98993 - STATISTICAL ANALYSIS

Learning outcomes

At the end of the course the student has acquired theoretical and computational approaches to study the properties of health related materials, including complex molecular architectures and the basic concepts of probability distributions, and statistical tools used in treating experimental data and tackle biological intrinsic variability.

Course contents

Chapter 1: Description of Errors

1.1 What is Error?

1.2 Random vs Systematic Error.

1.3 Quantifying error; Absolute vs Relative Error

1.4 Estimated Uncertainty

1.4.1 Mean Value, Variance and Standard Deviation (SD)

1.4.2 When Standard Deviation should be computed with denominator n or n-1

1.5 Standard Error (SE). When we use SE and when SD

1.6 Comparison of measured and accepted values.

1.7 Comparison of two Measured numbers

1.8 Examples

Chapter 2: Propagation of Errors

2.1 Uncertainties in Direct Measurements

2.2 The Square-Root Rule for a Counting Experiment

2.3 Addition and Subtraction

2.4 Multiplication and Quotients

2.5 Arbitrary Functions of One Variable

2.6 Propagation of errors step by step

2.7 Examples

Chapter 3: Statistical Analysis of Random Uncertainties

3.1 Random and Systematic Errors

3.2 The Standard Deviation as the Uncertainty in a single measurement

3.3 The Standard Deviation of the Mean

3.4 Systematic Errors

3.5 Examples

Chapter 4. The Normal Distribution

4.1 Histograms and Distributions

4.2 Limiting Distributions

4.3 The Normal Distribution

4.4 The Standard Deviation as 68% Confidence Limit

4.5 Justification of the Mean as Best Estimate

4.6 Standard Deviation of the Mean and acceptability of a Measured Answer

4.7 Examples

Chapter 5: Probability Distributions

5.1 Definition of the Binomial Distribution

5.2 Properties of the Binomial Distribution

5.3 The Gauss Distribution for Random errors

5.4 Definition of the Poisson Distribution

5.5 Properties of the Poisson Distribution

5.6 Applications; Testing of Hypotheses, subtracting a background

5.7 Examples

Chapter 6: The Chi-Squared Test for a Distribution

6.1 Introduction to Chi squared

6.2 General definition of Chi squared

6.3 Degrees of freedom and reduced Chi-squared

6.4 Probabilities for Chi-Squared

6.5 Examples

Chapter 7: Treating data

7.1 The problem of data rejection – Chauvenet’s Criterion

7.2 Combining separate measurements; Weight average

7.3 Least- Squares fitting; Calculations of constants and associated errors

7.4 Covariance in error propagation

7.5 Coefficient of linear correlation, Quantitative significance of r

7.6 Examples

Readings/Bibliography

1. John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical measurements, 2^nd Edition, University Science Books Sausalito, California (1997).

2. Lecture Notes.

Teaching methods

Teaching in classroom.

Assessment methods

Written examination at the end of the semester, with theoretical questions and problems to solve (each one corresponding to a specific maximum score if correctly answered, for a total of 33 points equivalent to a final mark of 30 with Lode). The exam is passed with a minimum score of 18/30.

Teaching tools

Lessons and exercises in the classroom for the theory (3 CFU).

Office hours

See the website of Evangelos Bakalis