sampling technology, and the basic theory of sampling technology is abstract. Such as sampling error, sampling average error and sampling organization. According to the characteristics of the research object, specific problems must be analyzed. The calculation of sampling error involves both the calculation of average index and the calculation of standard deviation. How to train students to use old and new knowledge to calculate, analyze and solve problems is a difficult point in teaching.
In order to break through this difficulty, I used the sample survey case of straw mat quality in teaching. This case reflects from the determination of investigation methods, the classification of main signs and the principle of simple random sampling in the investigation scheme to the implementation steps of the investigation: straw mat width classification, original grade registration, numbering, determination of sampling population, calculation of full-grade population standard deviation, determination of sampling number, design of calculation table, determination of sample number, and unified grade standard for on-site investigation.
Grading process: Grading is carried out by five people respectively, and the number of grades is finally determined by the mode method, subject to the grading standard of three people in the five-person grading. The above are the representative sampling survey methods introduced earlier, which adopt the methods of average index and mode. At the same time, when calculating the average grade of straw mats, the average index of quality mark value is also used, that is, the grade quality mark value is converted into quantity mark, and the average grade of straw mats of different sizes is calculated, and then the error between sampling index and original grade inspection index is calculated.
Such a complicated sampling investigation process and the calculation results of indicators clearly tell students the problems that need to be explained and solved: when buying straw mats, the error of the judgment standard of the grade inspector brings about the difference between the grade error and the price of straw mats. However, due to the existence of errors, there is a huge difference between the total inventory value of the whole batch of straw mats calculated according to the results of this sampling survey and the actual value. Through the analysis of policy market and human factors, this paper finds out the reasons and puts forward practical solutions to promote the purchase price of straw mats to be consistent.
Through the design of teaching methods in the above aspects, students can have a more comprehensive understanding of statistics and a general grasp of the basic content of the subject, so that students feel vague concepts and complex theories in the learning process, and gradually become clear after repeated consolidation and practice in these teaching links, and their comprehensive application ability of statistical knowledge has been greatly improved.