Descriptive statistics

This work has been designed with a view to facilitating the learning of DESCRIPTIVE STATISTICS for people who relate it to their projects or interests.

Our aspiration is to present an easily understood work, freed from an overwhelming theory, with statistical procedures that require a minimum mathematical level, without this naturally affecting the precision of the results or the scientific rigor, which we always encourage to exist.

Aware that much can be learned and well in action, we have proposed to include in each Unit several resolved exercises and proposals as a matter of real applications that allow the reader to adopt a positive attitude to the usefulness of the study of STATISTICS.

We consider that self-criticism and self-evaluation are essential in the learning of STATISTICS and its facilitation; For this reason, we also record in the unit the corresponding REVIEW and SELF-ASSESSMENT, which will allow us to drive and drive and strengthen our learning and, above all, trust among ourselves is the hot cause of our best achievements.

For the review we suggest to you, dear reader, that you observe the following working methodology:

Study the themes and subtopics of each Unit. When developing the scheduled review that exists in each Unit, cover with a sheet of paper the answer is that the margin is stated; then contrast them with those that are the fruit of their own elaboration. The spaces and consist of unfilled in each of the sentences are in relation to the word or words that must be written in response.

The methodology that we propose in this work requires a positive disposition regarding the study, strengthened with the presence of factors of such importance as:

Clarity in goals and objectives Will, as a force that drives us to carry out our work Perseverance, defined as the ability that allows us to stay on task.

Trust in ourselves and we are not capable of you. Interest, which translates into motivation and enthusiasm for studying.

The authors present in advance their thanks to the students of the STATISTICS, for which they are kind enough to send us and that allow this work in future deliveries to improve in quality.

Below the complete document, at the bottom of this page you can find a link to download the original file

Descriptive statistics

In addition, as a complement to the study, we invite you to consult the following descriptive statistics video course through which you will reinforce the concepts covered in this document, 13 videos.

Content

Introduction

First unit

PRELIMINARY NOTIONS

SPECIFIC OBJECTIVES:

PRELIMINARY NOTIONS

POPULATION AND SAMPLE

VARIABLE

DISCRETE VARIABLE

CONVENTIONAL SYSTEM

REVIEW

SELF APPRAISAL

SELF-ASSESSMENT 2

RECOMMENDED READING.

Annexes.

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First unit

PRELIMINARY NOTIONS

CONTENTS:

1.1. POPULATION AND SAMPLE

1.2. DESCRIPTIVE STATISTICS AND INFERENTIAL STATISTICS

1.3. VARIABLES

1.4. DATA ROUNDING

SPECIFIC OBJECTIVES: Upon completion of the study of this unit, you will be able to:

1. Distinguish the meaning of population and sample.

2. Explain the difference between descriptive and inferential statistics.

3. Describe what the descriptive and inferential statistics method consists of.

4. Identify various examples of variables.

5. Round numeric values.

To achieve these goals, you must:

1. Answer the self-assessment that appears at the end of each unit. Contrast your answers with those we offer at the end of the book until you reach an efficiency level equivalent to one hundred percent (100%).

2. Carry out the review of the unit, checking your answers with those that we record on the margin of the pages. We believe that this constitutes a reinforcing material that will allow you to review the fundamental aspects.

3. Solve the proposed exercises, since they constitute the necessary practice to strengthen your degree of understanding. PRELIMINARY NOTIONS

POPULATION AND SAMPLE 1.1.1. POPULATION

It is the set of elements motives of an investigation.

Parameters. - are the numerical values that correspond to the characteristics of the population.

1.2.1. SHOWS.

It is a part of the population whose analysis can obtain characteristics that correspond to the population.

Statistics. - are the numerical elements that correspond to the characteristics of the sample.

Below, we propose population and sample examples:

Population. - teachers of secondary education in the province of Loja.

Parameter. - arithmetic mean of the age of secondary education teachers in the province of Loja. Shows. - Middle school teachers from the Loja canton.

Statistical. - arithmetic mean of the age of middle school teachers in the canton of Loja.

1.2. DESCRIPTIVE AND INFERENTIAL STATISTICS

1.2.1. DESCRIPTIVE STATISTICS

It deals with the presentation and analysis of facts and things, explaining its different parts, but without drawing conclusions that can be generalized to a whole.

1.2.1.1. Descriptive Statistics Method. This branch of statistics, to fulfill its own objectives, uses the following method:

- Data collection. - Consists of obtaining data related to the problem being studied, using instruments such as: questionnaires, interviews, reports, memories, etc. Thus, for example: among the students who belong to the specialty of agronomy of a school "x", we collect the data corresponding to: origin of the students, current resident, schools from which they come.

- Analysis of data. - it is done taking into account factors, such as: indignation and study of each student, and then note the value of each aspect; So:

ORIGIN OF STUDENTS CURRENT RESIDENCE SCHOOL FROM WHICH THEY COME

PAYROLL

LOJA PROVINCE

OTHER PARISH

NN…

 Differentiation of the Presidency and the current place of residence of each individual. In the case, reason for analysis, such as the origin of the students, we note the number of students from: Loja, Zamora, Morona Santiago, Tungurahua, Chimborazo; and also students who do not answer.

In relation to the spectrum <>, it is appreciated that there are students who have completed primary studies in the city of Loja, in the province of Loja and in other provinces.

1. Data tabulation classification. That is, order them and express them by means of tables, like this:

ORIGIN OF THE STUDENTS PROVINCE NUMBER OF STUDENTS Loja 66 Zamora 9 Morona Santiago 1 Tungurahua 1 Chimborazo 1 No answer 1

TOTAL 79

SCHOOL OF THOSE COMING FROM NOMINE NUMBER OF STUDENTS City of Loja 36 Other cantons of Loja 36 Other provinces 7 TOTAL 79

 Determination of the numerical values that correspond to the population.

 The data presented in the tables, for better understanding and dissemination, can be represented by graphs.

1.2.2 INFERENTIAL STATISTICS This branch of Statistics draws from the sample valid conclusions that, once taught and analyzed, can provide us with certain common characteristics of the population.

1.2.2.1. Inferential Statistics Method. - In the field of research, inferential statistics using the following method:

 Formulation of the hypothesis. - Once the problem for study has been delimited, it is essential to make hypotheses in a client or statements that will be checked in parentheses, so that they are admitted to rejected.  Preparation of the research plan. - It is the planning of the research work, emphasizing: in the delimitation of the group or groups subject of the research, in the information search instruments, and in a schedule that guarantees compliance with the research stages. The initial work plan can be modified, depending on circumstances AND on the go.  Data collection. - Certain search instruments, surveys, interviews, etc. that are going to be used in the study of the problem, they are applied by the group or groups chosen as a sample. Data analysis. - The information collected is arranged in tables, then the calculation of certain statistics is made and the statistical test that best suits the research is chosen, it can be: difference in measurements, difference in proportions, t-student, Chi square, etc.. The choice of this statistical test will, of course, be subject to the type of data available, the size and number of samples of the samples.  Acceptance or rejection of the hypothesis. - Once the appropriate statistical test has been applied, the hypothesis test must be carried out, by means of which the proposed hypothesis is accepted or rejected, taking into account the level of confidence, that is, <Then the calculation of certain statistics is made and the statistical test that best suits the research is chosen, it can be: difference in measurements, difference in proportions, t-student, Chi square, etc. The choice of this statistical test will, of course, be subject to the type of data available, the size and number of samples of the samples.  Acceptance or rejection of the hypothesis. - Once the appropriate statistical test has been applied, the hypothesis test must be carried out, by means of which the proposed hypothesis is accepted or rejected, taking into account the level of confidence, that is, <Then the calculation of certain statistics is made and the statistical test that best suits the research is chosen, it can be: difference in measurements, difference in proportions, t-student, Chi square, etc. The choice of this statistical test will, of course, be subject to the type of data available, the size and number of samples of the samples.  Acceptance or rejection of the hypothesis. - Once the appropriate statistical test has been applied, the hypothesis test must be carried out, by means of which the proposed hypothesis is accepted or rejected, taking into account the level of confidence, that is, <the size and number of samples of the samples.  Acceptance or rejection of the hypothesis. - Once the appropriate statistical test has been applied, the hypothesis test must be carried out, by means of which the proposed hypothesis is accepted or rejected, taking into account the level of confidence, that is, <the size and number of samples of the samples.  Acceptance or rejection of the hypothesis. - Once the appropriate statistical test has been applied, the hypothesis test must be carried out, by means of which the proposed hypothesis is accepted or rejected, taking into account the level of confidence, that is, <

Conclusions. - Under the assumption of not having incurred the failures due to the technical conduct of the investigation, we would have to make the final decisions that are reliable and that, after the statistical analysis of the hypotheses, help us to solve the problem that caused the investigation. It is also imperative that the researcher in announced own goals inferred from the research.

VARIABLE It is a quantitative qualitative characteristic, which can take different values for each of the elements of the population. According to its values, the variable is classified as: DISCRETE AND CONTINUOUS.

DISCREET VARIABLE. Represents a quantitative characteristic that cannot take values between two consecutive integers.

For example:

The number of constitutional presidents of Ecuador.2

The number of presidents can be: 0, 1, 2,…, but it is evident that there are no 20, 5 presidents. consequently, this variable cannot take values that are between two integers.

1 Morales V. Pedro. Techniques of Operational Research in Education and Psychology. Page 8. 2 Integers:…, -2, -1, 0, 1, 2,…. 1.2.3. CONTINUOUS VARIABLE

Represents a quantitative characteristic that can take any numerical value.

for example:

The age of the constitutional presidents of Ecuador.

The age can be given in years, months, days, etc. The age of a president can be expressed like this: 50.2 years, that is, we always have to find another value between two integers that can be taken by the variable.

DATA ROUNDING

Today, with the use of computers, thousands of decimal places or interest can be obtained. Today, but in this one, absolute precision is not required, but rather the order wave approximation of certain values. The following systems are used to make the rounding approximation:

CONVENTIONAL SYSTEM according to which:

1.3.1.1. If the last digit is less than five, it is deleted, and in the resulting quantity it is the same. Examples: 7.23 rounded to the tenth

10,284 rounded to the hundredth is

137.4 approximated to unity is a.230 approximated to hundreds is 1.4.1.2. If the last digit is greater than or equal to 5, it is deleted, and the previous digit is rounded to the next higher number.

Examples:

8,277 rounded to the hundredth is

112.38 rounded to the tenth is

14,375 rounded to the hundredth ima is

7,350 rounded to hundreds is

1.4.2. INTERNATIONAL SYSTEM (YES)

Take the following examples:

a) 1.1425 rounded to two decimal places is b) 126.641 rounded to three whole numbers is c) 48.85 rounded to two whole numbers is d) 39.5 rounded to two whole numbers is e) 74.5 rounded to two whole numbers is

We conclude that:  If the decimal fraction is less than 5, it is left at the same number, or it is not taken into account to be retained as in example a).  If the decimal fraction is greater than 5, the first number retained is increased by 1 United, as examples b) and c).  If the decimal fraction is exactly 5 and if it is preceded by 5 an odd number is increased by 1 more unit. Example d)  if the decimal fraction is exactly 5 and a number is preceded by 5 so the number does not change. Example e).

In the calculations carried out in this work, we have used the CONVENTIONAL SYSTEM for data rounding.

Exercises solved

1.3.2. Is the qualification in the subject of statistics a continuous or discrete variable? It is a continuous variable.

1.3.3. Using the CONVENTIONAL SYSTEM, round the following numbers:

5.32 round to the tenth 8,373 round to the hundredth 249.2 will round the unit 6,540 round to hundreds Development: 5.32 rounded to the tenth is 5.3 8.3373 round to the tenth hundred is 8.37 249.2 rounded to the unit is 249 6,540 rounded to hundreds is 6,500

1.3.4. Using the CONVENTIONAL SYSTEM, round the following numbers:

5,246 rounded to the hundredth 324.37 rounded to the tenth 4,260 rounded to hundreds Development: 5,246 rounded to the hundredth is 5.25 324.47 rounded to the hundredth is 324 4,260 rounded to hundreds is 4,300

1.3.5. With the help of the CONVENTIONAL SYSTEM, round the following numbers:

3.1238 round to two decimal places 328.641 round to three whole numbers 68.5 round to two whole numbers 83.5 round to two whole numbers Development 3.238 rounded to two decimal places is 3.12 328.641 rounded to three whole numbers is 329

68.5 rounded to two whole numbers is 68 83.5 rounded to two whole numbers is 84

PROPOSED EXERCISES

1.3.6. Is the weight variable continuous or discrete?

1.3.7. Round the following numbers using the CONVENTIONAL SYSTEM:

234.28 round to the tenth 139.3 round to the unit 34,184 round to the hundredth 2,470 round to the hundreds 1.3.8. Round the following numbers using the CONVENTIONAL SYSTEM

42.5 round to two whole numbers 87.5 round to two whole numbers 7.1125 round to two decimal places 328.634 round to three whole numbers

REVIEW

1.3.9. The numerical values of the characteristics

Population Is Called Parameters

And the numerical values of the characteristics of the

Sample Take Name From Statisticians

1.3.10. Descriptive Statistics Deal Analysis

From Representation And Acts, extracts

Instead, the Inferential Statistics conclusions

1.3.11. The Number Of Deputies To The National Congress Is A Variable

discreet

1.3.12. The Age Of The Inhabitants Of Vilcabamba Continues

It is a Variable_

1.3.13. Among the instruments used for data collection we have: reports sheets

The Questionnaire, The Interview And observation

1.3.14. Does the data analysis include tabulation and statistical calculations?

. Yes.

1.3.15. Graphical representations are used as means of dissemination of statistical data.

1.3.16. When formulating the hypothesis, are we interested in proving something? (Yes or no). Yes.

1.3.17. Any systematic elaboration on organizing a job is called

research plan

1.3.18. When certain measurement instruments are used, such as surveys, interviews, etc., data is being collected

1.3.19. Is one group enough or two groups better to be able to compare the results in an investigation two groups

1.3.20. The statistical process by which the hypothesis is to be accepted or rejected is called:

hypothesis testing

1.3.21. For the conclusions to be correct among others, we must take into account the _ confidence level

1.3.22. Rounding data is used for decimal or integer figures. approximation SELF EVALUATION Mark with an x (x) the correct statement of each one of the following propositions:

1.3.23. The population refers to:

a) A set of mathematical elements

b) A set of all the elements of an investigation c) A meeting of characteristics d) A set of parameters

1.3.24. One of the following propositions defines what sample is:

a) The set of elements of an investigation b) The statistical set c) A part of the population

d) A consequence extracted from the population

1.3.25. DESCRIPTIVE STATISTICS tries to:

a) The representation and analysis of the population b) Characteristics of the sample c) Data of a sample to analyze them d) Valid conclusions of the sample 1.3.26. DESCRIPTIVE STATISTICS extracts:

a) Conclusions from the population b) Characteristics of the sample c) Data from a sample to analyze them d) Valid conclusions from the sample 1.3.27. Point out that elements do not correspond to the descriptive statistics method:

a) Data collection

b) Hypothesis formulation c) Data analysis d) Graphical representation

1.3.28. For the analysis of the results in the Inferential Statistics method, one of the following aspects must be taken into account: a) Selection of the sample

b) Application of the survey c) Statistical calculations d) Test of the hypothesis

1.3.29. The statistical variable is defined as: a) A set of elements that can take different values b) A set of literals c) A group of statistics

d) The frequency of a population

1.3.30. Analyze the following statistical variables and select the continuous variable:

a) Countries of the Andean pact

b) Weight of the Ecuadorian ladies

c) Parents of college students

d) Ministers of State that form the Ecuadorian government

1.3.31. Check the correct statement:

a) 7.283 approximate to the hundredth is equal to 7.29 b) 16.395 approximate to the tenth is equal to 16.3 c) 18.935 approximate to the unit is equal to 18 d) None of the previous approximations is correct

Check the SELF-ASSESSMENT responses on page 339

SELF-ASSESSMENT 2 1. Mark with an x (x) the correct statements according to the International System of Rounding of Figures: a) The number 37.5 rounded to two whole numbers equal to 38

b) The number 129,145 rounded to two decimal places is equal to 129.15 c) The number 130.37 rounded to one decimal place is equal to 130.3 d) The number 5,284 rounded to two decimal places is equal to 5.28

2. The parameters are:

a) A part of the population

b) The values that correspond to the characteristics of the sample

c) The numerical values that represent the characteristics of the population d) The elements that are reasons for an investigation 3. Statistics are the numerical values:

a) They correspond to the population

b) They represent a qualitative characteristic

c) They correspond to the characteristics of the population d) They correspond to the characteristics of the sample 4. Indicate which of the following variables are discrete

a) The weight of the students of a college b) The illiterate population of Ecuador c) The average age of the students of a university

d) The number of teachers of basic general education in Ecuador

5. By means of the sample valid conclusions are drawn for

a) A part of the population b) A part of the sample c) The population

d) A set of statistics

6. The student performance in the subject of statistics is:

a) An attribute

b) A discrete variable c) A continuous variable d) None of the previous propositions 7. A quantitative statistical series is one that:

a) Represents exact values

b) It is based on a continuous and discrete characteristic c) It includes numerical figures d) The previous propositions are false.

Check the answers for SELF-TEST 2 ON PAGE 340. RECOMMENDED READINGS

The books

Jaime Bustamante G.

"I hate books; to teach only to talk about what is not known. " EMILIO ROUSSEAU

In the course of time, the book has had opponents and supporters. Eminent modern and contemporary pedagogues have fixed their position in front of the book; Thus, Rousseau is mentioned, as the prototype of the teacher who whipped the books with great ardor. But, advantageously, we also find prominent teachers who defend the book in an eloquent way, valuing it in its proper measure and recognizing it as the very memory of humanity.

We consider that, in the current and future times, the book is and will continue to be the resource of greatest importance in education; It is specified that our students teach him reading techniques, in short, a good intellectual work methodology that allows them to read reflectively, as well as interest them in reading and in the books that constitute the source of knowledge.

The book has unlimited perspectives, but the misuse of books is what really must be fought, for example: there are some professors, including at the level of colleges and universities, who simply read a book during class; book that on the other hand they take it well lined, so that their students do not find out where the science is. We must also emphasize that the acquisition of books is frequently recommended, but, the students never get to use them, since the same teacher uses another book as a source of consultation.

In the youth from Loja and in people of all ages there is an urgent need to study, to know. Day and night we observe through our city the hasty walk of people with books and notebooks destined for study centers, from which the renewed desire of distant men and women for science, art and culture can be deduced. They aspire to an intellectual improvement, but they must contribute a book industry, ready to carry out quality work and in the shortest possible time.

Publishing industries that exist in the national territory must deserve economic incentives from the Ecuadorian State, reducing for example the import tariffs of materials and releasing once the import taxes on modern machinery, which allows them in the best conditions to guarantee the success in the composition, printing and binding of his books.

In short, the State could very much be along this line, which is inherent to the very development of the country. The lowering of the cost of the book will be achieved with a skilful policy that will help massively culturalize the Ecuadorian people. Convinced of the transcendent role that the book plays, we must proclaim and make our own the thought that "The books teach those who live and those who are to be born the inheritance of those who were" and therefore we must all sponsor their production and dissemination. What is statistics? To start these notes, it is very appropriate to consider what is statistics, since the word can be used with several senses.

It should be noted, first of all, that most people associate the word statistics with census publications or news that collect figures on production, birth, my admission to the University, traffic accidents, etc., Or with the charts and graphs that appear in magazines or newspapers; or with the figures or percentages that politicians use in their speeches, presided over by the ritual phrase: "statistics show that…". This concept corresponds to the plural statistics, which is used to indicate a set of figures, statistical data, which are organized and presented to show the characteristics or behaviors of a certain phenomenon of interest. And this is what you have in mind when you talk about population statistics, educational statistics, industrial production statistics,statistics of a with a soccer championship, etc.

Statistics, however, is not simply a set of figures, nor is it interested only in collecting and presenting data. When speaking in these statistical notes, what will be kept in mind is the singular statistic, which refers to a field of knowledge, to a discipline developed to deal with numerical or quantitative data, obtained by observation or experimentation. As a scientific discipline, statistics only have the purpose of collecting and presenting data; this orientation was predominant in the early stages of its development, in which a great effort was put in the collection of large masses of data in the summarization and presentation of this information through tables and graphs and in the calculation of percentages, averages and other types of measures.Basically, there was some interest in describing the characteristics and relationships of the data sets and it was considered essential to collect all or a large part of the data of interest, in order to guarantee that the results derived from statistical analysis were valid.

Subsequently, the development of scientific experimentation, especially in the biological and agricultural fields, raises the problem of how to reach basic conclusions or generalizations for a population from the study of only a small group (sample) of the elements that comprise it. This gave rise to the development of statistical inference, which rests on the theory of probabilities.

More modernly, statistical inference received a new impulse when a series of needs for obtaining reliable information, in the social and economic fields, led to the development of the sample surveys that are so widely used today by researchers, bureaucrats, businessmen, etc. And, to collect data on family and personal income, agricultural and industrial production, employment, household characteristics, public opinion, etc. Also statistical inference techniques are widely used to face decision problems under uncertainty conditions, typical of business.

Currently, statistics is a discipline dedicated to the development of theories and appropriate techniques to carry out, in a systematic and reliable way, both in the collection, classification, presentation, analysis and interpretation of numerical data sets produced by observation or experimentation, such as using this information to make valid and useful inferences for the population from which it comes.

This special nature of statistics, together with the general trend that exists in the modern world towards the quantification and collection of data, has made it very useful in practically all fields of human activity, making it a fundamental tool of empirical scientific research. This explains the immense development that statistics have had and the widespread use that is made of them; and underlines the need for a greater number of people every day to have a clear idea of what it consists of and what can be done for it and what are its technical and applicable principles. It is important to distinguish between descriptive statistics and inductive inferential statistics.Descriptive statistics means that technique and instruments that are used when you only want to describe and analyze a set of data, no matter the depth and detail with which it is done, but it is not intended from those data to make generalizations or inferences for a greater set. The making of charts and graphs, the construction of frequency distributions, the calculation of averages, variances and correlation coefficients, are examples of techniques routinely used within descriptive statistics.Variations and correlation coefficients are examples of techniques routinely used within descriptive statistics.Variations and correlation coefficients are examples of techniques routinely used within descriptive statistics.

By statistical inference to or inductive is understood the techniques or procedures that are used when the pursued purpose is not only to describe the data but to generalize what was observed in them for a larger set or universe, from which they were selected. Statistical inference is an inductive process: it starts from a sample and its results are generalized for the set or universe from which it was selected. As all inference implies a probability of error or uncertainty, this is a measure used by the theory of probabilities. The nature of statistical inference will be explained in detail later, but it is important to note that in a number of fields of action of the human being, It is necessary, with great frequency,make decisions or generalize from incomplete or sample-based information: a physician must decide on the effectiveness of a vaccine based on what is observed in a certain number of patients; an industrialist must decide whether to accept or reject a batch of raw material based on the study of a batch part; a biologist must decide whether to generalize the results observed in a sample of rabbits to all rabbits of the Breed studied to those of other breeds; A social worker must decide if the problems that affect a sample of elderly people in a city can be generalized - or considered valid enough - for all elderly people in the city or in urban areas of the country;an official from the Ministry of Agriculture should be an estimate of the bean harvest with the information from a sample of farms; An advisor to the Ministry of Public Education must decide, based on an experiment carried out on a sample of students taken from a representative sample of schools, whether or not a certain programmed procedure for teaching chemistry is superior to the traditional method; etc.

The modern shift in emphasis on statistics from descriptive to inference to reflects its ability to deal with the kinds of problems cited above.

A distinction is also made between mathematical statistical theory and applied statistics. The first, using certain basic principles AND mathematical elements such as probability theory, is concerned with studying the behavior of random processes and deriving laws (or principles and procedures) that allow inferences about a population from a random sample., and I gave myself the confidence that these inferences deserve. Applied statistics deals with the application of these techniques developed for theoretical statistics, for the solution of concrete problems that occur in reality.

(TAKEN FROM DESCRIPTIVE STATISTICS OF MIGUEL GOMEZ BARRANTES, PAGES 11, 12, 13, AND 14)

Second unity

FREQUENCIES

CONTENTS:

2.1. FREQUENCIES

2.2. MANAGEMENT OF DATA IN FREQUENCY TABLES 2.2.1. TOTAL WIDTH OR ROUTE OF THE VARIABLE 2.2.2. CLASS INTERVAL

2.2.3. DATA TABULATION

2.3. ACCUMULATED FREQUENCY

2.4. RELATIVE FREQUENCY

2.5. PERCENTAGE OF THE FREQUENCY SPECIFIC OBJECTIVES:

Upon completion of the study of this Unit, you will be able to:

Distinguish the meaning of FREQUENCY.

Calculate the actual CLASS limits.

Calculate the width of the INTERVAL.

Write the class mark of an INTERVAL.

Calculate the number of intervals in a SERIES.

Tabulate statistical data.

Write the accumulated frequency of a SERIES.

Calculate the relative frequency of a data set.

Calculate the percentage of FREQUENCY.

To achieve these goals, you must:

Answer the self-evaluation, with an efficiency level equivalent to one hundred percent (100%), comparing your solutions with those that we offer at the end of the book.

Review this Unit by answering the reinforcement material that we propose.

Solve the proposed exercises. FREQUENCIES

FREQUENCY

It is the number of times that the same variable value is repeated. For example:

Number of students from Ecuador distributed by levels of study:

Levels1 Frequency (f) pre primary 28,504 Primary 1338 119.00 Medium 469,968

TOTAL 1,836,591

DATA MANAGEMENT IN FREQUENCY TABLES

There are many types of data that can be collected in various ways; But, it is essential to proceed with its ordering, in order to provide greater comfort in the analysis and in the extraction of conclusions.

In the process of an investigation, this Unit is of utmost importance, since, through it, we continuously highlight the various phases that must be taken into account to comply with the ordering of the data.

TOTAL WIDTH OR TRAVEL OF THE VARIABLE

From the data of a survey on the ages of people, the following values were obtained:

41 39 37 20 56 25 27 32 31 28 19 47 38 43 21 32 35 34 47 49 18 25 37 29 20 43 37 40 32 31 35 46 30 32 53 50 42 31 44 37

This data set shows that the highest age is 56 and the lowest is

18 years. The difference between these two values is 38 (56-18 = 38). This value 38 constitutes the total amplitude or path of the variable, and is defined as the difference that is established between the greatest and least value of the variable.

=? -? Being:

a = amplitude Xmayor = highest value Xmenor = smallest value

Class interval

The extreme numbers and those included in them are called the class interval.

For example: the interval 80 -86 is made up of the following numerals

80, 81, 82, 83, 84, 85 and 86

Class limits. - are the extreme values that make up the interval

So for example the interval 70-75 means that it starts at 70 and ends at 75; but these limits are not true, therefore, the interval 70-75 varies from 69.5 to 75.5 What are the real limits, respectively. The first is called the lower real limit (Li) and the second, the upper real limit (Ls).

Width of the interval. - If the interval 1820 is proposed, taken from a statistical series on school grades, the size or width of the class interval is taken when establishing the difference between its real limits, thus: 20.5 - 17.5 = 3

That is to say:

? = -?

upper range Where:

i = width of

Ls = actual limit

Li = lower real limit

When it comes to statistical series, the width of the interval is an assumed integer of odd preference so that its class mark is an integer

Class mark. - it is the average value of each interval to determine it, the extreme values of the interval are added and this result is divided by two.

This relationship expressed by the formula is as follows:

? + 1? = 2

For example:

interval Being:

Xm = class mark li = lower limit of the interval ls = upper limit of the

In a school through a survey a study was carried out on a technical specialty as a variable the age of parents.

The table of age values that would give like this:

Intervals

(x) Class mark

(Xm) Frequencies

(F)

75 - 79 77 1

70 - 74 72 0

65 - 69 67 5

60 - 64 62 4

55 - 59 57 8

50 - 54 52 22

45 - 49 47 15

40 - 44 42 11

35 - 39 37 8

30 - 34 32 1

TOTAL 75

Number of intervals: it constitutes an integer that reflects the totality of classes to determine the number of intervals in a series, the width or distance is divided for the width of the interval and this quotient is added to the unit.

It is announced translated into a formula that looks like this:

? = + 1?

Being:

ni = number of intervals i = width of interval a = width

It is convenient to use a number of intervals not less than 5 nor greater than 15 if the number of intervals is less than 5, the frequencies would be highly concentrated, which means that a more realistic analysis of the data is not allowed. Frequencies would be widely dispersed, making it difficult to prepare the table, its graphic representation, and its mathematical calculations.

Thus: In a class of 36 students the following grades have been obtained in the subject of physics:

18 15 19 16 17 15 12 13 14 12 13 14 13 12 9 11 13 14 9 6 5 13 9 14 11 10 7 13 14 9 10 7 13 10 14 9

The amplitude the width of the interval and the number of intervals would be in their order:

1. A = 19 - 5 = 14, which constitutes the amplitude of the series.

2. We decided that the width of the interval is 3. 3.? =? + 1 1? = 14 + 1 = 5.6 = 6 intervals 3 This means that the series will have 6 intervals.

Data tabulation This is the process by which the material is ordered and conveniently grouped according to the purposes of the research.

Statistical series. - constitutes a set of values of a variable that are ordered in ascending or descending order.

For example: the weights in kilograms of 10 people are as follows 49 - 52 - 60 -

55 - 54 - 65 - 70 - 58 - 57 - 62.

Ordering them into a statistical series in form, they would be:

Weights X 70

49 Frequency statistics series. - is the ordering of the variable in ascending or descending form. And in which there are some repeated values. Repeated values must be expressed in Tables.

The procedure to be used in the formation of the statistical frequency series

is the following:

- The variable is ordered in ascending or descending order.

- Repeated values are written in Tables using vertical or horizontal stripes. - Add the number of lines that exist to form the column of frequencies.

Example:

Order in a statistical series of frequency the following data corresponding to the height in centimeters of 25 people:

159 161 165 163 167 160 160 161 163 163 167 166 163 160 162 162 165 161 160 164 161 164 166 164 162

First. - we order the variable

167 - 166 - 165 - 164 - 163 - 162 - 161 - 160 - 159

Second. - we write in Tables the repeated values

Third. - we build the frequency column

So, the table of values looks like this:

x Repeating values Frequency (f)

167 II 2

166 II 2

165 II 2

164 III 3

163 IIII 4

162 III 3

161 IIII 4

160 IIII 4

159 I 1

TOTAL 25

Statistical interval series. - it is a set of values ordered in ascending or descending order according to the class intervals that have been previously determined. The process used to form a statistical series of intervals is as follows:

- Find the amplitude or path of the variable

- the width of the interval is proposed

- The number of intervals that the statistical series will have is calculated

- The column of the intervals is constructed by making the upper limit of the first interval the largest of the variable. The width of the interval is reduced to the upper limit and 1 is added, obtaining the lower limit, thus determining the first interval - To obtain the second interval, the width of the interval is subtracted from the limits of the first interval, and so on - The lowest value of the variable must be included in the last interval

- We make the location and count of the repeated values

- We build the column of frequencies

For example:

The following data were obtained in a survey of students in a specialty of the baccalaureate cycle of a school in the city of Loja, regarding the age of the parents:

34 40 41 48 49 51 50 55 67 36 41 41 47 49 52 49 60 79 36 42 45 46 49 51 55 60 45 37 40 46 45 49 51 56 61 38 40 45 46 49 52 55 61 39 40 45 45 50 51 56 65 37 43 47 45 50 50 55 65 36 44 45 49 50 41 47 66 37 41 45 49 50 50 57 66

We sort these data into a statistical series of intervals.

First. - find the amplitude of the series: a = 79 - 34 = 45

Second. - we propose that the width of the interval be 5: i = 5

Third. - the number of intervals is calculated:? =? + 1?

? = 45 + 1 5

? = 9 + 1 = 10

That is, the number of intervals is 10.

Fourth. - the column of the intervals is formed: If the upper limit of the first interval is 79, what is the greatest value of the variable, then.

79 - 5 + 1 = 75

which is the lower limit of the first interval: (75-79)

The remaining intervals will be formed by decreasing the width of the interval (i = 5) to the two limits of the previous interval.

So:

So the second interval is: 70 - 74 75 - 5 = 70 79 - 5 = 74

70 - 75 = 65 74 - 5 = 69

The third interval is: 65 - 69, and so on.

Fifth. - the location and counting of the repeated values is carried out.

Sixth. - the column of the frequencies is constructed by corresponding to each interval a frequency.

Accumulated frequency It is the sum of the frequencies from the lowest value of the variable. X

75 - 79

70 - 74

65 - 69

60 - 64

55 - 59

50 - 54

45 - 49

40 - 44

35 - 39

30 - 34 The previous table remains and with the accumulated frequencies

X f Accumulated frequency (fa)

75 - 79 1 75 (74 + 1)

70 - 74 0 74 (74 + 0)

65 - 69 5 74 (69 + 5)

60 - 64 4 69 (65 + 4)

55 - 59 8 65 (57 + 8)

50 - 54 22 57 (35 + 22)

45 - 49 15 35 (20 + 15)

40 - 44 11 20 (9 + 11)

35 - 39 8 9 (1 + 8)

30 - 34 1 1

TOTAL 75

Relative frequency X Repeating values f

75 - 79 I 1

70 - 74 0

65 - 69 IIIII 5

60 - 64 IIII 4

55 - 59 IIIIIIII 8

50 - 54 IIIIIIIIIIIIIIIIIIIIII 22

45 - 49 IIIIIIIIIIIIIII 15

40 - 44 IIIIIIIIIII 11

35 - 39 IIIIIIII 8

30 - 34 I 1

TOTAL 75 Is the ratio established by dividing the frequency of the variable by the total number of

cases.

Relative frequency = frequency total number of cases

That is to say:

f fr = N

Where: fr = relative frequency f = frequency n = number of cases

Example:

Determine the relative frequency for the population data of a school. POPULATION: number of students in a college. VARIABLE: number of students per course XF fr

Eighth of Basic 68 0.143

Ninth of Basic 93 0.195

Tenth of Basic 77 0.162

First of Basic 87 0.183

Second of Basic 84 0.176

Third of Basic 67 0.141

TOTAL 476 1,000

Example:

Relative frequency for eighth grade

f fr = N

fr = 68 476

fr = 0.143

Percentage of the frequency It is the value that corresponds to each frequency and that is given for every 100 cases of

an investigated fact. To calculate the frequency percentage, the formula is used:

? =?. ?

frequency Where:

p = percentage of the

f = frequency

Example: N = total number of cases

Percentage of teaching staff selected by the Ministry of Education, by province, for the seminar-workshop on reform of study programs. PROVINCES f% Esmeraldas 23 5.53 Manabí 31 7.45 Guayas 76 18.27 El Oro 38 9.13 Carchi 22 5.29 Imbabura 17 4.09 Pichincha 40 9.62 Cotopaxi 22 5.29 Tungurahua 47 11.30 Chimborazo 24 5.77 Bolívar 25 6.00 Cañar 2 0.48 Azuay 23 5.53 Loja 23 5.53 Napo - Tena 2 0.48 Zamora - Chinchipe 1 0.24 TOTAL 416 100%

If f = 23 and N = 416, then the percentage of emeralds according to the formula:

? =?. ?

So:

2. 3. 100 p = 416

p = 5.53

For some investigations it is necessary to determine the percentage of the accumulated frequencies, for which the formula used is as follows

? = ??. ?

frequency Where:

p = percentage of the

fa = cumulative frequency N = total number of cases

For example:

Qualifications of a course in the subject of mathematics X f fa p

17 1 32 100

16 2 31 96.88

15 3 29 90.63

14 4 26 81.25

13 8 22 68.75

12 3 14 43.75

11 4 11 34.38

10 3 7 21.88

9 2 4 12.50

8 2 2 6.25

TOTAL 32

If we take fa = 32 and if p = fa. 10 0 N

So p = 32. 10 0 = 100 32

Also if: fa = 31

So p = 32. 100 = 96.88 32 Exercises solved

The National language teacher of a school in the city of Loja, after the first quarterly evaluation, obtains the following grades from the eighth grade students of basic

20 18 17 16 15 14 13 12 17

16 14 12 11 13 14 15 15 14

13 12 10 12 14 15

a) I ordered in a statistical series of frequency

b) Construct the column of the accumulated frequency c) Obtain the percentages of the frequency x Repeating values f fa% f 20 I 1 24 4.17 19 0 23 0 18 I 1 23 4.17 17 II 2 22 8, 33 16 II 2 20 8.33 15 IIII 4 18 16.67 14 IIIII 5 14 20.83 13 III 3 9 12.50 12 IIII 4 6 16.67 11 I 1 2 8.33 10 I 1 1 4.17 TOTAL 24 100%

Some students, when asked about their height, gave the following data in centimeters:

149 147 165 160 161 164 168 169 170 159 158 164 162 170 160 157 149 162 165 171 168 167 151 152 154 149 153 153 154 162 169 168 167 164 168 167 168 161 150 163 167 167 165 166 169

Decide:

to. The interval statistical series. b. The amplitude. c. The number of intervals.

d. The midpoints or class mark of the intervals. and. Relative frequency. F. Cumulative frequency percentage.

a = 171 - 147 = 24 if: i = 3 and ni = a + 1 i

So: ni = 24 + 1 3

ni = 8 + 1

ni = 9

xf Xm fr fa% f 169 - 171 6 170 0.13 45 100 166 - 168 11 167 0.24 39 88.67 163 - 165 7 164 0.16 28 62.22 160 - 162 7 161 0.16 21 46, 67 157 - 159 3 158 0.07 14 31.11 154 - 156 2 155 0.04 11 24.44 151 - 153 4 152 0.09 9 20.00 148 - 150 4 149 0.09 5 11.11 145 - 147 1 149 0.02 1 2.22 TOTAL 45 100%

The ages of a group of people have been tabulated in the following table.

Get. xf 52 - 55 2 48 - 51 3 44 - 47 8 40 - 43 14 36 - 39 12 32 - 35 7 28-31 4

to. The amplitude

b. Interval width

c. The class mark for the intervals d. The accumulated frequency e. The product f. Xm.

F. Frequency percentage

If a = 55 - 28 = 27

i = 55 - 52 + 1 = 4 xf Xm fr fa% f 52 - 55 2 53.5 50 107.0 4 48 - 51 3 49.5 48 148.5 6 44 - 47 8 45.5 45 364.0 16 40 - 43 14 41.5 37 581.0 28 36 - 39 12 37.5 23 450.0 24 32 - 35 7 33.5 11 234.5 14 28 - 31 4 29.5 4 118.0 8 TOTAL 50 100%

In a course t 36 students there are five students not promoted with Professor X, and in another class of 25 Professor Y has 4 students not promoted. Indicate with which teacher there are more non-promoted students in relation to the group

The percentage of students not promoted in the 36 course is:

. 100 =

5. 100 = 36

0 13.89

That is, what with teacher x there is 13 as 89% of students not promoted.

The percentage of students not promoted in the course of 25 is:

Four. 100 = 25

= 16%

That is, with professor x there is 13 as 89%, while with professor Y

16% of students are not promoted.

Then, it is with the teacher And that there is a greater number of students not promoted, according to the percentage previously established

Proposed exercises

The traditional export products of Ecuador: Coffee, Ivana cocoa did not register the following sales volume in 1979: coffee, 74 million kilos; cocoa, 14.3 million kilos; and, banana, 1,368.8 million kilos Determine:

to. The percentage

Compile data related to the number of births that occurred at your place of residence in the last five years and determine:

to. Relative frequencies

b. Accumulated frequencies

c. The percentages of the accumulated frequencies

Knowing the number of inhabitants of your province of origin, distributed by cantons, find:

to. The frequency percentage b. Cumulative frequency

Collect the grades obtained on a math test by your classmates and, using the appropriate class intervals, find:

to. The class mark b. Frequency c. Relative frequency

d. Cumulative frequency

and. The percentage of the accumulated frequency

In a third year of high school the following set of data was recorded, which are related to the height given in centimeters:

162 155 147 161 163 160 159 155 154 154 166 154 157 156 164 157 153 158 152 160 145 153 157 153 162 158 157 160 160 162 165 161 162 162 160 153 153 150 153 157 157 160 158 155 152 Find:

to. The ordering according to a statistical series of frequency b. The relative frequency c. Cumulative frequency

d. Frequency percentage

Review Frequency is the number of times a statistical phenomenon is repeated

The difference between the largest value and the smallest value of the variable in a series is called the total amplitude. To obtain the width of the interval, the difference between the real limits of each interval is established.

The class mark is obtained from the half-sum of the limits of each interval

Next, 3 statistical series are proposed. Write next to them the corresponding name.

ab XX f 20 18 - 20 8 19 15 - 17 10 18 12 - 14 14 17 9 - 11 10 16 6 - 8 3 15 3 - 5 2

18 2

17 5

16 3

15 2

13 1

12 1

to. Statistical series. b. Interval statistical series c. Statistical frequency series

To calculate the relative frequency we must use the formula.

Determine the cumulative frequency for the following statistical series

xf fa

170 1 fa 169 2 10 168 1 9 7 167 3 6 166 2 3 165 1 1 Determine the percentage of the frequency for the next statistical series of intervals

xf fa

140 - 144 2

135 - 139 7

130 - 134 12

125 - 129 15

120 - 124 10

115 - 119 9

110 - 114 7

105 - 109 6 100 - 104 5 TOTAL 100%

% f 2.74 9.59 16.44 20.55 13.70 12.33 9.59 8.22

Self appraisal

Instructions. - This self-assessment aims to inform you how far you have learned the various topics in this unit. Answer each question in this test by tracing an X in the appropriate letter box. After you reflect enough on each of the proposed topics, compare your solutions with our answer sheet

One of the following propositions is true

to. Extreme numbers are called class intervals.

b. The number of times that the same variable value is repeated is called frequency c. The difference between any two values is the total amplitude d. The above propositions are false

The word tabulation means

to. Data count b. Statistical table c. Material sorting

d. Set of values of a variable

You want to tabulate 20 values in one variable, which type of frequency table would you prefer.

to. A statistical table

b. A statistical frequency table c. An interval statistical table d. No use would be made. Point out the correct proposition:

to. The width of the interval is proposed by the author of the book b. The width of the interval is proposed by a professor of statistics c. The width of the interval is decided based on the width or path of variable d. The width of the interval is decided by the student

Identify the actual class limits of the following interval 18 20

to. 17.5-18.5 b. 17.5-19.5 c. 17.5-21.5 d. 17.5 - 20.5

What is the width of the following interval 171 - 177

to. I = 7 b. i = 6 c. i = 8 d. i = 6.5

The class mark for the following interval 140 151 is:

to. 145.5 b. 140.5 c. 147.5 d. 151.5

The following proposed example, sustained cumulative frequency. Determine which is the true column (fa):

abcd

xf fa fa fa fa 70 1 17 18 15 1 69 2 16 15 14 3 68 3 12 13 12 6 67 2 9 9 9 8 66 2 7 7 7 10 65 3 5 5 5 13 64 2 2 2 2 15 abcd

What is the error that has been incurred in determining the relative frequencies in the following statistical table ?:

to. 0.20 b. 0.60 c. 0.30 d. 0.10 xf fr 120 2 0.20 119 4 0.60 118 3 0.30 117 1 0.10 10

The formula to find the percentage of the frequency is:

to. =?. 10? b. = ??. 100? c. =?. ? 100 d. =?. 100?

Check the SELF-ASSESSMENT answers on page 339. OTHER APPLICATION ACTIVITIES

FIRST AND SECOND UNITS

EXERCISE No. 1

1. Carry out the following operations indicated

to. 7 - 6 + 8 - 10 + 3 - 15 = b. 2 - 5 3 + 7 - 9 = 8 10 40

c. (3

(- 5

(- 2

1) = 5 3 7 3

d. (- 4 9

÷ (7) = 6

2. Round the following numbers to two decimal places

to. 7,705

b. 176,089 c. 521.0258 d. 72,2606

3. Write in less than 100 words your critical judgment regarding the following definition of statistics: “Statistics studies the behavior of mass phenomena. Like all sciences, it seeks the general characteristics of a group and disregards the particulars of each element of said group ”

ALFONSO BARRANCHO Y

4. In less than 200 words write your criteria regarding the importance of statistics

5. Through a synthesis establish the difference between descriptive and inductive statistics 6. Through exercises describe the meaning of DISCRETE VARIABLE and CONTINUE

7. The following data correspond to the grades obtained by a group of students in the subject of statistics 15 12 16 18 17 19 20 14

12 19 14 20 11 18 16 15

13 16 18 17 12 11 12 14

17 15 13 14 16 12 11 18

to. Find the amplitude

b. I ordered the data in a statistical series in descending order

c. Write in a statistical table the columns corresponding to:

- Repeating values

- Frequencies

- The accumulated frequency

ADVISORY:

ACCUMULATED FREQUENCY is the sum of the frequencies from the lowest value of the variable

Example:

xf fa

18 1 15

17 3 14

16 4 11

15 5 7

14 2 2

fifteen

8. With the help of the following statistical table, find the columns of:

- The midpoints or class marks.

- The real limits of each class.

28 - 32 0

23 - 27 10

18 - 22 15

13 - 17 12

08 - 12 5 9. If the height in centimeters of a group of students is:

96 110 105 85 95 98 115 112 100 115

116 105 80 118 119 102 86 94 92 99

108 89 120 117 93 97 107 113 111 101

91 82 114 103 88 106 117 103 106 92

96 105

to. To find the amplitude

b. I ordered the data in descending order with intervals of 5 c. Determine the column of: - The frequencies

- The percentages of each class with two decimal places

- The accumulated frequency

- The percentages of the accumulated frequency with two decimal places

EXERCISE NO 2

1. Solve the following indicated operations:

to. - 12 + 8 - 18 + 15 - 32 + 17 =

b. 3 - 5 4 6 - 12 + 10 1 - 7 = 2 12) 5) (- 8) = 5 3 9

d. 11 ÷ 4 = 15 9

2. Round the following numbers to two decimal places

to. 3.1815 =

b. 1,536,845 = c. 2,343,375 = d. 421,2494 =

3. Synthetically write your critical judgment related to the following statement: “One of the reasons that makes statistics very useful is the fact that it has techniques that allow us to reach valid conclusions, even when the data has been collected following procedures wrong ”ALFONSO BARRANCHO

4. Write three examples of discrete variable and 2 examples of continuous variable

5. Below are the grades obtained by a group of students in the mathematics course 20 15 12 16 18 12 11 9 14 10 19 16 14 20 17 15 17 16 14 11 12 15 19 18 20 13 16 17 11 18 14 17 18 11 13 15 14 19 10 12

to. Determine the amplitude

b. Prepare a statistical table in which include the columns corresponding to: - The values of the variable ordered in ascending order

- Repeating values

- Frequencies the accumulated frequency

ADVISORY:

ACCUMULATED FREQUENCY is the sum of the frequencies from the lowest value of the variable

Example

Determine the accumulated frequency for the statistical series found in the following table

Frequency Rating 20 2 19 4 18 5 17 12 16 17 40

Development: The ratings are ordered in descending order, so the lowest value of the variable is 16.

Consequently, the column that corresponds to the accumulated frequency is:

fa 40 38 34 29 17

6. With the data found in the following statistical table:

150 5

149 7

148 14

147 11

146 6

145 2

Decide:

to. The percentage of the frequencies with two decimal places of approximation b. The accumulated frequency column c. The percentages of the accumulated frequency

7. The weight in kilograms of a group of people is:

45 52 60 65 48 54 62 46 68 65

50 55 63 69 62 46 59 68 60 56

70 61 68 49 50 57 54 59 68 48

61 68 66 51 47 49 50 66 64 70

47 69 53 46 45 58 68 48 63 60

70 62 65 49 58

to. Find the amplitude

b. Sorts the data in ascending order using 5 c intervals. Determine the number of intervals d. Determine the columns of - Frequencies

- The percentages of the frequencies with two decimal places of approximation.

- The accumulated frequency

- The percentages of the accumulated frequency with two decimal places

8. With the help of the following statistical table Determine the columns of a. The class mark for each interval b. The actual limits of each class

c. Frequency products for each class brand

145 - 147 2

142 - 144 9

139 - 141 11

136 - 138 15

133 - 135 16

130 - 132 8

127 - 129 6 SELF-ASSESSMENT 2

1. The frequency is:

to. The average value of each interval b. A part of the population c. The number of times the same variable value is repeated

d. A qualitative or quantitative characteristic that can take different values

2. Point out the propositions that are correct:

to. Total width or range of the variable is the largest value of variable b. The class mark is the average value of each interval c. The number of intervals is obtained by dividing the amplitude by the width of the interval d. Class limits are the extreme values that make up an interval

3. The accumulated frequency is:

to. The number of times the same value of variable b is repeated. A set of values of a variable c. The sum of the frequencies from the lowest value of the variable

d. A set of values ordered in ascending or descending order

4. Is the width of the interval 36 - 40 is

to. 3 b. 5 c. 7 d. None of the above values.

5. Point out the formula used to calculate the number of intervals:

to. ? =? +? 2 b. ? =? +? 2 C. ? =? + 1 2 d. None of the above values.

6. 21 of what number is 7%?

to. 120 b. 147 c. 300 d. 310

7. The relative frequency is obtained

to. Multiplying the frequency by the total value of cases b. Multiplying the frequency by 100 c. Dividing the frequency by the total number of cases d. None of the above propositions

8. The class mark for the following interval 55 - 59 is

to. 57 b. 58 c. 114 d. None of the above propositions

9. With the help of the following statistical table Determine the columns corresponding to:

to. The class mark of each interval b. The accumulated frequencies X f Xm fa 75 - 79 1 70 - 74 5 65 - 69 4 60 - 64 8 55 - 59 10 50 - 54 6 34 10. With the information found in the following statistical table.

Determine the percentage of frequencies with two decimal places of approximation

xf% 20 1 19 5 18 10 17 12 16 9 12 2 39 Second Unit

FREQUENCIES

CONTENTS:

2.1. FREQUENCIES

2.2. MANAGEMENT OF DATA IN FREQUENCY TABLES 2.2.1. TOTAL WIDTH OR ROUTE OF THE VARIABLE 2.2.2. CLASS INTERVAL

2.2.3. DATA TABULATION

2.3. ACCUMULATED FREQUENCY

2.4. RELATIVE FREQUENCY

2.5. PERCENTAGE OF THE FREQUENCY SPECIFIC OBJECTIVES:

Upon completion of the study of this Unit, you will be able to:

Distinguish the meaning of FREQUENCY.

Calculate the actual CLASS limits.

Calculate the width of the INTERVAL.

Write the class mark of an INTERVAL.

Calculate the number of intervals in a SERIES.

Tabulate statistical data.

Write the accumulated frequency of a SERIES.

Calculate the relative frequency of a data set.

Calculate the percentage of FREQUENCY.

To achieve these goals, you must:

Answer the self-evaluation, with an efficiency level equivalent to one hundred percent (100%), comparing your solutions with those that we offer at the end of the book.

Review this Unit by answering the reinforcement material that we propose.

Solve the proposed exercises.

Check the answers to SELF-TEST 2 on page 340. RECOMMENDED READING Brief history of statistics

Statistics has a long history. Perhaps the first time it was used was when a primitive Caudillo tried to know the number of warriors available in the tribes at a certain time. Hello, it would take to defeat the enemy; or perhaps when a king of remote Antiquity wanted to find out the changes in the number of his subjects or how much he could collect in the form of taxes. In more recent times, for example, the Statistics used to quantify death rates during the great plague suffered in London, and in the first studies of natural resources. These uses of statistics that constitute a wide field of activity that can be called "governmental arithmetic" are purely descriptive.

In the 17th and 18th centuries, professional gamblers asked some mathematicians who developed some principles that could improve the chances of winning with cards and dice. The two most notable mathematicians involved in that first and most important study of probability were Bernoulli and DeMoivre in the 1730s, the second, DeMoivre, developed the equation of the normal distribution curve during the first two decades of the 19th century. two other mathematicians la plaza and gauss carried out important works on the calculation of probabilities. His work consisted of applying the principles of probability to Astronomy.

During the 18th century, Statistical science had applications of a mathematical, political and governmental nature. At the beginning of the 19th century, Quetelet, a famous Belgian researcher, applied Statistics in the investigation of social and educational problems. Walker (1929) attributes to Quetelet the development of statistical theory as a general research method applicable to all observational sciences. Without a doubt, the person who had the greatest influence on the introduction and use of Statistics in the Social Sciences was Francis Galton. In the course of his long life he contributed notably in the studies of heredity and eugenics, Psychology, Anthropometry and Statistics, and is attributed with the current knowledge about correlation, that is,the measure of agreement between two variables. The mathematician Pearson collaborated with Galton in later years, and was involved in creating many of the regression and correlation formulas that are used today. Among Galton's important contributions are the development of the centiles (or percentiles).

The famous American psychologist James Mckeen Cattell studied in Europe in the 1880s and was in communication with Galton and other European statisticians. Upon his return to the United States, he and his disciples, including EL Thorndike, began to apply statistical methods to psychological and educational problems. The influence of these men was important; after a few years, theoretical and applied statistics courses will be taught at universities in that country.

In the 20th century, new techniques and methods have been applied in the study of small samples. The main contributions to the theory of small samples was that of the English statistician RA Fisher. Although most of his methods were developed in the agricultural or biological field, it was not long before sociologists recognized their usefulness and applied their ideas. Currently, Statistics is the main methodological tool for research in the social sciences. The reader or student interested in the history of statistics is recommended to consult a brief but completed article by the authors Dudycha and Dudycha (1972).

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