Statistics
A.Y. 2024/2025
Learning objectives
The goals of the course are:
- To provide the students with methods and procedures to build summaries and graphical representations of a set of data
- To introduce the students to the tools (technical language and models) to represent and analyze random phenomena
- To introduce the students to the fundamental methods of statistical inference
- To enable the students to apply the statistical methods and techniques object of the course to real data through the use of an appropriate statistical software
- To provide the students with methods and procedures to build summaries and graphical representations of a set of data
- To introduce the students to the tools (technical language and models) to represent and analyze random phenomena
- To introduce the students to the fundamental methods of statistical inference
- To enable the students to apply the statistical methods and techniques object of the course to real data through the use of an appropriate statistical software
Expected learning outcomes
Knowledge and understanding.
Having successfully taken the course examination, the students are expected to
- know the foundational principles of probability and statistics;
- know the correct terminology of probability and statistics and to be able to correctly argument the theory.
The students' knowledge and comprehension is expected not be limited to the enunciation of definitions and results and to the solution of standard exercises, but rather to be critical and enabling to distinguish among different situations and to take appropriate choices, justifying the followed procedures.
Ability of applying knowledge and understanding.
Having successfully taken the course examination, the students are expected to
- be able to apply the attained knowledge to specific problems of data analysis;
- be able to select the principles which are useful to obtain the solutions to the problems at hand;
- be able to elicit relevant indicators and summaries from large amounts of data.
Having successfully taken the course examination, the students are expected to
- know the foundational principles of probability and statistics;
- know the correct terminology of probability and statistics and to be able to correctly argument the theory.
The students' knowledge and comprehension is expected not be limited to the enunciation of definitions and results and to the solution of standard exercises, but rather to be critical and enabling to distinguish among different situations and to take appropriate choices, justifying the followed procedures.
Ability of applying knowledge and understanding.
Having successfully taken the course examination, the students are expected to
- be able to apply the attained knowledge to specific problems of data analysis;
- be able to select the principles which are useful to obtain the solutions to the problems at hand;
- be able to elicit relevant indicators and summaries from large amounts of data.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Lesson period
First semester
Course syllabus
· Summarizing data and descriptive analysis: types of data, frequency distributions, position and shape indexes (mean, median, quantiles, variance, standard deviation, range, interquartile range,..), histograms, boxplots and other frequency graphs.
· Probability and random variables: properties of probability, discrete random variables (Bernoulli, Binomial, Poisson distributions), continuous random variables (Exponential, Gaussian, t-Student distributions), mean and variance, properties of means and variances, joint distributions and independence, Law of Large Numbers, Central Limit Theorem.
· Estimation: sampling distributions, properties of estimators (bias and MSE), confidence intervals, testing a hypothesis, principles of significance tests, significance levels and types of error, power of a test.
· Comparing samples: comparing the means of two independent Gaussian samples (t-test), comparing the means of two dependent Gaussian samples (paired t-test).
· Regression models: univariate and multivariate linear models, least squares estimators of the parameters of a linear models, tests and confidence intervals for the parameters of a linear model, prediction of a new observation, goodness of fit methods, analysis of the residuals
· Probability and random variables: properties of probability, discrete random variables (Bernoulli, Binomial, Poisson distributions), continuous random variables (Exponential, Gaussian, t-Student distributions), mean and variance, properties of means and variances, joint distributions and independence, Law of Large Numbers, Central Limit Theorem.
· Estimation: sampling distributions, properties of estimators (bias and MSE), confidence intervals, testing a hypothesis, principles of significance tests, significance levels and types of error, power of a test.
· Comparing samples: comparing the means of two independent Gaussian samples (t-test), comparing the means of two dependent Gaussian samples (paired t-test).
· Regression models: univariate and multivariate linear models, least squares estimators of the parameters of a linear models, tests and confidence intervals for the parameters of a linear model, prediction of a new observation, goodness of fit methods, analysis of the residuals
Prerequisites for admission
The course makes use of the mathematical formalism taught in a basic course in mathematics.
Teaching methods
Class lectures, exercise sessions and lab sessions. During the lab sessions, students will be illustrated the use of the program language R to analyze real dataset, by using their laptop.
Teaching Resources
All the bibliographical suggestions as well as additional material will be available onWeBeep (webeep.polimi.it), the portal for the network activities of students and professors at Politecnico di Milano; students registered to the course for the current academic year can access it.
Assessment methods and Criteria
The course assessment will consist of two parts, namely a written exam and a team project. Both parts are mandatory.
The written exam will be taken in one of the dates scheduled by the School within the academic year; it will consist of 2 exercises, to be solved autonomously in maximum 2:00 hours. At the end of the exam the student will decide whether or not to have their exam evaluated. The written exam will be evaluated with a score expressed in a scale from 0 to 30, the maximum evaluation being 32/30. The written exam will be passed upon obtaining a score greater than or equal to 18/30. The exam evaluation will account for the degree of clarity of the exposition and for the correctness of computations. During the examination, the students will not be allowed to use books or notes, nor to use the mobile phone or other electronic devices. The students will be allowed to use the calculator, the statistical tables and a formulary of A4 format containing any material deemed useful by the student.
The team project will consist of an analysis of a real dataset, to be conducted in teams of 2 to 4 students, using the models and methods introduced in the course. The team projects will be presented at the end of the course in a seminar during an open workshop that will take place after the end of the semester. Each team will receive an evaluation in a scale from 0 to 30.
The final evaluation of the course will be obtained as a weighted average of the scores obtained by the student in the two parts of the assessment, with weights 0.7 (written exam) and 0.3 (team project).
During the exam, the students will have to
- Demonstrate the degree of knowledge and comprehension of the key aspects of the course, presenting the used methodologies in a clear and exhaustive way;
- Demonstrate their ability to apply the learned notions to solve exercises and real problems, on any of the topics covered in the course.
The written exam will be taken in one of the dates scheduled by the School within the academic year; it will consist of 2 exercises, to be solved autonomously in maximum 2:00 hours. At the end of the exam the student will decide whether or not to have their exam evaluated. The written exam will be evaluated with a score expressed in a scale from 0 to 30, the maximum evaluation being 32/30. The written exam will be passed upon obtaining a score greater than or equal to 18/30. The exam evaluation will account for the degree of clarity of the exposition and for the correctness of computations. During the examination, the students will not be allowed to use books or notes, nor to use the mobile phone or other electronic devices. The students will be allowed to use the calculator, the statistical tables and a formulary of A4 format containing any material deemed useful by the student.
The team project will consist of an analysis of a real dataset, to be conducted in teams of 2 to 4 students, using the models and methods introduced in the course. The team projects will be presented at the end of the course in a seminar during an open workshop that will take place after the end of the semester. Each team will receive an evaluation in a scale from 0 to 30.
The final evaluation of the course will be obtained as a weighted average of the scores obtained by the student in the two parts of the assessment, with weights 0.7 (written exam) and 0.3 (team project).
During the exam, the students will have to
- Demonstrate the degree of knowledge and comprehension of the key aspects of the course, presenting the used methodologies in a clear and exhaustive way;
- Demonstrate their ability to apply the learned notions to solve exercises and real problems, on any of the topics covered in the course.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 1
SECS-S/01 - STATISTICS - University credits: 5
SECS-S/01 - STATISTICS - University credits: 5
Lectures: 48 hours
Professor:
Menafoglio Alessandra