Laboratory work No. 3. Statistical data processing in the MatLab system
General statement of the problem
The main purpose of execution laboratory work is to familiarize yourself with the basics of working with statistical data processing in the MatLAB environment.
Primary statistical data processing
Statistical data processing is based on primary and secondary quantitative methods. The purpose of the primary processing of statistical data is to structure the information received, which implies grouping the data into summary tables according to various parameters. Primary data must be presented in a format that allows a person to make an approximate assessment of the resulting data set and to identify information about the data distribution of the resulting data sample, such as the homogeneity or compactness of the data. After the primary data analysis, methods of secondary statistical data processing are applied, on the basis of which statistical patterns in the existing data set are determined.
Carrying out primary statistical analysis on a data array allows you to gain knowledge about the following:
Which value is most typical for the sample? To answer this question, measures of central tendency are defined.
How large is the spread of data relative to this characteristic value, i.e., what is the “fuzziness” of the data? In this case, measures of variability are determined.
It is worth noting the fact that statistical indicators of central tendency and variability are determined only on quantitative data.
Measures of central tendency– a group of values around which the rest of the data is grouped. Thus, measures of central tendency generalize the array of data, which makes it possible to form conclusions both about the sample as a whole and to conduct comparative analysis different samples with each other.
Suppose we have a data sample, then measures of central tendency are assessed by the following indicators:
1. Sample mean is the result of dividing the sum of all sample values by their number. Determined by formula (3.1).
(3.1)
Where - i th element of the selection;
n– number of sample elements.
The sample mean provides the greatest accuracy in the process of estimating central tendency.
Let's say we have a sample of 20 people. The sample elements are information about the average monthly income of each person. Let's assume that 19 people have an average monthly income of 20 thousand rubles. and 1 person with an income of 300 tr. The total monthly income of the entire sample is 680 rubles. The sample mean in this case is S=34.
2. Median– forms a value above and below which the number of different values is the same, i.e. this is the central value in a sequential series of data. Determined depending on the even/odd number of elements in the sample using formulas (3.2) or (3.3). Algorithm for estimating the median for a data sample:
First of all, the data is ranked (ordered) in descending/ascending order.
If the ordered sample has an odd number of elements, then the median coincides with the central value.
(3.2)
Where n
In the case of an even number of elements, the median is defined as the arithmetic mean of the two central values.
(3.3)
where is the average element of the ordered sample;
- element of the ordered selection next to ;
Number of sample elements.
If all sample elements are different, then exactly half of the sample elements are greater than the median, and the other half are less. For example, for the sample (1, 5, 9, 15, 16), the median is equal to element 9.
In statistical data analysis, the median helps identify sample elements that greatly influence the value of the sample mean.
Let's say we have a sample of 20 people. The sample elements are information about the average monthly income of each person. Let's assume that 19 people have an average monthly income of 20 thousand rubles. and 1 person with an income of 300 tr. The total monthly income of the entire sample is 680 rubles. The median, after ordering the sample, is defined as the arithmetic mean of the tenth and eleventh elements of the sample) and is equal to Me = 20 tr. This result is interpreted as follows: the median divides the sample into two groups, so that we can conclude that in the first group each person has an average monthly income of no more than 20 thousand rubles, and in the second group no less than 20 thousand rubles. IN in this example we can say that the median is characterized by how much the “average” person earns. While the value of the sample average is significantly exceeded S=34, which indicates the unacceptability of this characteristic when assessing average earnings.
Thus, the greater the difference between the median and the sample average, the greater the dispersion of the sample data (in the example considered, a person with an income of 300 rubles clearly differs from the average people in a particular sample and has a significant impact on the estimate of average income). What to do with such elements is decided in each individual case. But in the general case, to ensure the reliability of the sample, they are removed, since they have a strong influence on the assessment of statistical indicators.
3. Fashion (Mo)– generates the value that occurs most frequently in the sample, i.e. the value with the highest frequency. Mode estimation algorithm:
In the case when a sample contains elements that occur equally frequently, it is said that there is no mode in such a sample.
If two neighboring elements samples have the same frequency, which is greater than the frequency of the remaining elements of the sample, then the mode is defined as the average of these two values.
If two sample elements have the same frequency, which is greater than the frequency of the remaining sample elements, and these elements are not adjacent, then the sample is said to have two modes.
Mode in statistical analysis is used in situations where a quick assessment of the measure of central tendency is necessary and high accuracy is not required. For example, fashion (by size or brand) can be conveniently used to determine the clothes and shoes that are in greatest demand among customers.
Measures of scatter (variability)– a group of statistical indicators characterizing the differences between individual sample values. Based on the indicators of the measures of dispersion, the degree of homogeneity and compactness of the sample elements can be assessed. Measures of dispersion are characterized by the following set of indicators:
1. Range - this is the interval between the maximum and minimum values of the observation results (sample elements). The range indicator indicates the spread of values in the data set. If the range is large, then the values in the aggregate are very scattered, otherwise (the range is small) it is said that the values in the aggregate lie close to each other. The range is determined by formula (3.4).
(3.4)
Where 
- maximum sample element;
- minimum sample element.
2.Average deviation– arithmetic mean difference (in absolute value) between each value in the sample and its sample mean. The average deviation is determined by formula (3.5).
(3.5)
Where - i th element of the selection;
The sample mean value calculated using formula (3.1);
Number of sample elements.
Module 
necessary due to the fact that deviations from the average for each specific element can be both positive and negative. Consequently, if you do not take the module, then the sum of all deviations will be close to zero and it will be impossible to judge the degree of data variability (crowding of data around the sample mean). When performing statistical analysis, the mode and median may be taken instead of the sample mean.
3. Dispersion- a measure of dispersion that describes the comparative deviation between data values and the average value. It is calculated as the sum of the squared deviations of each sample element from the average value. Depending on the sample size, the variance is estimated different ways:
For large samples (n>30) according to formula (3.6)
(3.6)
For small samples (n<30) по формуле (3.7)
(3.7)
where X i is the i-th sample element;
S – sample mean;
Number of sample elements;
(X i – S) - deviation from the average value for each value of the data set.
4. Standard deviation-a measure of how widely scattered data points are relative to their mean.
The process of squaring individual deviations when calculating variance increases the degree of deviation of the resulting deviation from the original deviations, which in turn introduces additional errors. Thus, in order to bring the estimate of the spread of data points relative to their mean closer to the value of the mean deviation, the square root of the variance is taken. The extracted root of the variance characterizes a measure of variability called the root mean square or standard deviation (3.8).
(3.8)
Let's say you are the manager of a software development project. You have five programmers under your command. By managing the project execution process, you distribute tasks among programmers. To simplify the example, we will proceed from the fact that the tasks are equal in complexity and completion time. You decided to analyze the work of each programmer (the number of completed tasks during the week) over the last 10 weeks, as a result of which you received the following samples:
| Week Name | ||||||||||
By estimating the average number of completed tasks, you get the following result:
| Week Name | S | ||||||||||
| 22,3 | |||||||||||
| 22,4 | |||||||||||
| 22,2 | |||||||||||
| 22,1 | |||||||||||
| 22,5 |
Based on the S indicator, all programmers work on average with the same efficiency (about 22 tasks per week). However, the variability indicator (range) is very high (from 5 tasks for the fourth programmer to 24 tasks for the fifth).
| Week Name | S | P | ||||||||||
| 22,3 | ||||||||||||
| 22,4 | ||||||||||||
| 22,2 | ||||||||||||
| 22,1 | ||||||||||||
| 22,5 |
Let’s estimate the standard deviation, which shows how the values in the samples are distributed relative to the average, and specifically, in our case, estimate how large the spread in task completion is from week to week.
| Week Name | S | P | SO | ||||||||||
| 22,3 | 1,56 | ||||||||||||
| 22,4 | 1,8 | ||||||||||||
| 22,2 | 2,84 | ||||||||||||
| 22,1 | 1,3 | ||||||||||||
| 22,5 | 5,3 |
The resulting estimate of the standard deviation indicates the following (we will evaluate two extreme cases, programmers 4 and 5):
Each value in the sample of 4 programmers deviates on average by 1.3 assignments from the average value.
Each value in the programmer's sample 5 deviates on average by 5.3 items from the average value.
The closer the standard deviation is to 0, the more reliable the mean, since it indicates that each value in the sample is almost equal to the mean (in our example, 22.5 items). Therefore, programmer 4 is the most consistent, unlike programmer 5. The variability of task completion from week to week for the 5th programmer is 5.3 tasks, which indicates a significant spread. In the case of the 5th programmer, the average cannot be trusted, and, therefore, it is difficult to predict the number of completed tasks for the next week, which in turn complicates the planning procedure and adherence to work schedules. It doesn’t matter what management decision you make in this course. It is important that you receive an assessment on the basis of which you can make appropriate management decisions.
Thus, a general conclusion can be drawn that the average does not always evaluate the data correctly. The correctness of the average estimate can be judged by the value of the standard deviation.
The purpose of the lesson:
- creating conditions for mastering the topic at the level of comprehension and primary memorization;
- for the formation of mathematical competence of the student’s personality.
Lesson Objectives
Educational: form an idea of statistics as a science; familiarize students with the concepts of basic statistical characteristics; develop the ability to find the range and mode of a series, analyze data, and improve mental calculation skills.
Educational: promote mastery of concepts and their interpretation; development of supra-subject skills of analysis, comparison, systematization and generalization; continue the formation of subject language, promote the formation of key competencies (cognitive, informational, communicative) at various stages of the lesson, promote the formation of a unified scientific picture of the world among students by identifying interdisciplinary connections between statistics and various sciences.
Educational: cultivate interest in the subject being studied, information culture; readiness to comply with generally accepted norms and rules, high efficiency and organization.
Technologies used: Technology of student-centered learning, information and communication technologies.
Necessary equipment, materials: multimedia projector, computer, interactive whiteboard.
During the classes
1. Organizational moment.
Checking students' readiness for class
Checking Attendance
2. Goal setting.
Rationale for the need to study this topic
Involving students in the process of setting lesson goals
From what sources do we receive and collect information? (Suggested answers: radio, television, newspapers, magazines, telephone, people, Internet, letters).
Where do people store information? (Suggested answers : in memory and on external media).
Is studying at a technical school about obtaining information? At school you studied general education subjects, but when you study at a technical school, what else do you get? (Suggested answer: s professional knowledge). The more we learn, the more information our memory contains.
Today I offer you another piece of information. You are trained as a mining operator, you will work on EKG-8I excavators. What is the performance of this excavator. At my request, the plant provided me with the following information. (Excavator performance - table)
|
By waste rock (thousand tons) |
Guys, is a lot of information good? Can all information be useful and of high quality? What should we be able to do so as not to get lost in the maze of information? (Students’ expected answer: “Must be able to separate useful, high-quality information from low-quality information.”) Those. be able to process it.
CONCLUSION: today in lesson we will learn to process information.
3. Organization of activities to study new material.(students make notes in notebooks and complete assignments during the explanation process)
1. Definition of statistics
What are statistics? It is said that the English Prime Minister Benjamin Disraeli (1804 - 1881) answered this question as follows: “There are three types of lies: lies, damned lies and statistics.”
Statistics is an exact science that studies methods of collecting, analyzing and processing data that describe mass actions, phenomena and processes.
(An excerpt from the novel “The Twelve Chairs” by Ilf and Petrov is read out.
“Statistics knows everything”, it is known how much food the average citizen of the republic eats per year: it is known how many hunters, ballerinas: machines, bicycles, monuments, lighthouses and sewing machines in the country: How much life, full of ardor, passions and thoughts, looks at us from statistical tables!..”
Its name comes from the Latin word “status” - state, from this root the words stato (Italian), statistik (German), state (English) - state.
Statistics studies:
The main goals of studying the elements of statistics:
- the number of individual population groups of the country and its regions,
- production and consumption of various types of products,
- transportation of goods and passengers by various modes of transport,
- natural resources and much more.
Do you know in which country statistical practice began (in China); the country's first population censuses date back to the 5th century. II millennium BC
In the 19th century, it became possible to process data using formulas, mathematical laws and special characteristics. This?.... ( mat. statistics).
2. Math statistics
Math statistics is a branch of mathematics that studies methods of collecting, systematizing and processing the results of observations of random mass phenomena in order to identify existing patterns.
So why did Disraeli compare statistics to lies? (There was no scientific, rigorous processing of information; the data was interpreted by anyone as they wished).
Mathematical statistics has universal methods of information processing
This is what allowed the heroes of the film “Office Romance” to say the following words about statistics ( fragment of the film "Office Romance").
CONCLUSION: Statistics bring information into the system.
3. Graphical representation of information
Distribution polygon
Distribution histogram
Pie chart
4.
Measurement characteristics
1. A series of data is a series of results of any measurements.
For example: 1) measuring human height
2) Human (animal) weight measurements
3) Meter readings (electricity, water, heat...)
4) Results in the 100-meter dash
2. Volume of a data series - the volume of a data series is the amount of all data.
For example: given a series of numbers 1; 3; 6; -4; 0
its volume will be equal to 5. Why?
3. The range of a data series is the difference between the largest and smallest numbers from a data series.
For example: if given a series of numbers 1; 3; 6; -4; 0; 2, then scope this data series will be equal to 6 (since 6 - 0 = 6)
4. Mode of a data series - the mode of a data series is the number of the series that occurs most often in this series.
For example: p data poison may or may not have a mode.
Thus, in the data series 47, 46, 50, 52, 47, 52, 49, 45, 43, 53, each of the numbers 47 and 52 occurs twice, and the remaining numbers less than twice. In such cases, it was agreed that the series has two modes: 47 and 52.
5. Median of the series
A median with an odd number of terms is the number written in the middle.
Median with an even number of terms - this is the arithmetic mean of the two numbers written in the middle.
For example: determine the median of a series of numbers
16; -4; 5; -2; -3; 3; 3; -2; 3. Answer: -3
2) -1; 0; 2; 1; -1; 0;2; -1. Answer: 0
6. The arithmetic mean is the quotient of dividing the sum of the numbers in a series by their number.
For example: given a series of numbers -1; 0; 2; 1; -1; 0; 2; -1. Then the arithmetic mean will be equal to: (-1+0+2+1+(-1)+0+2+(-1)): 8 = 2: 8 = 0.25
4. Consolidation of the studied material.
Practical work
Exercise: characterize the performance of student Peter Ivanov in mathematics for the fourth quarter.
Completing of the work:
1. Collection of information:
The grades written out from the magazine are: 5,4,5,3,3,5,4,4,4.
2. Processing of received data:
Statistical processing of data is impossible without their ordering, generalization and analysis. Any results obtained must first be brought into such a form that maximum useful information can be extracted from them. If there is too much data received, then it needs to be grouped or summarized.
So, for grouping it is necessary to determine the norms in accordance with which the obtained data will be distributed. In this case, not only the clarity, but also the potential usefulness of the information received will depend on the chosen method. Properly grouped research results are much more convenient to study and analyze.
Data processing can be applied in many areas of human activity. They can be divided into 3 main types:
1) universal methods that can be used regardless of the scope of application;
2) methods for certain areas of activity involved in the study of real processes or phenomena;
3) methods for studying certain data.
It is clear that the more accurate the method by which statistical data processing is carried out, the more effective the analysis of a specific situation will be. If the first method is applicable for scientific results, the significance of which will be assessed only by general scientific criteria, then the third method is used only for solving certain problems in a specific area.
In addition to general knowledge of the methods by which data is processed, it is also important to know how best to work with the results obtained. Statistical data processing involves creating tables or graphs to visualize the information obtained.
At the initial stage, the information can be summarized in a table. For example, statistical processing of experimental data recorded in tabular form allows researchers to save from additional unnecessary recordings of indicators, measurement values, and additional factors influencing the course of the experiment. In tables it is convenient to record not only the data of a study or experiment, but also to summarize intermediate and main results. True, to construct them correctly, it is necessary to think in advance about the required number of rows and columns, and write down all the necessary parameters.
You can simply make a table on a piece of paper or directly enter the data into the computer. The second option will allow you to quickly sort the received data in the desired way, find the largest or, conversely, smallest value, summarize or search for a selected group of results.
Do not forget that if competent statistical processing of data requires several tables, then they must be numbered and a unique name must be invented for each one.
A more visual way to record data is with graphs. They visually show the relationship between different quantities, making it easier to understand the research results.
Knowing the basic principles of constructing tables and graphs, you can quickly and efficiently process the received data.
Slide 2
- Statistics is an exact science that studies methods of collecting, analyzing and processing data that describe mass actions, phenomena and processes
- Mathematical statistics is a branch of mathematics that studies methods of collecting, systematizing and processing the results of observations of random mass phenomena in order to identify existing patterns.
Slide 3
Statistics studies
- the number of individual population groups of the country and its regions,
- production and consumption of various types of products,
- transportation of goods and passengers by various modes of transport,
- natural resources and much more.
- The results of statistical studies are widely used for practical and scientific conclusions.
- Currently, statistics begins to be studied in secondary school; in universities it is a compulsory subject, because it is associated with many sciences and fields.
- To increase the number of sales in a store, to improve the quality of knowledge in school, to move the country towards economic growth, it is necessary to conduct statistical studies and draw appropriate conclusions. And everyone should be able to do this.
Slide 4
The main goals of studying the elements of statistics
- Formation of skills in primary processing of statistical data;
- image and analysis of quantitative information presented in different forms (in the form of tables, diagrams, graphs of real dependencies);
- developing ideas about important statistical ideas, namely: the idea of estimation and the idea of testing statistical hypotheses;
- developing the ability to compare the probabilities of random events occurring with the results of specific experiments.
Slide 5
- Data series
- Data series volume
- Range of data series
- Data series mode
- Median of the series
- Average
- Ordered data series
- Data distribution table
- Let's sum it up
- Nominative data series
- Result Frequency
- Percent frequency
- Grouping data
- Data processing methods
- Let's sum it up
Slide 6
Definition
- A data series is a series of results of some measurements.
- For example:1) measuring human height
- 2) Human (animal) weight measurements
- 3) Meter readings (electricity, water, heat...)
- 4) Results in the 100-meter dash
- Etc.
Slide 7
- The volume of a data series is the amount of all data.
- For example: given a series of numbers 1; 3; 6; -4; 0
- its volume will be equal to 5. Why?
Slide 8
Complete the task
- Determine the volume of this series.
- Answer: 10
Slide 9
Definition
- Range is the difference between the largest and smallest numbers in a data series.
- For example: if given a series of numbers 1; 3; 6; -4; 0; 2, then the range of this data series will be equal to 6 (since 6 – 0 = 6)
Slide 10
Complete the task
- At the institute we took a test in higher mathematics. There were 10 people in the group, and they received the corresponding ratings: 3, 5, 5, 4, 4, 4, 3, 2, 4, 5.
- Determine the range of this series.
- Answer: 3
Slide 11
Definition
- The mode of a data series is the number of the series that occurs most often in this series.
- A data series may or may not have a mode.
- Thus, in the data series 47, 46, 50, 52, 47, 52, 49, 45, 43, 53, each of the numbers 47 and 52 occurs twice, and the remaining numbers less than twice. In such cases, it was agreed that the series has two modes: 47 and 52.
Slide 12
Complete the task
- So, in the data series
- 47, 46, 50, 52, 47, 52, 49, 45, 43, 53 each of the numbers 47 and 52 appears twice, and the remaining numbers less than twice. In such cases, it was agreed that the series has two modes: 47 and 52.
- At the institute we took a test in higher mathematics. There were 10 people in the group, and they received the appropriate ratings:
- 3, 5, 5, 4, 4, 4, 3, 2, 4, 5.
- Determine the mode of this series.
- Answer: 4
Slide 13
Definition
- A median with an odd number of terms is the number written in the middle.
- A median with an even number of terms is the arithmetic mean of the two numbers written in the middle.
- For example: determine the median of a series of numbers
- 16; -4; 5; -2; -3; 3; 3; -2; 3. Answer: -3
- 2) -1; 0; 2; 1; -1; 0;2; -1. Answer: 0
Slide 14
Complete the task
- At the institute we took a test in higher mathematics. There were 10 people in the group, and they received the corresponding ratings: 3, 5, 5, 4, 4, 4, 3, 2, 4, 5.
- Determine the median of this series.
- Answer: 4
Slide 15
Definition
- The arithmetic mean is the quotient of dividing the sum of the numbers in a series by their number.
- For example: given a series of numbers -1; 0; 2; 1; -1; 0; 2; -1. Then the arithmetic mean will be equal to: (-1+0+2+1+(-1)+0+2+(-1)):8 =2:8=0.25
Slide 16
Complete the task
- At the institute we took a test in higher mathematics. There were 10 people in the group, and they received the corresponding ratings: 3, 5, 5, 4, 4, 4, 3, 2, 4, 5.
- Determine the arithmetic mean of this series.
- Answer: 3.9
Slide 17
Practical work
- Assignment: characterize student Ivanov’s performance in mathematics for the fourth quarter.
- COMPLETING OF THE WORK:
- 1.Collection of information:
- The grades written out from the magazine are: 5,4,5,3,3,5,4,4,4.
- 2.Processing of received data:
- volume = 9
- range = 5 - 3 = 2
- fashion = 4
- median = 3
- arithmetic mean =(5+4+5+3+3+5+4+4+4) : 9 ≈ 4
- Characteristics of academic performance: the student is not always ready for the lesson.
- Mostly he studies with grades "4". A quarter comes out to “4”.
Slide 18
On one's own
- We need to find the volume of the series, the range of the series, the mode, the median and the arithmetic mean:
- Card 1. 22.5; 23; 21.5; 22; 23.
- Card 2. 6; -4; 5; -2; -3; 3; 3; -2; 3.
- Card 3. 12.5; 12; 12; 12.5; 13; 12.5; 13.
- Card 4. -1; 0; 2; 1; -1; 0; 2; -1.
- Card 5. 125; 130; 124; 131.
- Card 6. 120; 100; 110.
Slide 19
Let's check
- Card 1.
- row volume = 5
- range of row = 10
- fashion = 23
- median = 21.5
- arithmetic mean = 13.3
- Card 3.
- row volume = 7
- range of series = 1
- mode = 12.5
- median = 12.5
- arithmetic mean = 12.5
- Card 2.
- row volume = 9
- range of row = 10
- fashion = 3
- median = -3
- arithmetic mean = 1
- Card 4.
- row volume = 8
- range of row = 3
- mode = -1
- median = 0
- arithmetic mean = 0.25
Slide 20
- Card 5.
- row volume = 4
- row range = 7
- fashion = no
- median = 127
- arithmetic mean =127.5
- Card 6.
- row volume = 3
- range range = 20
- fashion = no
- median = 100
- arithmetic mean = 110
Slide 21
Definition
- Ordered data series are series in which the data is arranged according to some rule.
- How to arrange a series of numbers? (Write the numbers so that each subsequent number is no less (no more) than the previous one); or write down some names “alphabetically”...
Slide 22
Complete the task
- Given a series of numbers:
- -1;-3;-3;-2;3;3;2;0;3;3;-3;-3;1;1;-3;-1
- Sort it in ascending numbers.
- Solution:
- -3;-3;-3;-3;-3;-2;-1;-1;0;1;1;2;3;3;3;3
- The result is an ordered series. The data itself has not changed, only the order in which they appear has changed.
Slide 23
Definition
- A data distribution table is a table of an ordered series in which, instead of repeating the same number, the number of repetitions is recorded.
- Conversely, if the distribution table is known, then an ordered series of data can be compiled.
- For example:
- From it we get the following ordered series:
- -3;-3;-3;-1;-1;-1;-1;5;5;7;8;8;8;8;8
Slide 24
Complete the task
- In a women's shoe store, statistical research was carried out and a corresponding table was compiled for the price of shoes and the number of sales:
- Price (RUB): 500 1200 1500 1800 2000 2500
- Quantity: 8 9 14 15 3 1
- For these indicators, you need to find statistical characteristics:
- create an ordered series of data
- volume of data series
- series range
- fashion series
- median of the series
- arithmetic mean of a data series
Slide 25
And answer the following questions
- From these price categories, at what price should the store not sell the shoes?
- Shoes, at what price should it be distributed?
- What price should you aim for?
Slide 26
Let's sum it up
- We got acquainted with the initial concepts of how statistical data processing occurs:
- data is always the result of some kind of measurement
- A number of some data can be found:
- volume, range, mode, median and
- average
- 3) any data series can be
- organize and compose
- data distribution table
Slide 27
Definition
- The nominative series of data is NOT NUMERICAL DATA, but, for example, names; titles; nominations...
- For example: list of World Cup finalists since 1930: Argentina, Czechoslovakia, Hungary, Brazil, Hungary, Sweden, Czechoslovakia, Germany, Italy, Netherlands, Netherlands, Germany, Germany,
- Argentina, Italy, Brazil, Germany, France
Slide 28
Complete the task
- Find from previous example:
- volume of row 2) mode of row
- 3) create a distribution table
- Solution: volume = 18; fashion – German team.
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